Introduction to rational and irrational numbers

WEBVTT
Kind: captions
Language: en

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So let's talk a little bit
about rational numbers.

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And the simple way to think
about it is any number that

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can be represented as
the ratio of two integers

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is a rational number.

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So for example, any integer
is a rational number.

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1 can be represented as 1/1 or
as negative 2 over negative 2

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or as 10,000/10,000.

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In all of these cases, these are
all different representations

00:00:39.790 --> 00:00:42.260
of the number 1,
ratio of two integers.

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And I obviously can
have an infinite number

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of representations
of 1 in this way,

00:00:46.170 --> 00:00:48.760
the same number over
the same number.

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The number negative 7 could be
represented as negative 7/1,

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or 7 over negative 1, or
negative 14 over positive 2.

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And I could go on, and
on, and on, and on.

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So negative 7 is definitely
a rational number.

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It can be represented as
the ratio of two integers.

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But what about things
that are not integers?

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For example, let us imagine--
oh, I don't know-- 3.75.

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How can we represent that as
the ratio of two integers?

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Well, 3.75, you
could rewrite that

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as 375/100, which is the
same thing as 750/200.

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Or you could say, hey,
3.75 is the same thing as 3

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and 3/4-- so let
me write it here--

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which is the same
thing as-- that's 15/4.

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4 times 3 is 12, plus 3 is
15, so you could write this.

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This is the same thing as 15/4.

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Or we could write this as
negative 30 over negative 8.

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I just multiplied the
numerator and the denominator

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here by negative 2.

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But just to be clear,
this is clearly rational.

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I'm giving you multiple
examples of how

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this can be represented as
the ratio of two integers.

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Now, what about
repeating decimals?

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Well, let's take
maybe the most famous

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of the repeating decimals.

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Let's say you have 0.333, just
keeps going on and on forever,

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which we can denote by
putting that little bar on top

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of the 3.

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This is 0.3 repeating.

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And we've seen--
and later we'll show

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how you can convert
any repeating decimal

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as the ratio of two integers--
this is clearly 1/3.

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Or maybe you've seen things like
0.6 repeating, which is 2/3.

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And there's many, many,
many other examples of this.

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And we'll see any
repeating decimal, not just

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one digit repeating.

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Even if it has a million
digits repeating,

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as long as the pattern
starts to repeat itself

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over and over and
over again, you

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can always represent that as
the ratio of two integers.

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So I know what you're
probably thinking.

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Hey, Sal, you've
just included a lot.

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You've included all
of the integers.

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You've included all of finite
non-repeating decimals,

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and you've also included
repeating decimals.

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What is left?

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Are there any numbers
that are not rational?

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And you're probably
guessing that there are,

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otherwise people
wouldn't have taken

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the trouble of trying to
label these as rational.

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And it turns out-- as you
can imagine-- that actually

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some of the most famous
numbers in all of mathematics

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are not rational.

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And we call these numbers
irrational numbers.

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And I've listed there
just a few of the most

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noteworthy examples.

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Pi-- the ratio of
the circumference

00:04:07.355 --> 00:04:12.100
to the diameter of a circle--
is an irrational number.

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It never terminates.

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It goes on and on and on
forever, and it never repeats.

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e, same thing-- never
terminates, never repeats.

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It comes out of continuously
compounding interest.

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It comes out of
complex analysis.

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e shows up all over the place.

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Square root of 2,
irrational number.

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Phi, the golden ratio,
irrational number.

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So these things that
really just pop out

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of nature, many of these
numbers are irrational.

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Now, you might say, OK,
are these irrational?

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These are just these
special kind of numbers.

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But maybe most
numbers are rational,

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and Sal's just picked out
some special cases here.

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But the important thing to
realize is they do seem exotic,

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and they are exotic
in certain ways.

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But they aren't uncommon.

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It actually turns out
that there is always

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an irrational number between
any two rational numbers.

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Well, we could go on and on.

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There's actually
an infinite number.

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But there's at least one,
so that gives you an idea

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that you can't
really say that there

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are fewer irrational numbers
than rational numbers.

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And in a future
video, we'll prove

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that you give me two rational
numbers-- rational 1,

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rational 2-- there's going to be
at least one irrational number

00:05:21.640 --> 00:05:23.810
between those, which
is a neat result,

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because irrational
numbers seem to be exotic.

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Another way to think about it--
I took the square root of 2,

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but you take the square root
of any non-perfect square,

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you're going to end up
with an irrational number.

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You take the sum
of an irrational

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and a rational number-- and
we'll see this later on.

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We'll prove it to ourselves.

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The sum of an irrational
and a rational

00:05:42.610 --> 00:05:44.310
is going to be irrational.

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The product of an
irrational and a rational

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is going to be irrational.

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So there's a lot, a lot, a
lot of irrational numbers

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out there.


Proof that there is an irrational number between any two rational numbers

WEBVTT
Kind: captions
Language: en

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What I want to do in this video
is prove that between any two

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rational numbers-- so let's say
that's a rational number there,

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and then let's say that this
is another rational number that

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is larger than
this one right over

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here-- that between any
two rational numbers,

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you can find an
irrational number.

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So that number right
over there is irrational.

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You can find at least
one irrational number.

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And that's kind of
crazy, because there's

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a lot of rational numbers.

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There's an infinite number
of rational numbers.

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So we're saying between any
two of those rational numbers,

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you can always find
an irrational number.

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And we're going to start
thinking about it by just

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thinking about the
interval between 0 and 1.

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So if we think about the
interval between 0 and 1,

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we know that there are
irrational numbers there.

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In fact, one of them
that might pop out at you

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is 1 over the square
root of 2, which

00:01:05.860 --> 00:01:09.440
is the same thing as the
square root of 2 over 2,

00:01:09.440 --> 00:01:11.520
is equal-- I
shouldn't say equal,

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is roughly, is approximately
equal to 0.70710678118.

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And I could just
keep going on and on

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and on and on and on and on.

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This thing does not repeat.

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But the important point is,
it's clearly between 0 and 1.

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So I could write 1 over
the square root of 2

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is clearly between 0 and 1.

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So the way that I'm going
to prove that there's

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an irrational number between
any two rational numbers

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is I'm going to start with
this set of inequalities,

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and I'm going to
manipulate it so I end up

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with an r1 over here
and an r2 over here.

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And then from 1 over
the square root of 2,

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I would have manipulated
this to construct

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that irrational-- at least
one of the irrational numbers

00:02:00.570 --> 00:02:04.280
that's between those
two rational ones.

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So instead of making this
an interval between 0 and 1,

00:02:08.180 --> 00:02:10.660
let's make this an
interval between 0

00:02:10.660 --> 00:02:13.110
and the difference
between these two numbers.

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So the distance between
r1 and r2 is r2 minus r1.

00:02:18.640 --> 00:02:20.510
So let's multiply
both sides of this--

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or all three parts
of this inequality,

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I guess I could say, by
r2 times r2 minus r1.

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So let's do that.

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So if you multiply this,
0 times r2 minus r1, well

00:02:33.210 --> 00:02:35.830
you're just still
going to have 0 there,

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is less than-- And we know
that r2 is greater than r1,

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so r2 minus-- let me make
it clear what we're doing.

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We're going to multiply
everything times r2 minus r1.

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r2, we're assuming,
is greater than r1,

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so this thing right over here
is going to be greater than 0.

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So if you multiply
the different sides

00:02:55.195 --> 00:02:57.070
of an inequality by
something greater than 0,

00:02:57.070 --> 00:02:58.730
you don't switch the inequality.

00:02:58.730 --> 00:03:01.740
So 0 times that is 0, 1 over
the square root of 2 times

00:03:01.740 --> 00:03:05.620
that is going to be 1
over the square root of 2

00:03:05.620 --> 00:03:12.330
times r2 minus r1.

00:03:15.130 --> 00:03:18.790
And then that's going
to be less than-- well,

00:03:18.790 --> 00:03:27.510
1 times that is just
going to be r2 minus r1.

00:03:32.050 --> 00:03:35.710
And now, we just have to kind
of shift everything over.

00:03:35.710 --> 00:03:39.280
So let's add r1 to
all sides of this.

00:03:39.280 --> 00:03:42.250
So if we add something to
all parts of the inequality,

00:03:42.250 --> 00:03:44.470
then that's also not going
to change the inequality.

00:03:44.470 --> 00:03:47.990
So we're going to
add r1 over here.

00:03:47.990 --> 00:03:50.020
We can add r1 over here.

00:03:50.020 --> 00:03:52.629
And we can add r1 over there.

00:03:52.629 --> 00:03:54.170
And so on the
left-hand side, we have

00:03:54.170 --> 00:04:01.094
r1 is less than r1 plus--
let me just copy and paste

00:04:01.094 --> 00:04:03.260
all of this so I don't have
to keep changing colors.

00:04:03.260 --> 00:04:05.900
Whoops, that's not
what I wanted to do.

00:04:05.900 --> 00:04:07.660
Let me do this.

00:04:07.660 --> 00:04:08.320
There you go.

00:04:08.320 --> 00:04:09.220
All right.

00:04:09.220 --> 00:04:11.040
That should be pretty good.

00:04:11.040 --> 00:04:14.240
So copy and paste that.

00:04:14.240 --> 00:04:21.310
r1 plus this, plus that--
let me write the plus down--

00:04:21.310 --> 00:04:28.830
plus that, is less
than-- that one

00:04:28.830 --> 00:04:32.320
is a different shade of
blue-- is less than-- well,

00:04:32.320 --> 00:04:34.190
what's r1 plus r2 minus r1?

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Well, that's just
going to be r2.

00:04:37.080 --> 00:04:39.370
So I've just shown
you that you give me

00:04:39.370 --> 00:04:41.550
any two rational
numbers, and I'm

00:04:41.550 --> 00:04:44.520
assuming r2 is greater
than r1, I have just

00:04:44.520 --> 00:04:46.970
constructed an
irrational number that's

00:04:46.970 --> 00:04:50.750
going to be between those
two rational numbers.

00:04:50.750 --> 00:04:54.040
You take r1, you take the
lower of the rational numbers,

00:04:54.040 --> 00:04:55.880
and to that you add
1 over square root

00:04:55.880 --> 00:04:58.450
of 2 times the difference
between those two

00:04:58.450 --> 00:05:00.320
rational numbers,
and you are going

00:05:00.320 --> 00:05:05.060
to get this right over here
is an irrational number.

00:05:05.060 --> 00:05:08.120
You're saying hey, how do
I know that this thing--

00:05:08.120 --> 00:05:11.970
how can I be satisfied that
this thing is irrational?

00:05:11.970 --> 00:05:13.430
Well, we've already seen.

00:05:13.430 --> 00:05:17.620
You take the product of an
irrational and a rational,

00:05:17.620 --> 00:05:19.750
you get an irrational number.

00:05:19.750 --> 00:05:22.480
You take the sum of
an irrational number

00:05:22.480 --> 00:05:25.570
and a rational number, you
get an irrational number.

00:05:25.570 --> 00:05:28.430
So we've constructed
an irrational number

00:05:28.430 --> 00:05:31.660
that's between
these two rationals.


Sum and product of rational numbers

WEBVTT
Kind: captions
Language: en

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What I want to do in
this video is think about

00:00:02.080 --> 00:00:05.430
whether the product or
sums of rational numbers

00:00:05.430 --> 00:00:07.550
are definitely going
to be rational.

00:00:07.550 --> 00:00:09.850
So let's just first
think about the product

00:00:09.850 --> 00:00:10.939
of rational numbers.

00:00:10.939 --> 00:00:13.230
So if I have one rational
number and-- actually, let me

00:00:13.230 --> 00:00:14.938
instead of writing
out the word rational,

00:00:14.938 --> 00:00:17.890
let me just represent it
as a ratio of two integers.

00:00:17.890 --> 00:00:20.010
So I have one rational
number right over there.

00:00:20.010 --> 00:00:22.390
I can represent it as a/b.

00:00:22.390 --> 00:00:26.240
And I'm going to multiply it
times another rational number,

00:00:26.240 --> 00:00:30.350
and I can represent that as a
ratio of two integers, m and n.

00:00:30.350 --> 00:00:32.280
And so what is this
product going to be?

00:00:32.280 --> 00:00:36.975
Well, the numerator,
I'm going to have am.

00:00:36.975 --> 00:00:40.640
I'm going to have a times m.

00:00:40.640 --> 00:00:43.690
And in the denominator, I'm
going to have b times n.

00:00:47.290 --> 00:00:49.004
Well a is an integer,
m is an integer.

00:00:49.004 --> 00:00:50.670
So you have an integer
in the numerator.

00:00:50.670 --> 00:00:52.600
And b is an integer
and n is an integer.

00:00:52.600 --> 00:00:54.770
So you have an integer
in the denominator.

00:00:54.770 --> 00:00:58.140
So now the product is a ratio
of two integers right over here,

00:00:58.140 --> 00:01:00.770
so the product is also rational.

00:01:00.770 --> 00:01:03.480
So this thing is also rational.

00:01:03.480 --> 00:01:07.370
So if you give me the product
of any two rational numbers,

00:01:07.370 --> 00:01:10.020
you're going to end up
with a rational number.

00:01:10.020 --> 00:01:12.380
Let's see if the same thing
is true for the sum of two

00:01:12.380 --> 00:01:13.690
rational numbers.

00:01:13.690 --> 00:01:18.020
So let's say my first
rational number is a/b, or can

00:01:18.020 --> 00:01:21.330
be represented as a/b, and
my second rational number

00:01:21.330 --> 00:01:24.950
can be represented as m/n.

00:01:24.950 --> 00:01:26.750
Well, how would I add these two?

00:01:26.750 --> 00:01:28.340
Well, I can find a
common denominator,

00:01:28.340 --> 00:01:31.380
and the easiest
one is b times n.

00:01:31.380 --> 00:01:33.590
So let me multiply
this fraction.

00:01:33.590 --> 00:01:36.670
We multiply this one times
n in the numerator and n

00:01:36.670 --> 00:01:37.520
in the denominator.

00:01:37.520 --> 00:01:40.390
And let me multiply
this one times

00:01:40.390 --> 00:01:42.882
b in the numerator and
b in the denominator.

00:01:42.882 --> 00:01:44.340
Now we've written
them so they have

00:01:44.340 --> 00:01:46.610
a common denominator of bn.

00:01:46.610 --> 00:02:05.690
And so this is going to
be equal to an plus bm,

00:02:05.690 --> 00:02:08.256
all of that over b times n.

00:02:10.919 --> 00:02:13.120
So b times n, we've
just talked about.

00:02:13.120 --> 00:02:15.810
This is definitely going to
be an integer right over here.

00:02:15.810 --> 00:02:18.530
And then what do
we have up here?

00:02:18.530 --> 00:02:21.500
Well, we have a times
n, which is an integer.

00:02:21.500 --> 00:02:23.050
b times m is another integer.

00:02:23.050 --> 00:02:24.720
The sum of two integers
is going to be an integer.

00:02:24.720 --> 00:02:26.470
So you have an integer
over in an integer.

00:02:26.470 --> 00:02:27.940
You have the ratio
of two integers.

00:02:27.940 --> 00:02:31.380
So the sum of two
rational numbers

00:02:31.380 --> 00:02:33.290
is going to give you another.

00:02:33.290 --> 00:02:35.520
So this one right over
here was rational,

00:02:35.520 --> 00:02:37.500
and this one is right
over here is rational.

00:02:37.500 --> 00:02:39.040
So you take the product
of two rational numbers,

00:02:39.040 --> 00:02:39.840
you get a rational number.

00:02:39.840 --> 00:02:41.710
You take the sum of
two rational numbers,

00:02:41.710 --> 00:02:45.160
you get a rational number.


Proof that square root of prime number is irrational

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.970
In a previous video, we used
a proof by contradiction

00:00:02.970 --> 00:00:05.844
to show that the square
root of 2 is irrational.

00:00:05.844 --> 00:00:07.760
What I want to do in
this video is essentially

00:00:07.760 --> 00:00:10.470
use the same argument but
do it in a more general way

00:00:10.470 --> 00:00:14.860
to show that the square root of
any prime number is irrational.

00:00:14.860 --> 00:00:16.925
So let's assume that p is prime.

00:00:20.000 --> 00:00:22.990
And we're going to set this up
to be a proof by contradiction.

00:00:22.990 --> 00:00:28.080
So we're going to assume
that the square root of p

00:00:28.080 --> 00:00:33.070
is rational and see if this
leads us to any contradiction.

00:00:33.070 --> 00:00:36.630
So if something
is rational, that

00:00:36.630 --> 00:00:39.476
means that we can represent it
as the ratio of two integers.

00:00:39.476 --> 00:00:40.850
And if we can
represent something

00:00:40.850 --> 00:00:42.230
as the ratio of
two integers, that

00:00:42.230 --> 00:00:43.688
means that we can
also represent it

00:00:43.688 --> 00:00:45.800
as the ratio of two
co-prime integers,

00:00:45.800 --> 00:00:47.940
or two integers that have
no factors in common.

00:00:47.940 --> 00:00:50.820
Or that we can represent it as
a fraction that is irreducible.

00:00:53.790 --> 00:00:56.130
So I'm assuming
that this fraction

00:00:56.130 --> 00:00:57.510
that I'm writing
right over here,

00:00:57.510 --> 00:01:01.475
a/b, that this right over here
is an irreducible fraction.

00:01:01.475 --> 00:01:02.850
You say, well,
how can I do that?

00:01:02.850 --> 00:01:06.170
Well, this being rational says
I can represent the square root

00:01:06.170 --> 00:01:11.092
of p as some fraction, as
some ratio of two integers.

00:01:11.092 --> 00:01:13.550
And if I can represent anything
as a ratio of two integers,

00:01:13.550 --> 00:01:17.040
I can keep dividing both the
numerator and the denominator

00:01:17.040 --> 00:01:19.470
by the common factors
until I eventually

00:01:19.470 --> 00:01:23.150
get to an irreducible fraction.

00:01:23.150 --> 00:01:25.620
So I'm assuming that's
where we are right here.

00:01:25.620 --> 00:01:28.400
So this cannot be reduced.

00:01:28.400 --> 00:01:32.260
And this is important for our
proof-- cannot be reduced,

00:01:32.260 --> 00:01:35.270
which is another way of saying
that a and b are co-prime,

00:01:35.270 --> 00:01:38.800
which is another way of saying
that a and b share no common

00:01:38.800 --> 00:01:41.950
factors other than 1.

00:01:41.950 --> 00:01:44.450
So let's see if we can
manipulate this a little bit.

00:01:44.450 --> 00:01:46.440
Let's take the
square of both sides.

00:01:46.440 --> 00:01:52.320
We get p is equal to-- well,
a/b, the whole thing squared,

00:01:52.320 --> 00:01:54.690
that's the same thing as
a squared over b squared.

00:01:57.850 --> 00:02:00.010
We can multiply both
sides by b squared,

00:02:00.010 --> 00:02:07.010
and we get b squared times
p is equal to a squared.

00:02:07.010 --> 00:02:09.270
Well, what does this
tell us about a squared?

00:02:09.270 --> 00:02:13.100
Well, b is an integer, so b
squared must be an integer.

00:02:13.100 --> 00:02:15.940
So an integer times p
is equal to a squared.

00:02:15.940 --> 00:02:21.090
Well, that means that p must
be a factor of a squared.

00:02:21.090 --> 00:02:22.120
Let me write this down.

00:02:22.120 --> 00:02:34.780
So a squared is a multiple of p.

00:02:34.780 --> 00:02:36.280
Now, what does that
tell us about a?

00:02:36.280 --> 00:02:40.130
Does that tell us that a
must also be a multiple of p?

00:02:40.130 --> 00:02:41.850
Well, to think about
that, let's think

00:02:41.850 --> 00:02:44.400
about the prime
factorization of a.

00:02:44.400 --> 00:02:47.150
Let's say that a can
be-- and any number-- can

00:02:47.150 --> 00:02:49.400
be rewritten as a
product of primes.

00:02:49.400 --> 00:02:51.240
Or any integer, I should say.

00:02:51.240 --> 00:02:54.410
So let's write this out
as a product of primes

00:02:54.410 --> 00:02:55.230
right over here.

00:02:55.230 --> 00:02:58.450
So let's say that I have
my first prime factor times

00:02:58.450 --> 00:03:02.520
my second prime factor, all
the way to my nth prime factor.

00:03:02.520 --> 00:03:05.370
I don't know how many prime
factors a actually has.

00:03:05.370 --> 00:03:09.190
I'm just saying that a is
some integer right over here.

00:03:09.190 --> 00:03:11.590
So that's the prime
factorization of a.

00:03:11.590 --> 00:03:14.870
What is the prime factorization
of a squared going to be?

00:03:14.870 --> 00:03:17.470
Well, a squared
is just a times a.

00:03:17.470 --> 00:03:22.110
Its prime factorization is
going to be f1 times f2,

00:03:22.110 --> 00:03:23.820
all the way to fn.

00:03:23.820 --> 00:03:30.490
And then that times f1 times
f2 times, all the way to fn.

00:03:30.490 --> 00:03:32.050
Or I could rearrange
them if I want.

00:03:32.050 --> 00:03:40.970
f1 times f1 times f2 times f2,
all the way to fn times fn.

00:03:40.970 --> 00:03:44.330
Now, we know that a squared
is a multiple of p. p

00:03:44.330 --> 00:03:48.360
is a prime number,
so p must be one

00:03:48.360 --> 00:03:51.350
of these numbers in the
prime factorization.

00:03:51.350 --> 00:03:55.120
p could be f2, or p
could be f1, but p

00:03:55.120 --> 00:03:58.730
needs to be one of these numbers
in the prime factorization.

00:03:58.730 --> 00:04:01.494
So p needs to be one
of these factors.

00:04:01.494 --> 00:04:03.160
Well, if it's, let's
say-- and I'll just

00:04:03.160 --> 00:04:04.760
pick one of these arbitrarily.

00:04:04.760 --> 00:04:07.790
Let's say that p is f2.

00:04:07.790 --> 00:04:13.560
If p is f2, then that means
that p is also a factor of a.

00:04:13.560 --> 00:04:26.660
So this allows us to deduce
that a is a multiple of p.

00:04:26.660 --> 00:04:29.180
Or another way of saying
that is that we can represent

00:04:29.180 --> 00:04:35.530
a as being some integer times p.

00:04:40.260 --> 00:04:41.570
Now, why is that interesting?

00:04:41.570 --> 00:04:44.270
And actually, let
me box this off,

00:04:44.270 --> 00:04:47.760
because we're going to
reuse this part later.

00:04:47.760 --> 00:04:48.980
But how can we use this?

00:04:48.980 --> 00:04:51.870
Well, just like we did in the
proof of the square root of 2

00:04:51.870 --> 00:04:54.390
being irrational, let's
now substitute this back

00:04:54.390 --> 00:04:56.570
into this equation
right over here.

00:04:56.570 --> 00:04:58.870
So we get b squared times p.

00:04:58.870 --> 00:05:05.740
We have b squared times
p is equal to a squared.

00:05:05.740 --> 00:05:07.770
Well, a, we're now
saying we can represent

00:05:07.770 --> 00:05:10.600
that as some integer k times p.

00:05:10.600 --> 00:05:19.890
So we can rewrite that as
some integer k times p.

00:05:22.980 --> 00:05:25.660
And so, let's see, if we
were to multiply this out.

00:05:25.660 --> 00:05:28.860
So we get b squared times
p-- and you probably

00:05:28.860 --> 00:05:33.700
see where this is going--
is equal to k squared

00:05:33.700 --> 00:05:36.810
times p squared.

00:05:36.810 --> 00:05:40.760
We can divide both
sides by p, and we

00:05:40.760 --> 00:05:45.290
get b squared is equal
to p times k squared.

00:05:45.290 --> 00:05:46.742
Or k squared times p.

00:05:50.360 --> 00:05:55.510
Well, the same
argument that we used,

00:05:55.510 --> 00:05:58.320
if a squared is equal to b
squared times p, that let us

00:05:58.320 --> 00:06:00.260
know that a squared
is a multiple of p.

00:06:00.260 --> 00:06:02.310
So now we have it
the other way around.

00:06:02.310 --> 00:06:05.072
b squared is equal to
some integer squared,

00:06:05.072 --> 00:06:07.030
which is still going to
be an integer, times p.

00:06:07.030 --> 00:06:09.760
So b squared must
be a multiple of p.

00:06:09.760 --> 00:06:17.323
So this lets us know that b
squared is a multiple of p.

00:06:20.041 --> 00:06:22.430
And by the logic
that we applied right

00:06:22.430 --> 00:06:25.265
over here, that lets us know
that b is a multiple of p.

00:06:29.559 --> 00:06:31.600
And that's our contradiction,
or this establishes

00:06:31.600 --> 00:06:36.440
our contradiction that we
assumed at the beginning.

00:06:36.440 --> 00:06:39.670
We assumed that a
and b are co-prime,

00:06:39.670 --> 00:06:43.220
that they share no factors
in common other than 1.

00:06:43.220 --> 00:06:46.000
We assumed that this
cannot be reduced.

00:06:46.000 --> 00:06:48.010
But we've just established,
just from this,

00:06:48.010 --> 00:06:51.200
we have deduced that
is a multiple of p

00:06:51.200 --> 00:06:52.654
and b is a multiple of p.

00:06:52.654 --> 00:06:54.570
Which means that this
fraction can be reduced.

00:06:54.570 --> 00:06:57.970
We can divide the numerator
and the denominator by p.

00:06:57.970 --> 00:06:59.270
So that is our contradiction.

00:06:59.270 --> 00:07:01.560
We started assuming
it cannot be reduced,

00:07:01.560 --> 00:07:04.325
but then we showed that, no,
it must be able to be reduced.

00:07:04.325 --> 00:07:05.700
The numerator and
the denominator

00:07:05.700 --> 00:07:08.750
have a common factor of p.

00:07:08.750 --> 00:07:11.720
So our contradiction
is established.

00:07:11.720 --> 00:07:13.940
Square root of p
cannot be rational.

00:07:13.940 --> 00:07:16.347
Square root of p is irrational.

00:07:16.347 --> 00:07:17.430
Let me just write it down.

00:07:17.430 --> 00:07:25.370
The square root of p
is irrational because

00:07:25.370 --> 00:07:27.460
of the contradiction.


Proof that square root of 2 is irrational

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.700
What I want to do in this
video is prove to you

00:00:03.700 --> 00:00:08.160
that the square root
of 2 is irrational.

00:00:08.160 --> 00:00:12.280
And I'm going to do this through
a proof by contradiction.

00:00:12.280 --> 00:00:15.970
And the proof by
contradiction is set up

00:00:15.970 --> 00:00:18.690
by assuming the opposite.

00:00:18.690 --> 00:00:21.860
So this is our goal, but
for the sake of our proof,

00:00:21.860 --> 00:00:23.550
let's assume the opposite.

00:00:23.550 --> 00:00:27.330
Let's assume that square
root of 2 is rational.

00:00:27.330 --> 00:00:30.017
And then we'll see if we
lead to a contradiction,

00:00:30.017 --> 00:00:31.600
that this actually
cannot be the case.

00:00:31.600 --> 00:00:33.516
And if it cannot be the
case that is rational,

00:00:33.516 --> 00:00:37.330
if we get to a contradiction by
assuming the square root of 2

00:00:37.330 --> 00:00:40.360
is rational, then we have to
deduce that the square root

00:00:40.360 --> 00:00:42.760
of 2 must be irrational.

00:00:42.760 --> 00:00:44.500
So let's assume the opposite.

00:00:44.500 --> 00:00:48.220
Square root of 2 is rational.

00:00:48.220 --> 00:00:50.860
Well, if the square
root of 2 is rational,

00:00:50.860 --> 00:00:53.900
that means that we can
write the square root of 2

00:00:53.900 --> 00:01:00.310
as the ratio of two
integers, a and b.

00:01:00.310 --> 00:01:04.250
And we can also
assume that these

00:01:04.250 --> 00:01:05.654
have no factors in common.

00:01:05.654 --> 00:01:07.820
Let's say that they did
have some factors in common.

00:01:07.820 --> 00:01:09.990
If we divided the numerator
and the denominator

00:01:09.990 --> 00:01:12.660
by those same factors, then
you're getting into a situation

00:01:12.660 --> 00:01:14.230
where they have no
factors in common.

00:01:14.230 --> 00:01:17.826
Or another way of saying is
that a and b are co-prime.

00:01:17.826 --> 00:01:19.200
Or another way of
saying it is we

00:01:19.200 --> 00:01:21.060
could write this as a
ratio of two integers

00:01:21.060 --> 00:01:24.590
where this is irreducible,
where these no longer share

00:01:24.590 --> 00:01:25.690
any factors.

00:01:25.690 --> 00:01:28.180
If you can write anything as
the ratio of two integers,

00:01:28.180 --> 00:01:30.770
then you could obviously
simplify it further,

00:01:30.770 --> 00:01:33.730
factor out any common
factors to get it to a point

00:01:33.730 --> 00:01:35.600
where it is irreducible.

00:01:35.600 --> 00:01:38.400
So I'm going to assume
that my a and b,

00:01:38.400 --> 00:01:41.720
that this fraction right
over here, is irreducible.

00:01:41.720 --> 00:01:45.770
And this is important for
setting up our contradiction.

00:01:45.770 --> 00:01:48.483
So I'm going to assume that this
right over here is irreducible.

00:01:51.960 --> 00:01:53.909
a and b have no
factors in common.

00:01:53.909 --> 00:01:55.450
Let me write that
down because that's

00:01:55.450 --> 00:01:57.120
so important for this proof.

00:01:57.120 --> 00:01:59.450
a and-- want to do
that same color--

00:01:59.450 --> 00:02:11.135
a and b have no factors in
common, other than, I guess, 1.

00:02:19.380 --> 00:02:20.630
So this is irreducible.

00:02:20.630 --> 00:02:24.350
These two numbers are co-prime.

00:02:24.350 --> 00:02:25.590
So what does that do for us?

00:02:25.590 --> 00:02:28.130
Well, let's just try to
manipulate this a little bit.

00:02:28.130 --> 00:02:30.810
Let's square both
sides of this equation.

00:02:30.810 --> 00:02:33.210
So if you square the
principal root of 2,

00:02:33.210 --> 00:02:35.040
you're going to get 2.

00:02:35.040 --> 00:02:39.520
And that's going to be equal
to a squared over b squared.

00:02:42.930 --> 00:02:44.970
And that just comes
from a over b squared

00:02:44.970 --> 00:02:48.800
is the same thing as a
squared over b squared.

00:02:48.800 --> 00:02:52.020
And now we can multiply both
sides of this by b squared.

00:02:52.020 --> 00:03:02.370
And so we get 2 times b
squared is equal to a squared.

00:03:02.370 --> 00:03:06.210
Now, what does this
tell us about a squared?

00:03:06.210 --> 00:03:10.230
Well, a squared is some
number, b squared times 2.

00:03:10.230 --> 00:03:14.170
So anything times 2 is going--
this is going to be an integer.

00:03:14.170 --> 00:03:17.890
We assumed b is an integer, so
b squared must be an integer,

00:03:17.890 --> 00:03:20.120
and so you have an
integer times 2.

00:03:20.120 --> 00:03:21.910
Well, that must give
you an even number.

00:03:21.910 --> 00:03:23.860
That must give you
an even integer.

00:03:23.860 --> 00:03:25.990
So this right over
here, a squared,

00:03:25.990 --> 00:03:35.830
must be-- so this tells us
that a squared must be even.

00:03:35.830 --> 00:03:38.240
Now, why is that interesting?

00:03:38.240 --> 00:03:40.890
Well, a squared is the
product of two numbers

00:03:40.890 --> 00:03:42.810
or is the product
of the same number.

00:03:42.810 --> 00:03:44.260
It's a times a.

00:03:44.260 --> 00:03:49.850
So this is another way of
saying that a times a is even.

00:03:49.850 --> 00:03:52.160
So what does that
tell us about a?

00:03:52.160 --> 00:03:53.520
Let's just remind ourselves.

00:03:53.520 --> 00:03:55.210
a is either going to
be-- we're assuming

00:03:55.210 --> 00:03:58.960
a is an integer-- a is either
going to be even or odd.

00:03:58.960 --> 00:04:01.020
We just have to remind
ourselves if we multiply

00:04:01.020 --> 00:04:07.120
an even times an even,
we get an even number.

00:04:07.120 --> 00:04:13.800
If we multiply an odd times
an odd, we get an odd number.

00:04:13.800 --> 00:04:15.710
So we have a number
times itself.

00:04:15.710 --> 00:04:16.700
We got an even number.

00:04:16.700 --> 00:04:20.140
Well, the only way to get that
is if that number is even.

00:04:20.140 --> 00:04:27.110
So this tells us that a is even.

00:04:27.110 --> 00:04:29.960
And another way of
saying that a is even

00:04:29.960 --> 00:04:34.640
is to say that a
can be represented

00:04:34.640 --> 00:04:39.210
as the product of 2
times some integer.

00:04:39.210 --> 00:04:43.510
So let's say some integer k.

00:04:43.510 --> 00:04:45.300
So where is all of this going?

00:04:45.300 --> 00:04:47.400
Well, as you'll
see, we can then use

00:04:47.400 --> 00:04:50.100
this to show that b
must also be even.

00:04:50.100 --> 00:04:52.350
So let's think about
that a little bit.

00:04:52.350 --> 00:04:54.660
So let's go back to this
step right over here.

00:04:54.660 --> 00:04:57.122
If we say that a
can be represented

00:04:57.122 --> 00:04:58.830
as two times the
product of some integer,

00:04:58.830 --> 00:05:02.140
and that comes out of
the fact that a is even.

00:05:02.140 --> 00:05:05.280
Then we can rewrite this
expression right over here

00:05:05.280 --> 00:05:09.980
as 2-- I'll do it
over here-- 2 times

00:05:09.980 --> 00:05:17.600
b squared is equal
to 2k squared.

00:05:17.600 --> 00:05:21.230
Instead of a squared, I
could write 2k squared.

00:05:21.230 --> 00:05:23.160
We're claiming,
or we're deducing,

00:05:23.160 --> 00:05:26.790
that, assuming everything we've
just assumed, that a is even.

00:05:26.790 --> 00:05:28.920
So if a is even, it
can be represented

00:05:28.920 --> 00:05:31.840
as a product of 2
and some integer.

00:05:31.840 --> 00:05:38.970
And then we can write
that 2 times b squared

00:05:38.970 --> 00:05:42.670
is equal to 4k squared.

00:05:46.190 --> 00:05:48.790
And then you divide
both sides by 2.

00:05:48.790 --> 00:05:53.655
You get b squared is
equal to 2k squared.

00:05:58.790 --> 00:06:01.310
And this tells us
that, well, k squared

00:06:01.310 --> 00:06:03.300
is going to be an integer.

00:06:03.300 --> 00:06:04.903
You take any integer
times 2 you're

00:06:04.903 --> 00:06:06.080
going to get an even value.

00:06:06.080 --> 00:06:07.970
So this tells us that
b squared is even.

00:06:13.080 --> 00:06:16.020
So that tells us that
b squared is even.

00:06:16.020 --> 00:06:19.380
Well, if b squared is even, by
the same logic we just used,

00:06:19.380 --> 00:06:20.960
that tells us that b is even.

00:06:23.470 --> 00:06:25.530
So here's our contradiction.

00:06:25.530 --> 00:06:29.950
We assumed, in the
beginning, that a and b

00:06:29.950 --> 00:06:32.640
have no common
factors other than 1.

00:06:32.640 --> 00:06:36.360
We assumed that this fraction
right over here, a/b,

00:06:36.360 --> 00:06:37.960
is irreducible.

00:06:37.960 --> 00:06:40.577
But from that and
the fact that a/b

00:06:40.577 --> 00:06:42.160
must be equal to the
square root of 2,

00:06:42.160 --> 00:06:46.770
we were able to deduce that
a is even and b is even.

00:06:46.770 --> 00:06:48.670
Well, if a is even
and b is even,

00:06:48.670 --> 00:06:50.350
and they both have
2 as a factor,

00:06:50.350 --> 00:06:51.710
and then this isn't irreducible.

00:06:51.710 --> 00:06:54.810
You could divide the numerator
and the denominator by 2. a

00:06:54.810 --> 00:06:58.050
and b have a common factor of 2.

00:06:58.050 --> 00:06:59.260
So let me write this down.

00:06:59.260 --> 00:07:00.960
So this is just
to make it clear.

00:07:00.960 --> 00:07:16.500
So from this and this, we have
a and b have common factor of 2,

00:07:16.500 --> 00:07:19.690
which means that a
over b is reducible.

00:07:19.690 --> 00:07:21.115
And so that's the contradiction.

00:07:24.350 --> 00:07:26.950
So you assume that
square root of 2

00:07:26.950 --> 00:07:30.840
can be represented as an
irreducible fraction a/b,

00:07:30.840 --> 00:07:34.260
irreducible because you can
say ratio of two integers

00:07:34.260 --> 00:07:36.870
right over here, that leads
you to the contradiction

00:07:36.870 --> 00:07:39.000
that, no, it actually
can be reducible.

00:07:39.000 --> 00:07:42.080
So, therefore, you cannot
make this assumption.

00:07:42.080 --> 00:07:43.280
It leads to a contradiction.

00:07:43.280 --> 00:07:46.012
Square root of 2
must be irrational.


Recognizing rational and irrational expressions example

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.010
Let's think about whether
each of these expressions

00:00:03.010 --> 00:00:06.022
produce rational or
irrational numbers.

00:00:06.022 --> 00:00:07.730
And just as a reminder,
a rational number

00:00:07.730 --> 00:00:10.680
is one-- so if you have
a rational number x,

00:00:10.680 --> 00:00:16.710
it can be expressed as the
ratio of two integers, m and n.

00:00:16.710 --> 00:00:21.140
And if you have an irrational
number, this cannot happen.

00:00:21.140 --> 00:00:24.120
So let's think
about each of these.

00:00:24.120 --> 00:00:26.250
So 9 is clearly a
rational number.

00:00:26.250 --> 00:00:31.230
You can express 9 as
9/1, 18/2, or 27/3.

00:00:31.230 --> 00:00:35.410
So it can clearly be expressed
as the ratio of two integers.

00:00:35.410 --> 00:00:37.445
But what about the
square root of 45?

00:00:37.445 --> 00:00:39.070
So let's think about
that a little bit.

00:00:39.070 --> 00:00:41.530
Square root of 45.

00:00:41.530 --> 00:00:45.310
That's the same thing as the
square root of 9 times 5,

00:00:45.310 --> 00:00:47.180
which is the same thing
as the square root

00:00:47.180 --> 00:00:50.430
of 9 times the square root of 5.

00:00:50.430 --> 00:00:52.730
The principal root
of 9 is 3, so it's

00:00:52.730 --> 00:00:54.960
3 times the square root of 5.

00:00:54.960 --> 00:00:59.420
So this is going to be 9 plus
3 times the square root of 5.

00:00:59.420 --> 00:01:02.000
So the square root
of 5 is irrational.

00:01:02.000 --> 00:01:05.280
You're taking the square root
of a non-perfect square right

00:01:05.280 --> 00:01:06.100
over here.

00:01:06.100 --> 00:01:07.350
Irrational.

00:01:07.350 --> 00:01:10.395
3 is rational, but the product
of a rational and an irrational

00:01:10.395 --> 00:01:12.060
is still going to be irrational.

00:01:12.060 --> 00:01:13.750
So that's going
to be irrational.

00:01:13.750 --> 00:01:15.630
And then you're taking
an irrational number

00:01:15.630 --> 00:01:17.030
and you're adding 9 to it.

00:01:17.030 --> 00:01:18.760
You're adding a
rational number to it.

00:01:18.760 --> 00:01:21.800
But you add a rational
to an irrational,

00:01:21.800 --> 00:01:24.040
and you're still going
to have an irrational.

00:01:24.040 --> 00:01:26.530
So this whole thing
is irrational.

00:01:29.500 --> 00:01:34.970
Now let's think about this
expression right over here.

00:01:34.970 --> 00:01:36.470
Well, the numerator
can be rewritten

00:01:36.470 --> 00:01:40.400
as the square root of 9 times
5 over 3 times the square root

00:01:40.400 --> 00:01:41.320
of 5.

00:01:41.320 --> 00:01:43.236
Well, that's the same
thing as the square root

00:01:43.236 --> 00:01:47.300
of 9 times the square root of
5 over 3 times the square root

00:01:47.300 --> 00:01:48.230
of 5.

00:01:48.230 --> 00:01:52.200
Well, that's the same thing as
3 times the square root of 5

00:01:52.200 --> 00:01:55.420
over 3 times the
square root of 5.

00:01:55.420 --> 00:01:57.890
Well, that's just
going to be equal to 1.

00:01:57.890 --> 00:01:59.350
Or you could view it as 1/1.

00:01:59.350 --> 00:02:01.430
And 1 is clearly
a rational number.

00:02:01.430 --> 00:02:05.880
You could write it as 1/1,
2/2, 3/3, really any integer

00:02:05.880 --> 00:02:07.070
over itself.

00:02:07.070 --> 00:02:11.245
So this is going to be rational.

00:02:11.245 --> 00:02:13.370
Now, let's do this last
expression right over here.

00:02:13.370 --> 00:02:15.440
3 times the principal root of 9.

00:02:15.440 --> 00:02:17.280
Well, what's the
principal root of 9?

00:02:17.280 --> 00:02:18.000
Well, it's 3.

00:02:18.000 --> 00:02:21.490
So this is going to be 3
times 3, which is equal to 9.

00:02:21.490 --> 00:02:23.280
And we've already
talked about the fact

00:02:23.280 --> 00:02:25.880
that 9 can clearly be
expressed as the ratio of two

00:02:25.880 --> 00:02:34.933
integers-- 9/1, 27/3, 45/5, all
different ways of expressing 9.


Proof that rational times irrational is irrational

WEBVTT
Kind: captions
Language: en

00:00:00.145 --> 00:00:01.520
What I want to do
with this video

00:00:01.520 --> 00:00:05.820
is do a quick proof that if
we take a rational number,

00:00:05.820 --> 00:00:11.220
and we multiply it times
an irrational number,

00:00:11.220 --> 00:00:17.712
that this is going to give
us an irrational number.

00:00:17.712 --> 00:00:19.670
And I encourage you to
actually pause the video

00:00:19.670 --> 00:00:21.160
and try to think if you
can prove this on your own.

00:00:21.160 --> 00:00:22.201
And I'll give you a hint.

00:00:22.201 --> 00:00:25.690
You can prove it by a proof
through contradiction.

00:00:25.690 --> 00:00:28.240
Assume that a rational
times an irrational

00:00:28.240 --> 00:00:31.750
gets you a rational number, and
then see by manipulating it,

00:00:31.750 --> 00:00:34.980
whether you can establish that
all of a sudden this irrational

00:00:34.980 --> 00:00:37.484
number must somehow be rational.

00:00:37.484 --> 00:00:39.150
So I'm assuming you've
given a go at it.

00:00:39.150 --> 00:00:40.360
So let's think about
it a little bit.

00:00:40.360 --> 00:00:42.850
I said we will do it through
a proof by contradiction.

00:00:42.850 --> 00:00:56.360
So let's just assume that a
rational times an irrational

00:00:56.360 --> 00:00:57.820
gives us a rational number.

00:01:01.100 --> 00:01:05.530
So let's say that this-- to
represent this rational right

00:01:05.530 --> 00:01:07.870
over here, let's represent
it as the ratio of two

00:01:07.870 --> 00:01:10.140
integers, a over b.

00:01:10.140 --> 00:01:13.920
And then this irrational
number, I'll just call that x.

00:01:13.920 --> 00:01:20.220
So we're saying a/b times x can
get us some rational number.

00:01:20.220 --> 00:01:22.600
So let's call that m/n.

00:01:22.600 --> 00:01:25.940
Let's call this equaling m/n.

00:01:25.940 --> 00:01:29.620
So I'm assuming that a
rational number, which

00:01:29.620 --> 00:01:31.660
can be expressed as the
ratio of two integers,

00:01:31.660 --> 00:01:35.560
times an irrational number can
get me another rational number.

00:01:35.560 --> 00:01:38.160
So let's see if we can set
up some form of contradiction

00:01:38.160 --> 00:01:39.330
here.

00:01:39.330 --> 00:01:41.000
Let's solve for the
irrational number.

00:01:41.000 --> 00:01:42.570
The best way to
solve is to multiply

00:01:42.570 --> 00:01:46.970
both sides times the reciprocal
of this number right over here.

00:01:46.970 --> 00:01:55.640
So this, let's multiply
times b/a, times b/a.

00:01:55.640 --> 00:01:56.980
And what are we left with?

00:01:56.980 --> 00:02:05.509
We get our irrational number
x being equal to m times b.

00:02:05.509 --> 00:02:07.050
Or we could just
write that as mb/na.

00:02:17.340 --> 00:02:19.310
So why is this interesting?

00:02:19.310 --> 00:02:21.440
Well, m is an integer,
b is an integer,

00:02:21.440 --> 00:02:24.500
so this whole numerator
is an integer.

00:02:24.500 --> 00:02:27.560
And then this whole
denominator is some integer.

00:02:27.560 --> 00:02:30.690
So right over here, I have
a ratio of two integers.

00:02:30.690 --> 00:02:33.380
So I've just expressed
what we assumed

00:02:33.380 --> 00:02:36.900
to be an irrational number,
I've just it expressed it

00:02:36.900 --> 00:02:38.820
as the ratio of two integers.

00:02:38.820 --> 00:02:45.090
So now we have x
must be rational.

00:02:45.090 --> 00:02:47.330
And that is our
contradiction, because we

00:02:47.330 --> 00:02:51.220
assumed that x is irrational.

00:02:51.220 --> 00:02:54.330
And so therefore,
since this assumption

00:02:54.330 --> 00:03:01.330
leads to this contradiction
right over here,

00:03:01.330 --> 00:03:03.520
this assumption must be false.

00:03:03.520 --> 00:03:09.280
It must be that a rational times
an irrational is irrational.


Approximating irrational number exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.680
On the number line
below, which point

00:00:02.680 --> 00:00:06.895
more closely represents
the principal root of 43?

00:00:06.895 --> 00:00:10.195
Or you could view it as the
positive square root of 43.

00:00:10.195 --> 00:00:11.820
And I encourage you
to pause this video

00:00:11.820 --> 00:00:14.484
and, just looking at this,
try to approximate where A is

00:00:14.484 --> 00:00:15.607
and where B is.

00:00:15.607 --> 00:00:17.440
And let's think about
which of those numbers

00:00:17.440 --> 00:00:22.256
is closer to the
principal root of 43.

00:00:22.256 --> 00:00:23.630
So let's go through
each of them.

00:00:23.630 --> 00:00:26.750
So point A right over there--
it looks a little bit more

00:00:26.750 --> 00:00:31.100
than halfway along the
way between 6.5 and 6.6.

00:00:31.100 --> 00:00:34.950
Halfway looks like
it's right over there,

00:00:34.950 --> 00:00:37.470
so that would be 6.55.

00:00:37.470 --> 00:00:41.010
So I would estimate A to
be approximately-- A looks

00:00:41.010 --> 00:00:48.420
like it's approximately 6.56.

00:00:48.420 --> 00:00:53.760
So let's square that and see
what we get-- so 6.56 times

00:00:53.760 --> 00:00:55.330
6.56.

00:00:55.330 --> 00:00:57.080
And obviously, if
you had a calculator,

00:00:57.080 --> 00:00:59.240
you could just punch the
square root of 43 in,

00:00:59.240 --> 00:01:00.950
but we don't want to do that.

00:01:00.950 --> 00:01:03.690
Or this exercise doesn't
want us to do that,

00:01:03.690 --> 00:01:06.220
so I'm not going to use
a calculator for it.

00:01:06.220 --> 00:01:08.500
So 6 times 6 is 36.

00:01:08.500 --> 00:01:13.210
6 times 5 is 30, plus 3 is 33.

00:01:13.210 --> 00:01:17.490
6 times 6 is 36, plus 3 is 39.

00:01:17.490 --> 00:01:20.290
Now, you'd think about this 5.

00:01:20.290 --> 00:01:21.850
So let's throw a 0 here.

00:01:21.850 --> 00:01:23.663
5 times 6 is 30.

00:01:23.663 --> 00:01:25.010
Put a 3 up there.

00:01:25.010 --> 00:01:29.530
5 times 5 is 25, plus 3 is 28.

00:01:29.530 --> 00:01:33.580
5 times 6 is 30, plus 2 is 32.

00:01:33.580 --> 00:01:35.880
And then we've got
another 6 here.

00:01:35.880 --> 00:01:38.520
Well, we already figured
out what 6 times 6.65 is.

00:01:38.520 --> 00:01:41.400
It's 3,936, so let's
throw our two 0's here.

00:01:41.400 --> 00:01:45.460
And we're going to have
a 6, a 3, a 9, and a 3.

00:01:45.460 --> 00:01:50.480
And now we can add everything
together, and we get 6,

00:01:50.480 --> 00:01:52.000
3-- let's see.

00:01:52.000 --> 00:01:57.577
9 plus 8 is 17, plus 6 is 23.

00:01:57.577 --> 00:01:58.160
And let's see.

00:01:58.160 --> 00:02:01.380
2 plus 3 is 5.

00:02:01.380 --> 00:02:03.205
Another 5, you get 10.

00:02:03.205 --> 00:02:03.990
And let's see.

00:02:03.990 --> 00:02:07.020
This is 13, so you get 4.

00:02:07.020 --> 00:02:07.520
Let's see.

00:02:07.520 --> 00:02:09.280
And you have one,
two, three, four

00:02:09.280 --> 00:02:11.980
numbers behind the decimal.

00:02:11.980 --> 00:02:16.040
So this gets us to 43.0336.

00:02:16.040 --> 00:02:18.093
So this is pretty close to 43.

00:02:21.960 --> 00:02:24.700
Without even thinking
about what B squared is,

00:02:24.700 --> 00:02:26.460
I'm already
predisposed to saying

00:02:26.460 --> 00:02:28.980
that A is going to be
closer, but let's just try.

00:02:28.980 --> 00:02:30.650
Let's just think
about where B is.

00:02:30.650 --> 00:02:35.900
Actually, we don't even have to
look at B. Because A is pretty

00:02:35.900 --> 00:02:39.240
close, and A squared
is already larger.

00:02:39.240 --> 00:02:40.090
Let me write this.

00:02:40.090 --> 00:02:44.670
So A squared, if we approximated
it right, at 6.56-- so

00:02:44.670 --> 00:02:53.070
let me write this down--
squared, which is 43.0336,

00:02:53.070 --> 00:02:57.750
is already greater than 43.

00:02:57.750 --> 00:03:01.260
Or another way of
saying it, 6.56

00:03:01.260 --> 00:03:06.390
is already greater than
the principal root of 43.

00:03:06.390 --> 00:03:10.270
So A looks like it's
already-- it's pretty close.

00:03:10.270 --> 00:03:12.050
If anything, it's
a little bit larger

00:03:12.050 --> 00:03:13.660
than the square root of 43.

00:03:13.660 --> 00:03:17.410
So B is going to be that
much-- B is larger than A.

00:03:17.410 --> 00:03:19.690
So B is going to get
us even further above.

00:03:19.690 --> 00:03:22.580
If we square B, we're going
to get even further above 43.

00:03:22.580 --> 00:03:26.300
So we can rule out B, and that
saves us a lot of computation.

00:03:26.300 --> 00:03:28.030
And so we're going
to go with A. A is

00:03:28.030 --> 00:03:34.933
a pretty good approximation
of the square root of 43.


Proof that sum of rational and irrational is irrational

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.330
So I'm curious as
to what happens

00:00:02.330 --> 00:00:04.380
if I were to take
a rational number

00:00:04.380 --> 00:00:06.870
and I were to add it to
an irrational number.

00:00:09.870 --> 00:00:16.010
Is the resulting number going
to be rational or irrational?

00:00:16.010 --> 00:00:17.650
Well, to think about
this, let's just

00:00:17.650 --> 00:00:20.520
assume it's going to be
rational and then see

00:00:20.520 --> 00:00:24.040
if this leads to any
form of contradiction.

00:00:24.040 --> 00:00:25.750
So let's assume
that this is going

00:00:25.750 --> 00:00:30.850
to give us a rational number.

00:00:30.850 --> 00:00:32.790
So let's say that this
first rational number

00:00:32.790 --> 00:00:38.860
we can represent as the ratio
of two integers, a and b.

00:00:38.860 --> 00:00:42.720
Let's call this irrational
number, let's just call this x.

00:00:42.720 --> 00:00:46.890
And their sum gives us
another rational number.

00:00:46.890 --> 00:00:49.800
Well, let's express
that as the ratio of two

00:00:49.800 --> 00:00:53.070
other integers, m and n.

00:00:53.070 --> 00:01:00.990
So we're saying that a/b
plus x is equal to m/n.

00:01:03.674 --> 00:01:05.340
Well, another way of
thinking about it--

00:01:05.340 --> 00:01:07.800
we could subtract
a/b from both sides

00:01:07.800 --> 00:01:09.670
and we would get our
irrational number

00:01:09.670 --> 00:01:18.770
x is equal to m/n
minus a/b, which

00:01:18.770 --> 00:01:22.295
is the same thing as n
times b in the denominator.

00:01:26.210 --> 00:01:27.390
And then let's see.

00:01:27.390 --> 00:01:30.835
m/n is the same
thing as mb over nb.

00:01:30.835 --> 00:01:33.330
So this would be mb.

00:01:33.330 --> 00:01:34.990
I'm just adding
these two fractions.

00:01:34.990 --> 00:01:40.850
mb minus--

00:01:40.850 --> 00:01:45.580
Let's see. a/b is the same thing
as n times a over n times b.

00:01:45.580 --> 00:01:47.470
So minus n times a.

00:01:52.130 --> 00:01:54.020
All I did is I added
these two fractions.

00:01:54.020 --> 00:01:55.850
I found a common denominator.

00:01:55.850 --> 00:01:59.170
So to make it clear, I
multiplied this one b and b,

00:01:59.170 --> 00:02:02.190
and then I multiplied
this one n and n then

00:02:02.190 --> 00:02:05.250
I just added these two things,
and I got this expression right

00:02:05.250 --> 00:02:06.480
over here.

00:02:06.480 --> 00:02:08.909
So this denominator
is clearly an integer.

00:02:08.909 --> 00:02:10.630
I have the product
of two integers.

00:02:10.630 --> 00:02:12.380
That's going to be an integer.

00:02:12.380 --> 00:02:14.690
That's going to be an integer.

00:02:14.690 --> 00:02:17.520
And then this numerator,
mb, is an integer.

00:02:17.520 --> 00:02:18.509
na is an integer.

00:02:18.509 --> 00:02:19.800
The difference of two integers.

00:02:19.800 --> 00:02:21.591
This whole thing is
going to be an integer.

00:02:24.740 --> 00:02:29.290
So it looks like, assuming
that the sum is rational,

00:02:29.290 --> 00:02:31.290
that all of a sudden we
have this contradiction.

00:02:31.290 --> 00:02:35.380
We assumed that x is irrational,
we're assuming x is irrational,

00:02:35.380 --> 00:02:37.706
but, all of a sudden, because
we made that assumption,

00:02:37.706 --> 00:02:39.580
we're able to assume
that we can represent it

00:02:39.580 --> 00:02:42.510
as this ratio of two integers.

00:02:42.510 --> 00:02:45.265
So this tells us that
x must be rational.

00:02:48.720 --> 00:02:51.480
And that is the contradiction.

00:02:51.480 --> 00:02:54.450
That is a very large
contradiction right over there.

00:02:54.450 --> 00:02:57.140
The assumption was
that x is irrational.

00:02:57.140 --> 00:03:00.090
Now we got that x
must be rational.

00:03:00.090 --> 00:03:02.920
So, therefore, this
cannot be the case.

00:03:02.920 --> 00:03:05.980
A rational plus an
irrational must--

00:03:05.980 --> 00:03:08.460
so this is not right--
a rational plus

00:03:08.460 --> 00:03:10.530
an irrational must
be irrational.

00:03:10.530 --> 00:03:11.670
Let me write that down.

00:03:11.670 --> 00:03:22.870
So a rational plus an
irrational must be irrational.


Matching radical functions with graphs exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.660
Match each function
with its graph.

00:00:02.660 --> 00:00:05.190
And we have graph
D, A, B, and C.

00:00:05.190 --> 00:00:07.166
And let's just start
with the graph of B

00:00:07.166 --> 00:00:09.040
because, actually, this
one looks the closest

00:00:09.040 --> 00:00:12.270
to the square root of x, which
would look something like that.

00:00:12.270 --> 00:00:13.600
But it's clearly shifted.

00:00:13.600 --> 00:00:16.072
And it's flipped over
the horizontal axis.

00:00:16.072 --> 00:00:18.530
The fact that it's flipped over
the square horizontal axis,

00:00:18.530 --> 00:00:23.600
that means that we're taking
the negative square root.

00:00:23.600 --> 00:00:27.580
So it's going to be one of
these two cases right over here.

00:00:27.580 --> 00:00:30.210
We're shifting it down by 2.

00:00:30.210 --> 00:00:32.640
So we should have a negative
2, and both of these cases

00:00:32.640 --> 00:00:34.290
have a negative 2 here.

00:00:34.290 --> 00:00:36.750
And we're shifting it relative
to the square root of x.

00:00:36.750 --> 00:00:38.985
We're shifting it
to the right by 1.

00:00:38.985 --> 00:00:40.360
So if we're shifting
to the right

00:00:40.360 --> 00:00:44.470
by 1, what we see under the
radical should be x minus 1.

00:00:44.470 --> 00:00:47.210
And both of these have
an x minus 1 on it.

00:00:47.210 --> 00:00:49.360
So which of these is B?

00:00:49.360 --> 00:00:50.954
And which of these is C?

00:00:50.954 --> 00:00:52.620
And I encourage you
to think about that.

00:00:52.620 --> 00:00:54.870
Pause the video if you'd like.

00:00:54.870 --> 00:00:57.630
Well, the difference between
these two is this one has

00:00:57.630 --> 00:01:01.670
a scaling factor of 2 here
while this one doesn't.

00:01:01.670 --> 00:01:03.232
And of course, this
is going to turn

00:01:03.232 --> 00:01:04.690
into more and more
negative values.

00:01:04.690 --> 00:01:06.560
This is going to be
getting negative faster.

00:01:06.560 --> 00:01:09.580
And you see that graph C
here gets negative faster.

00:01:09.580 --> 00:01:11.350
And you could even
try some points.

00:01:11.350 --> 00:01:15.290
In either case, when
x is equal to 1,

00:01:15.290 --> 00:01:19.220
in either case, what we have
under the radical becomes 0.

00:01:19.220 --> 00:01:23.000
x minus 1 is 0. x minus 1
is 0 when x is equal to 1.

00:01:23.000 --> 00:01:26.220
And in either case, our
y value's negative 2.

00:01:26.220 --> 00:01:30.570
But then, you see,
as x increases,

00:01:30.570 --> 00:01:33.530
this one right over here
gets negative twice as fast.

00:01:33.530 --> 00:01:36.200
You see that right over here.

00:01:36.200 --> 00:01:42.100
This one right over here,
y has gone from negative 2

00:01:42.100 --> 00:01:44.570
to negative 4 here.

00:01:44.570 --> 00:01:49.040
For graph C, y has gone from
negative 2 to negative 6.

00:01:49.040 --> 00:01:50.330
So it's gone down by 4.

00:01:50.330 --> 00:01:52.570
This has only gone down by 2.

00:01:52.570 --> 00:01:55.650
So it's pretty clear
that graph B corresponds

00:01:55.650 --> 00:01:57.870
to the one that doesn't
have the 2 out front.

00:01:57.870 --> 00:02:00.400
So this is graph
B right over here.

00:02:00.400 --> 00:02:02.250
And graph C
corresponds to the one

00:02:02.250 --> 00:02:03.860
that has a negative 2 out front.

00:02:03.860 --> 00:02:05.330
So let me try that.

00:02:05.330 --> 00:02:08.130
The negative 2 out front.

00:02:08.130 --> 00:02:11.390
So now we just have to think
about these two equations

00:02:11.390 --> 00:02:13.460
and match them to
these two graphs.

00:02:13.460 --> 00:02:16.340
And so both of
these, they haven't

00:02:16.340 --> 00:02:18.180
been flipped around
the horizontal axis.

00:02:18.180 --> 00:02:20.650
They've been flipped
around the vertical axis.

00:02:20.650 --> 00:02:24.380
And that's why whatever
we have under the radical

00:02:24.380 --> 00:02:26.560
we've essentially taken
the negative of it.

00:02:26.560 --> 00:02:28.120
And actually, we
could figure out

00:02:28.120 --> 00:02:30.820
which ones these are just by
looking at how much they've

00:02:30.820 --> 00:02:33.480
been shifted in the y direction
relative to the square root

00:02:33.480 --> 00:02:39.605
of x-- which would look
something like that.

00:02:39.605 --> 00:02:42.800
I know I can't draw on
this one right over here.

00:02:42.800 --> 00:02:45.055
Well, this one has
been shifted up by 2.

00:02:45.055 --> 00:02:46.620
D has been shifted up by 2.

00:02:46.620 --> 00:02:49.780
This one over here, g of x
has been shifted up by 2.

00:02:49.780 --> 00:02:54.270
And you could also see that it's
been shifted to the left by 1.

00:02:54.270 --> 00:02:55.770
If you're shifting
to the left by 1,

00:02:55.770 --> 00:02:57.978
normally, under the radical,
you would have x plus 1.

00:02:57.978 --> 00:03:03.105
But then, we flipped it over
around the vertical axis.

00:03:03.105 --> 00:03:04.730
And so that takes
the negative of that.

00:03:04.730 --> 00:03:06.490
That's why it's
negative x minus 1.

00:03:06.490 --> 00:03:09.220
You could view this as
the negative of x plus 1.

00:03:09.220 --> 00:03:12.750
So either way, that
is D. Throw that

00:03:12.750 --> 00:03:15.020
under the D. And just
deductive reasoning

00:03:15.020 --> 00:03:17.630
tells us A. This is
A, and it makes sense.

00:03:17.630 --> 00:03:19.750
We see it has shifted up by 1.

00:03:19.750 --> 00:03:21.210
So it's plus 1.

00:03:21.210 --> 00:03:33.290
And we could view this as
the negative of x plus 4.

00:03:33.290 --> 00:03:35.380
So it's been shifted
4 to the left.

00:03:35.380 --> 00:03:37.290
And we see that is
definitely the case

00:03:37.290 --> 00:03:39.790
relative to the
square root of x.

00:03:39.790 --> 00:03:41.620
We got it right.


Equations for radical functions example

WEBVTT
Kind: captions
Language: en

00:00:00.734 --> 00:00:02.650
Let's see if we can come
up with the equations

00:00:02.650 --> 00:00:06.572
for these different functions--
function A, B, C, and D.

00:00:06.572 --> 00:00:09.030
And I encourage you to pause
the video right now and try it

00:00:09.030 --> 00:00:10.800
on your own.

00:00:10.800 --> 00:00:13.820
So it might jump out at you
that these all look a little

00:00:13.820 --> 00:00:18.400
bit like y is equal to the
square root of x, which

00:00:18.400 --> 00:00:20.390
would look something like this.

00:00:20.390 --> 00:00:22.580
And we've covered this
in previous videos.

00:00:22.580 --> 00:00:27.250
y equals the square root of
x looks something like this.

00:00:27.250 --> 00:00:30.590
Except what we have here--
all the other things look

00:00:30.590 --> 00:00:32.967
like at minimum we flipped
over the horizontal axis,

00:00:32.967 --> 00:00:34.425
and at least for
B and D we flipped

00:00:34.425 --> 00:00:36.140
over the horizontal
and vertical axis.

00:00:36.140 --> 00:00:39.240
And in all the cases
we have shifted it.

00:00:39.240 --> 00:00:41.980
So let's think about
how we do that.

00:00:41.980 --> 00:00:45.750
So let's first flip over
the horizontal axis.

00:00:45.750 --> 00:00:47.740
So where we would have
expected a positive y,

00:00:47.740 --> 00:00:49.640
we're now going to
have a negative y.

00:00:49.640 --> 00:00:51.340
So essentially we're
just going to do

00:00:51.340 --> 00:00:54.230
y is equal to the
negative square root of x

00:00:54.230 --> 00:00:57.300
to get this thing.

00:00:57.300 --> 00:01:01.910
This is y is equal to the
negative square root of x.

00:01:01.910 --> 00:01:03.810
So put that negative in
front of the radical.

00:01:03.810 --> 00:01:07.080
We flipped it over
the horizontal axis.

00:01:07.080 --> 00:01:12.550
So that is y is equal to the
negative square root of x.

00:01:12.550 --> 00:01:15.890
Now let's think about how
we can shift this thing

00:01:15.890 --> 00:01:18.702
to look like function
A. Well, there's

00:01:18.702 --> 00:01:20.660
two things about function
A that are different.

00:01:20.660 --> 00:01:26.920
It's shifted to the left by
3, and it's shifted up by 1.

00:01:26.920 --> 00:01:30.070
And we've seen this
in previous examples.

00:01:30.070 --> 00:01:31.950
If we want to get to
A-- so I could write

00:01:31.950 --> 00:01:36.720
a of x-- what we could do
to shift to the left by 3,

00:01:36.720 --> 00:01:41.300
we got to replace
our x with x plus 3.

00:01:41.300 --> 00:01:46.580
So this is going to be negative
square root of x plus 3.

00:01:46.580 --> 00:01:48.500
That shifted us to the left.

00:01:48.500 --> 00:01:50.160
And then to shift
up, we just have

00:01:50.160 --> 00:01:52.660
to add a 1 right over here.

00:01:52.660 --> 00:01:55.405
So that right over
there is our a of x.

00:01:58.890 --> 00:02:00.425
Now let's think about B of x.

00:02:00.425 --> 00:02:02.300
And actually maybe I
should move on this side

00:02:02.300 --> 00:02:07.890
so that I can clear my little
scribbles right over there.

00:02:07.890 --> 00:02:09.880
So that is our A of x.

00:02:09.880 --> 00:02:11.090
Now let's think about B of x.

00:02:11.090 --> 00:02:13.650
How is B of x
different than A of x?

00:02:13.650 --> 00:02:15.820
Well, B of x starts
at the same point,

00:02:15.820 --> 00:02:18.490
but it's flipped over
the vertical line

00:02:18.490 --> 00:02:20.650
x is equal to negative 3.

00:02:20.650 --> 00:02:22.910
So its domain is different.

00:02:22.910 --> 00:02:26.470
This function's
domain is defined

00:02:26.470 --> 00:02:29.200
for x is greater than
or equal to negative 3.

00:02:29.200 --> 00:02:32.160
B is defined for x is less
than or equal to negative 3.

00:02:32.160 --> 00:02:34.820
And we saw that also
in the previous video.

00:02:34.820 --> 00:02:38.240
What we need to do is flip
the sign of everything

00:02:38.240 --> 00:02:40.030
that's under the radical.

00:02:40.030 --> 00:02:45.600
So B of x is going to be equal
to the negative square root

00:02:45.600 --> 00:02:47.970
of-- and going from A
of x to B of x, we're

00:02:47.970 --> 00:02:50.640
going to flip the sign of
everything under the radical.

00:02:50.640 --> 00:02:53.620
So it's negative x plus 3.

00:02:53.620 --> 00:03:00.160
And then once again we have
the plus 1 right over there.

00:03:00.160 --> 00:03:04.370
Now, to go from B of x to D of
x is reasonably straightforward.

00:03:04.370 --> 00:03:07.300
We just have to change
how it's shifted.

00:03:07.300 --> 00:03:09.790
D is shifted relative to
the original function.

00:03:09.790 --> 00:03:13.170
It's shifted up by 3
and to the left by 1.

00:03:13.170 --> 00:03:17.900
So D of x is going to
look very similar to B

00:03:17.900 --> 00:03:21.170
of x except its shifting
is going to be different.

00:03:21.170 --> 00:03:22.710
So let me just
write the part that

00:03:22.710 --> 00:03:24.650
is going to be very
similar to D of x.

00:03:24.650 --> 00:03:30.070
It's negative x plus something,
and then plus something.

00:03:30.070 --> 00:03:33.260
D of x has been shifted--
relative to the square root

00:03:33.260 --> 00:03:36.110
of x-- to the left by 1.

00:03:36.110 --> 00:03:40.140
So this is going to be x plus
1 and it is shifted up by 3.

00:03:40.140 --> 00:03:43.810
So it's going to be
plus 3 just like that.

00:03:43.810 --> 00:03:51.300
And then finally C of x is just
like D of x, except everything

00:03:51.300 --> 00:03:57.050
under the radical has to
have its sign flipped.

00:03:57.050 --> 00:04:05.457
So it is going to be negative
square root of x plus 1 plus 3.

00:04:05.457 --> 00:04:07.290
Another way to think
about it, if you wanted

00:04:07.290 --> 00:04:11.120
to go from A of x to C of x,
they look very much the same

00:04:11.120 --> 00:04:13.160
except their shifting
is different.

00:04:13.160 --> 00:04:16.430
C of x is shifted up
by 3, instead of 1,

00:04:16.430 --> 00:04:17.619
so that's that difference.

00:04:17.619 --> 00:04:19.160
And it's shifted to
the left relative

00:04:19.160 --> 00:04:21.529
to the square root of
x by 1, instead of 3.

00:04:21.529 --> 00:04:24.307
So that's why we have x
plus 1, instead of x plus 3.

00:04:24.307 --> 00:04:25.890
The one thing I
always want to stress,

00:04:25.890 --> 00:04:29.194
it's very intuitive that
changing this value--

00:04:29.194 --> 00:04:31.360
if you make this larger
you're going to shift it up.

00:04:31.360 --> 00:04:33.568
If you make it smaller you're
going to shift it down.

00:04:33.568 --> 00:04:35.310
But it's a little
counterintuitive

00:04:35.310 --> 00:04:37.710
that when you make
this value larger

00:04:37.710 --> 00:04:39.560
you're shifting
more to the left.

00:04:39.560 --> 00:04:43.510
And when you make this value
smaller, or even negative,

00:04:43.510 --> 00:04:46.760
you start shifting
more to the right.

00:04:46.760 --> 00:04:50.560
And we go into this in some
depth in the previous video

00:04:50.560 --> 00:04:52.780
where we kind of start
talking about all of this,

00:04:52.780 --> 00:04:54.690
but it's just to think
about, especially

00:04:54.690 --> 00:04:57.520
relative to square root of x,
when you think about just y is

00:04:57.520 --> 00:04:59.990
equal to square root of
x, what makes y equal 0?

00:04:59.990 --> 00:05:02.180
What has to happen
under the radical first

00:05:02.180 --> 00:05:04.550
to be kind of at the starting
point right over here?

00:05:04.550 --> 00:05:07.199
Well, everything under the
radical has to be equal to 0.

00:05:07.199 --> 00:05:09.490
And so you could think about
that same thing over here.

00:05:09.490 --> 00:05:13.310
To be at this starting point,
everything under our radical

00:05:13.310 --> 00:05:14.840
has to be equal to 0.

00:05:14.840 --> 00:05:16.670
When does everything
under our radical

00:05:16.670 --> 00:05:19.470
equal 0 at x equal negative 1?

00:05:19.470 --> 00:05:22.500
Well, if everything under
the radical was x plus 1.

00:05:22.500 --> 00:05:24.970
So I encourage you to think
about that as deeply as you can

00:05:24.970 --> 00:05:27.178
and to play around with
things, and then hopefully it

00:05:27.178 --> 00:05:29.710
will start making a
little bit of sense.


Flipping and shifting radical functions

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.220
So let's think
about the graph of y

00:00:03.220 --> 00:00:05.115
is equal to the
principal root of x.

00:00:05.115 --> 00:00:06.990
And then we'll start
playing around with this

00:00:06.990 --> 00:00:09.070
and see what happens
to the graph.

00:00:09.070 --> 00:00:11.210
So y is equal to the
principal root of x.

00:00:11.210 --> 00:00:12.770
Well, this is going
to be undefined

00:00:12.770 --> 00:00:15.500
if we want to deal
with real numbers.

00:00:15.500 --> 00:00:17.820
For x being any negative value.

00:00:17.820 --> 00:00:21.420
So the domain here is really x
is greater than or equal to 0.

00:00:21.420 --> 00:00:24.800
When x is 0, y is
going to be equal to 0.

00:00:24.800 --> 00:00:29.230
When x is 1, the principal
root of 1 is positive 1.

00:00:29.230 --> 00:00:30.830
So it's going to be like that.

00:00:30.830 --> 00:00:33.800
When x is 4, the
principal root of 4 is 2.

00:00:33.800 --> 00:00:37.810
When x is 9, the
principal root of 9 is 3.

00:00:37.810 --> 00:00:41.470
So this is what it is
going to look like.

00:00:41.470 --> 00:00:46.630
Which is going to
look more like this.

00:00:46.630 --> 00:00:48.690
So it's going to look
something like that.

00:00:48.690 --> 00:00:53.820
That's my best attempt
at actually graphing it.

00:00:53.820 --> 00:00:55.460
Now let's think
about what happens

00:00:55.460 --> 00:00:57.110
if we wanted to
shift it in some way.

00:00:57.110 --> 00:00:59.680
So let's say we
wanted to shift it up.

00:00:59.680 --> 00:01:02.720
Let's say we wanted
to shift it up by 4.

00:01:02.720 --> 00:01:04.904
So how would we do that?

00:01:04.904 --> 00:01:06.570
Well, whatever value
we're getting here,

00:01:06.570 --> 00:01:08.960
we want y to be 4 higher.

00:01:08.960 --> 00:01:11.170
So we could just add 4 to it.

00:01:11.170 --> 00:01:18.730
So we could just use y is equal
to the square root of x plus 4.

00:01:18.730 --> 00:01:21.890
So that would be like taking
this graph right over here.

00:01:21.890 --> 00:01:24.834
So let me copy and
then let me paste it.

00:01:24.834 --> 00:01:26.500
So it's like taking
this graph and we're

00:01:26.500 --> 00:01:30.310
shifting it up 1, 2, 3, 4.

00:01:30.310 --> 00:01:34.320
It would look that.

00:01:34.320 --> 00:01:36.150
Well, that was pretty
straightforward.

00:01:36.150 --> 00:01:37.650
Let me do it in
that same blue color

00:01:37.650 --> 00:01:39.755
so that you
recognize that that's

00:01:39.755 --> 00:01:42.440
that one right over there.

00:01:42.440 --> 00:01:47.410
But what if we wanted to shift
it, let's say, to the left.

00:01:47.410 --> 00:01:50.240
Let's say we wanted to
do something like this.

00:01:50.240 --> 00:01:55.190
Let's say we wanted to
shift it to the left by 3.

00:01:55.190 --> 00:01:56.410
Like that.

00:01:56.410 --> 00:02:00.180
So how would we define
the function then?

00:02:00.180 --> 00:02:03.740
And I'll do this in
this orange color.

00:02:03.740 --> 00:02:07.190
We want to shift by 3.

00:02:07.190 --> 00:02:10.070
So you're shifting
to the left by 3.

00:02:10.070 --> 00:02:11.640
So think about it.

00:02:11.640 --> 00:02:15.250
This point, y equaled
0, right over here,

00:02:15.250 --> 00:02:19.600
where whatever you put under
the radical was equal to 0.

00:02:19.600 --> 00:02:24.140
So what do you have to put
under the radical here to get 0?

00:02:24.140 --> 00:02:27.660
Well, here x is negative 3.

00:02:27.660 --> 00:02:31.760
So if you put x plus
3 under the radical

00:02:31.760 --> 00:02:34.590
then you are going to get--
and you take the square root

00:02:34.590 --> 00:02:36.180
of that-- you're going to get 0.

00:02:36.180 --> 00:02:41.000
So this right over here, this
orange function, that is y.

00:02:41.000 --> 00:02:45.530
Let me do it over here. y
is equal to the square root

00:02:45.530 --> 00:02:47.965
of x plus 3.

00:02:47.965 --> 00:02:49.840
And once again it might
be counter-intuitive.

00:02:49.840 --> 00:02:52.900
We went from square root of
x to square root of x plus 3.

00:02:52.900 --> 00:02:56.080
When we added 4 outside of the
radical that shifted it up.

00:02:56.080 --> 00:03:00.736
But when we add 3
inside of the radical

00:03:00.736 --> 00:03:02.360
instead of it shifting
it to the right,

00:03:02.360 --> 00:03:04.970
instead of shifting it that
way, it shifted it to the left.

00:03:04.970 --> 00:03:08.330
It made this point go
from 0 to negative 3.

00:03:08.330 --> 00:03:09.830
And the important
thing to realize

00:03:09.830 --> 00:03:12.500
is what makes y equal to 0.

00:03:12.500 --> 00:03:15.340
Over here y equals
0 when x is 0.

00:03:15.340 --> 00:03:18.810
Over here and over
here, y is equal to 0

00:03:18.810 --> 00:03:22.990
when x plus 3 is equal to 0,
or x is equal to negative 3.

00:03:22.990 --> 00:03:24.720
And you could do
that for other points

00:03:24.720 --> 00:03:27.332
to see that it does
definitely shift to the left.

00:03:27.332 --> 00:03:29.040
And that's an important
thing to realize.

00:03:29.040 --> 00:03:30.998
This isn't actually just
for radical functions.

00:03:30.998 --> 00:03:32.837
This is actually for
functions in general.

00:03:32.837 --> 00:03:34.920
If you're just going to
add a 4-- add a number out

00:03:34.920 --> 00:03:38.000
here-- whatever you add is
going to shift it up or down.

00:03:38.000 --> 00:03:40.540
If this was a minus 4 it
would have shifted it down.

00:03:40.540 --> 00:03:43.970
But within, when you
replace the x with an x

00:03:43.970 --> 00:03:47.510
plus 3, when you replace
it with an x plus 3,

00:03:47.510 --> 00:03:50.840
this actually shifted
it to the left by 3.

00:03:50.840 --> 00:03:52.680
If you wanted to shift
it to the right by 3

00:03:52.680 --> 00:03:56.020
you would put an x
minus 3 over there.

00:03:56.020 --> 00:03:57.740
Well, that's all interesting.

00:03:57.740 --> 00:04:02.400
But let's say that I wanted
to flip this thing over.

00:04:02.400 --> 00:04:04.454
So I wanted it to
look like this.

00:04:04.454 --> 00:04:05.620
Let me see if I can draw it.

00:04:05.620 --> 00:04:11.550
I want this graph to
look something like that.

00:04:11.550 --> 00:04:16.519
I'll try my best to-- to
look something like that.

00:04:16.519 --> 00:04:19.300
So it's flipped around.

00:04:19.300 --> 00:04:21.440
I could do a better
job than that.

00:04:21.440 --> 00:04:24.630
So we have that point on it.

00:04:24.630 --> 00:04:27.570
And then we're going
to go 3 and then

00:04:27.570 --> 00:04:29.112
we're going to have
that point on it.

00:04:29.112 --> 00:04:30.486
So I want it to
look-- actually I

00:04:30.486 --> 00:04:32.420
did a decent job the
first time I drew it--

00:04:32.420 --> 00:04:34.100
I want to look
something like that.

00:04:34.100 --> 00:04:38.160
So essentially I want
to take its mirror image

00:04:38.160 --> 00:04:41.260
around the line x is equal to 3.

00:04:41.260 --> 00:04:43.210
How could I do that?

00:04:43.210 --> 00:04:45.340
Well, now my domain
is different.

00:04:45.340 --> 00:04:48.560
Now my domain-- it should
be undefined for anything

00:04:48.560 --> 00:04:51.960
where x is greater
than negative 3.

00:04:51.960 --> 00:04:55.290
And it should be defined
for any x that is less than

00:04:55.290 --> 00:04:57.527
or equal to negative 3.

00:04:57.527 --> 00:04:59.110
Or another way to
think about it is we

00:04:59.110 --> 00:05:00.790
need to flip the
sign of whatever we

00:05:00.790 --> 00:05:02.400
have under the radical.

00:05:02.400 --> 00:05:05.620
So this thing over here--
let me actually scroll over

00:05:05.620 --> 00:05:10.870
a little bit-- this
thing over here in green

00:05:10.870 --> 00:05:15.170
could be y is equal
to the square root

00:05:15.170 --> 00:05:18.800
of the negative of x plus 3.

00:05:18.800 --> 00:05:21.390
And I encourage you to try some
x values here to try it out.

00:05:21.390 --> 00:05:23.380
What we've done is
we've essentially

00:05:23.380 --> 00:05:26.290
flipped what happens
under the radical.

00:05:26.290 --> 00:05:30.870
Now in order to get a positive
value under the radical,

00:05:30.870 --> 00:05:34.680
now x plus 3 has to be negative.

00:05:34.680 --> 00:05:37.700
And the only way that
x plus 3 is negative,

00:05:37.700 --> 00:05:41.130
or the only way
that x plus 3 is,

00:05:41.130 --> 00:05:44.160
I guess you can
say nonpositive, is

00:05:44.160 --> 00:05:48.042
if x is less than negative 3.

00:05:48.042 --> 00:05:50.000
Now what if we wanted to
do something even more

00:05:50.000 --> 00:05:50.600
interesting?

00:05:50.600 --> 00:05:55.700
What if we wanted to flip
this one right over here

00:05:55.700 --> 00:06:00.880
over the horizontal
axis, over y equals 0?

00:06:00.880 --> 00:06:03.880
Well, then we're just
flipping what root we take.

00:06:03.880 --> 00:06:07.560
So that would be y is equal
to the negative square root

00:06:07.560 --> 00:06:10.380
of negative x plus 3.

00:06:10.380 --> 00:06:11.670
So it would look like this.

00:06:17.090 --> 00:06:18.565
It would look like that.

00:06:21.330 --> 00:06:22.970
And if we wanted to
shift that thing,

00:06:22.970 --> 00:06:25.420
we could just add or
subtract something outside

00:06:25.420 --> 00:06:26.630
of the radical.

00:06:26.630 --> 00:06:29.770
So let's say we wanted to do--
let me copy and paste this.

00:06:32.550 --> 00:06:36.560
So let's copy and
then let's see.

00:06:36.560 --> 00:06:38.780
Let's say that we wanted
to shift it down here.

00:06:38.780 --> 00:06:41.445
So instead of beginning at y
equals 0 right over here, where

00:06:41.445 --> 00:06:43.070
y equals negative 4,
then we would just

00:06:43.070 --> 00:06:45.990
subtract 4 outside
of the radical.

00:06:45.990 --> 00:06:49.410
So this thing and this thing--
I'm running out of colors

00:06:49.410 --> 00:06:52.940
here-- this thing
right over here

00:06:52.940 --> 00:06:58.980
would be y is equal to negative
square root of negative x

00:06:58.980 --> 00:07:04.640
plus 3 minus 4.

00:07:04.640 --> 00:07:06.650
And so we could keep
going on and on and on.

00:07:06.650 --> 00:07:08.150
But hopefully this
gives you a sense

00:07:08.150 --> 00:07:10.730
of the different ways you
could manipulate this thing.

00:07:10.730 --> 00:07:12.510
And we'll do some
more examples to get

00:07:12.510 --> 00:07:15.160
a better understanding of it.


Absolute value graphing exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.580
Graph f of x on
the given domain.

00:00:03.580 --> 00:00:06.390
So f of x is equal to 4
times the absolute value

00:00:06.390 --> 00:00:08.800
of x plus 6 minus 8.

00:00:08.800 --> 00:00:11.960
And the domain is x
between negative 8--

00:00:11.960 --> 00:00:12.940
including negative 8.

00:00:12.940 --> 00:00:15.481
This bracket means that we're
going to include this endpoint.

00:00:15.481 --> 00:00:16.437
And negative 2.

00:00:16.437 --> 00:00:18.020
And we're going to
include negative 2.

00:00:18.020 --> 00:00:19.390
The bracket means we're
going to include it.

00:00:19.390 --> 00:00:21.140
If we didn't want to
include the endpoints

00:00:21.140 --> 00:00:22.710
we would use parentheses.

00:00:22.710 --> 00:00:25.380
What are the coordinates
of the vertex?

00:00:25.380 --> 00:00:27.140
So let's think about
the vertex first.

00:00:27.140 --> 00:00:31.200
The vertex you can view it as
the corner of an absolute value

00:00:31.200 --> 00:00:31.990
function.

00:00:31.990 --> 00:00:34.560
And so it's either going
to be the maximum point

00:00:34.560 --> 00:00:36.970
or the minimum point
of the function.

00:00:36.970 --> 00:00:40.120
Now over here let's just
think about it a little bit.

00:00:40.120 --> 00:00:43.987
When would this hit a
maximum or a minimum point?

00:00:43.987 --> 00:00:46.070
Well, we know that the
absolute value of something

00:00:46.070 --> 00:00:48.040
is always going
to be nonnegative.

00:00:48.040 --> 00:00:48.770
It could be 0.

00:00:48.770 --> 00:00:50.530
That's why I'm not
saying that it's always

00:00:50.530 --> 00:00:51.405
going to be positive.

00:00:51.405 --> 00:00:53.320
It's going to be nonnegative.

00:00:53.320 --> 00:00:55.846
So we have something that's
nonnegative right over here.

00:00:55.846 --> 00:00:57.720
And then we're multiplying
it times something

00:00:57.720 --> 00:00:59.040
that is positive.

00:00:59.040 --> 00:01:01.320
So this thing, the whole
expression over here,

00:01:01.320 --> 00:01:06.050
nonnegative times a positive is
always going to be nonnegative.

00:01:06.050 --> 00:01:09.000
So it's either going
to be 0 or greater.

00:01:09.000 --> 00:01:10.940
So this thing is
always going to be

00:01:10.940 --> 00:01:13.140
adding to the value of f of x.

00:01:13.140 --> 00:01:16.500
So the minimum value
of f of x happens

00:01:16.500 --> 00:01:19.060
when this whole thing is 0.

00:01:19.060 --> 00:01:22.670
And then when this whole thing
is 0 then f of x is negative 8.

00:01:22.670 --> 00:01:25.820
That is the minimum value.

00:01:25.820 --> 00:01:29.630
At any other point this thing
is going to be larger than 0.

00:01:29.630 --> 00:01:33.200
So let's just think about when
does this whole thing equal 0?

00:01:33.200 --> 00:01:37.920
Well, this whole thing is going
to be equalling 0 if x plus 6

00:01:37.920 --> 00:01:39.130
is equal to 0.

00:01:39.130 --> 00:01:40.950
And when does x plus 6 equal 0?

00:01:40.950 --> 00:01:44.460
Well, when x is
equal to negative 6.

00:01:44.460 --> 00:01:47.560
When x is negative 6 then
this whole thing is 0

00:01:47.560 --> 00:01:49.920
and f of x is equal
to negative 8.

00:01:52.480 --> 00:01:53.422
So that's the vertex.

00:01:53.422 --> 00:01:55.130
And that's going to
be the minimum point.

00:01:55.130 --> 00:01:57.740
This is going to be an
upward opening absolute value

00:01:57.740 --> 00:01:58.690
function.

00:01:58.690 --> 00:01:59.774
So now let's just draw it.

00:01:59.774 --> 00:02:01.148
And what we could
do is, we could

00:02:01.148 --> 00:02:02.450
put some points at the vertex.

00:02:02.450 --> 00:02:04.221
So let's put some points.

00:02:04.221 --> 00:02:04.720
Let's see.

00:02:04.720 --> 00:02:08.440
Negative 6, negative 8.

00:02:08.440 --> 00:02:13.390
Negative 6, negative
8 is right over there.

00:02:13.390 --> 00:02:14.230
Let me scroll down.

00:02:14.230 --> 00:02:16.350
Make sure you can
see that point.

00:02:16.350 --> 00:02:18.647
Negative 6, negative
8 is right over there.

00:02:18.647 --> 00:02:21.230
And I actually want to make you
see the function definition as

00:02:21.230 --> 00:02:21.730
well.

00:02:21.730 --> 00:02:24.146
Now let's just think about
what it does at the boundaries.

00:02:24.146 --> 00:02:26.050
On either side it's
going to be a line.

00:02:26.050 --> 00:02:30.570
So when x is negative
8, what is f of x?

00:02:30.570 --> 00:02:34.750
Negative 8 plus 6 is negative 2.

00:02:34.750 --> 00:02:36.250
Take the absolute value of that.

00:02:36.250 --> 00:02:41.150
That is going to be 2
times 4 is 8 minus 8 is 0.

00:02:41.150 --> 00:02:44.380
So you have the point
negative 8 comma 0.

00:02:44.380 --> 00:02:47.200
And then let's think about what
happens at the other boundary.

00:02:47.200 --> 00:02:49.470
When x is negative 2.

00:02:49.470 --> 00:02:54.750
Negative 2 plus 6 is 4.

00:02:54.750 --> 00:02:56.465
The absolute value of 4 is 4.

00:02:56.465 --> 00:03:01.550
4 times 4 is going to
be 16 minus 8 is 8.

00:03:01.550 --> 00:03:06.560
So it's going to be the
point negative 2, 8.

00:03:06.560 --> 00:03:09.500
And just like that,
I think we're done.

00:03:09.500 --> 00:03:11.470
Yes, we are.


Graphs of absolute value functions

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.740
Let's think a little bit about
the graphs of absolute value

00:00:03.740 --> 00:00:04.340
functions.

00:00:04.340 --> 00:00:06.310
And I've defined one
right over here. f

00:00:06.310 --> 00:00:09.480
of x is equal to negative 3
times the absolute value of x

00:00:09.480 --> 00:00:11.220
minus 1 plus 9.

00:00:11.220 --> 00:00:13.040
And then we've
constrained its domain.

00:00:13.040 --> 00:00:16.140
This is for x
being-- or negative 4

00:00:16.140 --> 00:00:18.170
is less than or equal
to x, which is less than

00:00:18.170 --> 00:00:19.720
or equal to 5.

00:00:19.720 --> 00:00:21.360
And I encourage you
to pause this video

00:00:21.360 --> 00:00:23.795
and try to graph
this on your own.

00:00:23.795 --> 00:00:25.920
And clearly you could graph
this by just trying out

00:00:25.920 --> 00:00:28.830
different x-values and plotting
the corresponding x and f

00:00:28.830 --> 00:00:31.255
of x values, but really
think about the structure

00:00:31.255 --> 00:00:32.630
of this expression
to think about

00:00:32.630 --> 00:00:34.990
where does it hit a
maximum or minimum point?

00:00:34.990 --> 00:00:39.280
And where would the
vertex of it actually be?

00:00:39.280 --> 00:00:41.310
So let's look at this
just holistically

00:00:41.310 --> 00:00:43.810
before we even try some points.

00:00:43.810 --> 00:00:46.780
So we have an absolute value.

00:00:46.780 --> 00:00:48.570
This absolute value
here is clearly

00:00:48.570 --> 00:00:52.450
going to always be
non-negative, and then we're

00:00:52.450 --> 00:00:58.900
multiplying it by
a negative number.

00:00:58.900 --> 00:01:02.230
So this entire thing,
you have a negative times

00:01:02.230 --> 00:01:05.640
a non-negative, so this
whole thing is always

00:01:05.640 --> 00:01:08.430
going to be non-positive.

00:01:08.430 --> 00:01:10.610
The reason why I say
non-positive instead

00:01:10.610 --> 00:01:13.900
of negative is that
because it could be 0.

00:01:13.900 --> 00:01:16.050
So this whole part
of the expression

00:01:16.050 --> 00:01:18.470
is either going
to be 0, or it is

00:01:18.470 --> 00:01:21.480
going to be a negative number.

00:01:21.480 --> 00:01:24.070
Now, if this is
always non-positive,

00:01:24.070 --> 00:01:28.060
what is the maximum
value of this function?

00:01:28.060 --> 00:01:29.910
Well, the maximum
value happens when

00:01:29.910 --> 00:01:32.560
the absolute value
is equal to 0.

00:01:32.560 --> 00:01:36.420
When the absolute value part of
the expression is equal to 0,

00:01:36.420 --> 00:01:40.650
then this part is 0, and the
function hits its maximum value

00:01:40.650 --> 00:01:44.290
at f of x is equal to 9.

00:01:44.290 --> 00:01:47.960
So when does our absolute
value part of this expression

00:01:47.960 --> 00:01:49.010
equal 0?

00:01:49.010 --> 00:01:50.340
Well, let's think about it.

00:01:50.340 --> 00:01:55.630
When does the absolute
value of x minus 1

00:01:55.630 --> 00:02:01.350
is going to be equal to 0
if x minus 1 is equal to 0.

00:02:01.350 --> 00:02:06.450
If x minus 1 is equal to 0--
or if you add 1 to both sides--

00:02:06.450 --> 00:02:08.240
you get x is equal to 1.

00:02:08.240 --> 00:02:11.660
So when x equals 1, we
hit our maximum point.

00:02:11.660 --> 00:02:13.580
This thing becomes
0, which makes

00:02:13.580 --> 00:02:15.470
this whole thing equal to 0.

00:02:15.470 --> 00:02:20.328
And so we get the
point 1 comma 9.

00:02:22.970 --> 00:02:24.430
So let's graph that point.

00:02:24.430 --> 00:02:27.440
And actually, let me
do it in a color that's

00:02:27.440 --> 00:02:29.870
easier to see on a
white background.

00:02:29.870 --> 00:02:33.760
1 comma 9 is the maximum
point right over here.

00:02:33.760 --> 00:02:38.150
And you could also
view this as the vertex

00:02:38.150 --> 00:02:40.460
of this absolute value function.

00:02:40.460 --> 00:02:45.870
So 1 comma 9 is that
point right over there.

00:02:45.870 --> 00:02:48.120
And so we can already guess
the general shape of this.

00:02:48.120 --> 00:02:50.350
An absolute value
function either

00:02:50.350 --> 00:02:55.890
looks something like that, or
it looks something like that.

00:02:55.890 --> 00:02:59.226
It would look like this if the
coefficient right over here--

00:02:59.226 --> 00:03:01.350
or i guess maybe I shouldn't
call it a coefficient.

00:03:01.350 --> 00:03:04.850
If the thing multiplying the
absolute value was positive,

00:03:04.850 --> 00:03:06.600
then would it be upward
opening like that.

00:03:06.600 --> 00:03:10.520
But since it is negative, it is
going to be downward opening.

00:03:10.520 --> 00:03:14.400
We hit a maximum point
at the vertex in the case

00:03:14.400 --> 00:03:17.790
we're just talking about for
the reasons we just discussed.

00:03:17.790 --> 00:03:20.322
But then what happens there?

00:03:20.322 --> 00:03:22.030
What happens to the
right of that vertex,

00:03:22.030 --> 00:03:24.990
and what happens to the
left of this vertex?

00:03:24.990 --> 00:03:26.400
We don't have to do it this way.

00:03:26.400 --> 00:03:29.470
Once again, you could just
try out points if you like,

00:03:29.470 --> 00:03:32.090
but I like to really analyze
what's going on here.

00:03:32.090 --> 00:03:36.980
So let's break up this function
definition into two pieces.

00:03:36.980 --> 00:03:40.092
So let's say that f
of x is equal to--

00:03:40.092 --> 00:03:42.710
and we're going to break
it up into two pieces, one

00:03:42.710 --> 00:03:46.240
piece to the right
of the vertex and one

00:03:46.240 --> 00:03:48.590
piece to the left of the vertex.

00:03:48.590 --> 00:03:50.290
So to the right of
the vertex, that's

00:03:50.290 --> 00:03:56.380
going to be 1 is less than or
equal to x, which is less than

00:03:56.380 --> 00:03:57.810
or equal to 5.

00:03:57.810 --> 00:03:59.762
And to the left
of the vertex-- so

00:03:59.762 --> 00:04:01.220
this is to the
right of the vertex.

00:04:01.220 --> 00:04:04.163
To the left of the vertex,
that's going to be-- let

00:04:04.163 --> 00:04:05.750
me do this in a different color.

00:04:05.750 --> 00:04:09.750
That's going to be negative
4 is less than or equal to x,

00:04:09.750 --> 00:04:12.300
and I'll just say less than 1.

00:04:12.300 --> 00:04:16.250
So just to be clear,
our whole domain

00:04:16.250 --> 00:04:18.370
was between negative 4 and 5.

00:04:18.370 --> 00:04:23.227
Between negative 4 and
5 was our entire domain.

00:04:23.227 --> 00:04:25.310
Now, I'm breaking it up
to the right of the vertex

00:04:25.310 --> 00:04:27.060
and the left of the vertex.

00:04:27.060 --> 00:04:31.750
So this right over here is
to the right of the vertex,

00:04:31.750 --> 00:04:36.810
and this right over here is
to the left of the vertex.

00:04:36.810 --> 00:04:38.580
So to the right of
the vertex, for x

00:04:38.580 --> 00:04:42.930
being greater than or equal to
1, this absolute value-- or one

00:04:42.930 --> 00:04:45.600
way to think about it, when x
is greater than or equal to 1,

00:04:45.600 --> 00:04:47.881
x minus 1 is going
to be positive,

00:04:47.881 --> 00:04:49.630
and the absolute value
of a positive thing

00:04:49.630 --> 00:04:52.000
is just going to
be that same value.

00:04:52.000 --> 00:04:54.310
So for x is greater
than or equal to 1,

00:04:54.310 --> 00:05:01.310
our function is going to be
negative 3 times x minus 1 plus

00:05:01.310 --> 00:05:02.280
9.

00:05:02.280 --> 00:05:04.860
Or if you wanted to
distribute the negative 3,

00:05:04.860 --> 00:05:06.270
this would be the same thing.

00:05:06.270 --> 00:05:08.186
Actually, let me write
it over here so that we

00:05:08.186 --> 00:05:10.650
can write the
simplified version.

00:05:10.650 --> 00:05:13.320
So it would be-- let me
give myself some space.

00:05:13.320 --> 00:05:16.480
So for x is greater
than or equal to 1,

00:05:16.480 --> 00:05:18.480
we don't have to think
about the absolute value.

00:05:18.480 --> 00:05:22.400
So it would be negative
3 times x minus 1

00:05:22.400 --> 00:05:30.470
plus 9, which is negative
3x plus 3 plus 9,

00:05:30.470 --> 00:05:33.920
or negative 3x plus 12.

00:05:33.920 --> 00:05:36.900
So it's going to be
negative 3x plus 12

00:05:36.900 --> 00:05:39.010
when x is greater
than or equal to 1.

00:05:39.010 --> 00:05:41.100
And the really interesting
thing, the slope

00:05:41.100 --> 00:05:42.290
here is negative.

00:05:42.290 --> 00:05:44.517
We have a slope of negative 3.

00:05:44.517 --> 00:05:47.100
So starting from this point, if
we have a slope of negative 3,

00:05:47.100 --> 00:05:51.060
that means every time we move
1, increase in the x-direction,

00:05:51.060 --> 00:05:53.970
we're going to decrease
by 3 in the y-direction,

00:05:53.970 --> 00:05:58.300
increase by x,
decrease by y by 3.

00:05:58.300 --> 00:06:00.200
So it's a slope of negative 3.

00:06:00.200 --> 00:06:03.410
So it will look like that.

00:06:03.410 --> 00:06:05.480
And let me actually
do it in purple

00:06:05.480 --> 00:06:06.970
so that you can actually see it.

00:06:06.970 --> 00:06:08.509
That's not purple.

00:06:08.509 --> 00:06:09.050
There you go.

00:06:09.050 --> 00:06:10.340
That's purple.

00:06:10.340 --> 00:06:12.090
All right.

00:06:12.090 --> 00:06:15.815
So it would look like that.

00:06:15.815 --> 00:06:17.440
And actually let me
just make it clear.

00:06:17.440 --> 00:06:20.520
This is what I'm talking
about right over here.

00:06:20.520 --> 00:06:23.580
That piece of the function
is going to look like that.

00:06:23.580 --> 00:06:26.550
Now, what about when
x is less than 1?

00:06:26.550 --> 00:06:30.240
Well, when x is less
than 1, x minus 1

00:06:30.240 --> 00:06:33.770
is going to be negative,
and so the absolute value

00:06:33.770 --> 00:06:37.190
is going to be the negative
of that actual function.

00:06:37.190 --> 00:06:41.770
So when x is less than
1-- let me write this.

00:06:45.420 --> 00:06:47.340
The absolute value
of x minus 1 is

00:06:47.340 --> 00:06:51.300
equal to the negative of x
minus 1, which is the same thing

00:06:51.300 --> 00:06:52.900
as 1 minus x.

00:06:52.900 --> 00:07:01.500
So our function is going to
be negative 3 times 1 minus x

00:07:01.500 --> 00:07:05.730
plus 9, which is the
same thing as-- let's

00:07:05.730 --> 00:07:08.370
see, if you distribute the
negative 3, it's negative 3

00:07:08.370 --> 00:07:15.310
plus 3x plus 9, which
is equal to 3x plus 6.

00:07:15.310 --> 00:07:20.010
So the important
thing to realize here

00:07:20.010 --> 00:07:24.390
is that our slope is positive 3.

00:07:24.390 --> 00:07:27.530
And you could even say that we
have a y-intercept here at 6,

00:07:27.530 --> 00:07:29.534
so it would be this
point right over here,

00:07:29.534 --> 00:07:30.950
and the line would
look like this.

00:07:30.950 --> 00:07:34.270
When your slope is positive
3, you increase x by 1,

00:07:34.270 --> 00:07:35.800
you increase y by 3.

00:07:40.130 --> 00:07:43.390
Or another way of thinking about
it, if you decrease x by 1,

00:07:43.390 --> 00:07:45.690
then you decrease y by 3.

00:07:52.440 --> 00:07:56.570
So it would look like that.

00:07:56.570 --> 00:07:57.680
And we are done.

00:07:57.680 --> 00:08:02.010
We have graphed this
absolute value equation.

00:08:02.010 --> 00:08:03.510
Now, the whole
reason why I did this

00:08:03.510 --> 00:08:05.660
is just so we learn it
a little bit deeper.

00:08:05.660 --> 00:08:07.990
But there would have been
a faster way of doing this.

00:08:07.990 --> 00:08:09.448
You could have just
said, hey, this

00:08:09.448 --> 00:08:11.710
is going to hit a maximum
point when x is equal to 1

00:08:11.710 --> 00:08:14.430
because this non-positive
thing is going to hit 0.

00:08:14.430 --> 00:08:16.805
So you could have said
1 comma 9 is the vertex.

00:08:16.805 --> 00:08:19.180
And then you could have just
tried out the two endpoints,

00:08:19.180 --> 00:08:21.777
knowing that between the
endpoints and the vertex,

00:08:21.777 --> 00:08:23.610
you're essentially just
dealing with a line.

00:08:23.610 --> 00:08:26.785
So you could have just evaluated
f of negative 4 and f of 5.

00:08:26.785 --> 00:08:29.410
You could have figured out these
endpoints and drawn the lines.

00:08:29.410 --> 00:08:31.520
But now we understand it
deeper, and that's always

00:08:31.520 --> 00:08:33.620
a little bit more exciting.