i and Imaginary numbers

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on i and imaginary numbers.

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So let me just start
out with a definition.

00:00:08.200 --> 00:00:14.340
i is equal to the square
root of negative 1.

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Or to view it another way,
you could say that i squared

00:00:18.790 --> 00:00:22.030
is equal to negative 1.

00:00:22.030 --> 00:00:23.260
Now why is this special?

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Well, we knew or we've learned
that any number when you

00:00:27.440 --> 00:00:29.680
square it is equal to a
positive number, right?

00:00:29.680 --> 00:00:32.820
If I had negative 1
times negative 1, that

00:00:32.820 --> 00:00:34.460
equals positive 1.

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You don't have to write the
positive every time but that

00:00:36.150 --> 00:00:38.780
equal positive 1 and so
does 1 times 1, that

00:00:38.780 --> 00:00:40.570
equals positive 1.

00:00:40.570 --> 00:00:46.260
So if we think about the square
root of a number, so far all

00:00:46.260 --> 00:00:48.080
we've learned is taking the
square roots of positive

00:00:48.080 --> 00:00:52.880
numbers, and that makes sense
to us because the notion of a

00:00:52.880 --> 00:00:54.790
square root of a negative
number didn't really

00:00:54.790 --> 00:00:56.860
exist until now.

00:00:56.860 --> 00:01:00.960
So what we've done is we've set
this definition that the number

00:01:00.960 --> 00:01:04.760
i, and i isn't a variable,
it's an actual number.

00:01:04.760 --> 00:01:09.440
It's value is equal to the
square root of negative 1.

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Now I won't go into all of the
philosophical musings about

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whether i, as a number i, or
any imaginary numbers

00:01:18.570 --> 00:01:20.780
actually exist.

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Maybe I'll make another
presentation on that.

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But they exist enough to
be very useful to many

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engineers and physicists.

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So I'll leave you with that,
and I'll also just hint at --

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well, I won't go into the
whole e to the i pi

00:01:36.400 --> 00:01:37.170
equals negative 1.

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But that blows my mind,
but I won't go into that.

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And when you think about
whether i really exists, you

00:01:44.030 --> 00:01:47.900
should also think about whether
anything really exists.

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So, I've diversed for too long,
so let's get back to what I was

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saying before. i is equal to
the square root of

00:01:53.750 --> 00:01:57.030
negative 1, and i squared
equals negative 1.

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So let's think about the
implications of this.

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If I were to say -- well, i to
the first power, just like

00:02:04.730 --> 00:02:07.580
anything else, is equal
to itself, right?

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i squared, I've already said
using this definition, i

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squared is equal to negative 1.

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i to the third power, well
that would just equal i

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squared times i, right?

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And i squared is negative 1, so
it would be negative 1 times i.

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And that just
equals negative i.

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And i to the fourth would
equal i to the third times i.

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I'm just using my
exponent rules here.

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Well i to the third is negative
i times i, right? i to the

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third is negative i, and
then we just kept that i.

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Well what's negative i times i?

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Well that's the same thing
as negative 1 times

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i times i, right?

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And what's i times i?

00:03:02.690 --> 00:03:04.020
Well, the definition.

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i times i, i squared, is
equal to negative 1.

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So that equals negative 1 times
negative 1, which equals 1.

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Interesting.

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Let me clean this
up a little bit.

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Actually let me start with
i to the zero power.

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i to the zero power, well we
know anything to the zero

00:03:22.750 --> 00:03:23.690
power is equal to 1.

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So we'll keep that --
that still equals 1.

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i to the first power
is equal to i.

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i squared by definition
is equal to negative 1.

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i to the third power, I just
showed you, is equal to

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negative i, and that makes
sense because that's

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just i squared times i.

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And i to the fourth
power equaled 1 again.

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If I did i to the fifth power,
well that's just equal to i to

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the fourth power
times i, right?

00:04:01.110 --> 00:04:04.320
I'll write that down, i
to the fourth times i.

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i to the fourth is
equal to 1, right?

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This is equal to 1, so 1
times i is equal to i.

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Do you see a pattern here?

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i to the zero power equals 1.

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i to the negative
1 power equals i.

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i squared is equal
to negative 1.

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i to the third power
equals negative i.

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i the fourth equals 1 again.

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So i to the fourth equals i to
zero, and i to the one power

00:04:38.740 --> 00:04:41.280
is equal to i to the fifth.

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I think you'll find out that i
to the zero, and you could try

00:04:45.640 --> 00:04:49.410
this out if you don't believe
me, i to the zero is equal to i

00:04:49.410 --> 00:04:55.645
to the fourth, which equals i
to the eighth, which equals i

00:04:55.645 --> 00:04:57.250
to the twelfth, I think
you see the pattern.

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Any multiple of 4 equals 1.

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And i to the first equals i to
the fifth, equals i to the

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ninth equals i to the
thirteenth, equals i.

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So that's i to any power
that is a multiple

00:05:20.915 --> 00:05:23.110
of 4 plus 1, right?

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Because 5 is equal to 4 plus
1, nine is equal to 8 plus 1.

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And we could do a similar
pattern. i squared is equal to

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i to the sixth, equals i to
the tenth, and so on, and

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that equals negative 1.

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And finally i to the third is
equal to i to the seventh,

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which equals i to the eleventh,
and so on, equals negative i.

00:05:50.990 --> 00:05:52.060
So why is this useful?

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We see a pattern.

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It's a cycle of four.

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In this pattern, if we
look at this, we can use

00:05:57.940 --> 00:06:00.420
this to determine what
i to any power is.

00:06:00.420 --> 00:06:06.460
So if I were to ask you what i
to the hundredth power is,

00:06:06.460 --> 00:06:08.860
well, you could just work it
out, you could say, well that's

00:06:08.860 --> 00:06:10.810
just equal to i times i to the
ninety-ninth and

00:06:10.810 --> 00:06:11.500
so far so down.

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But if we use the cycle,
we see that hundred is

00:06:14.390 --> 00:06:17.490
a multiple of 4, right?

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4 times 25 is 100.

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So i to the hundredth will
fall into this category,

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this first one.

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It's a multiple of 4.

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So we know that i to the
hundredth power is equal to 1.

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Similarly, if I said i to the
hundred and one power, that's

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going to equal i, right?

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Because that equals 100 plus 1.

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So it puts you into
this category.

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Equal to a multiple of 4, 100
is a multiple of 4, and 101

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is a multiple of 4 plus 1.

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i to the hundred and second
power, similarly would

00:06:51.330 --> 00:06:54.120
equal negative 1.

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i to the hundred and third
power would equal negative i.

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I hope you understand what I'm
doing here, and all I did is I

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defined i as the square root of
negative 1, and then I

00:07:05.670 --> 00:07:08.560
kept multiplying i to
figure out a pattern.

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I said i to the zero equals 1,
i to the first equals i, i

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squared equals negative 1, i to
the third equals negative i,

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and i to the fourth
equals 1 again.

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And the pattern
repeated itself.

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And then I used that pattern to
be able to figure out i to any

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power, even it's a
very high number.

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So a very simple way to think
about it is if I had i to the

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three hundred and twenty three.

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What I do is I say if I
were to divide 4 into 323,

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what's the remainder?

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Well, I know 4 goes
into 320, right?

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4 times 80 is 320.

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So I know that when I divide 4
into 323, so 4 goes into 323 80

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times with a remainder
of 3, right?

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And the remainder is
what we care about.

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And this number is actually
called a modulus.

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Maybe I'll do another module on
modulus -- it's very important

00:08:05.440 --> 00:08:08.040
actually in computer
programming.

00:08:08.040 --> 00:08:10.580
But since we know that when you
divide this exponent by 4 the

00:08:10.580 --> 00:08:13.630
remainder is 3, we can say that
this is the same thing as i to

00:08:13.630 --> 00:08:17.490
the third, which we've
learned is negative i.

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Similarly, if I said i to
the five hundred and second

00:08:22.410 --> 00:08:27.220
power, well I know 500 is
divisible by 4, right?

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4 times 125 is 500.

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So the remainder is 2 if
I were divide it by 4.

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So I could say that this is
the same thing as i squared.

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And i squared we learned by
definition is negative 1.

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If I were to ask you i
to the thirty-seven?

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We know 36 was divisible by
4, so the remainder is 1.

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So it would be i to the
1, which equals i.

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Hopefully that gives you an
indication of what i is.

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It might have been confusing
the first time because we're

00:08:59.970 --> 00:09:03.940
dealing with a number that's
"imaginary," and I'm teaching

00:09:03.940 --> 00:09:05.960
the cycle property of it.

00:09:05.960 --> 00:09:08.710
What you might want to do is
review the video again, but

00:09:08.710 --> 00:09:10.990
then after that you could just
try the module on i, which

00:09:10.990 --> 00:09:12.930
essentially just keeps working
you through this

00:09:12.930 --> 00:09:14.070
type of problem.

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I hope you have fun.

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Bye.


Simplifying radicals

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on simplifying radicals.

00:00:04.270 --> 00:00:05.890
So let's get started with
a little terminology

00:00:05.890 --> 00:00:06.490
out of the way.

00:00:06.490 --> 00:00:08.310
You're probably just wondering
what a radical is and

00:00:08.310 --> 00:00:09.880
I'll just let you know.

00:00:13.190 --> 00:00:15.280
A radical is just that.

00:00:15.280 --> 00:00:17.470
Or you're probably more
familiar calling that

00:00:17.470 --> 00:00:18.830
the square root symbol.

00:00:18.830 --> 00:00:21.680
So with the terminology out of
the way, let's actually talk

00:00:21.680 --> 00:00:23.800
about what it means to
simplify a radical.

00:00:23.800 --> 00:00:25.020
And some people would argue
that what we're going to

00:00:25.020 --> 00:00:26.890
actually be doing is actually
making it more complicated.

00:00:26.890 --> 00:00:28.140
But let's see.

00:00:32.390 --> 00:00:36.900
So if I were to give you
the square root of 36.

00:00:36.900 --> 00:00:37.610
Hey, that's easy.

00:00:37.610 --> 00:00:41.790
That's just equal to 6 times
6 or you'd say the square

00:00:41.790 --> 00:00:43.850
root of 36 is just 6.

00:00:43.850 --> 00:00:50.390
Now, what if I asked you what
the square root of 72 is?

00:00:50.390 --> 00:00:54.590
Well we know that
72 is 36 times 2.

00:00:54.590 --> 00:00:55.680
Let's write that.

00:00:55.680 --> 00:01:00.850
Square root of 72 is the
same thing as the square

00:01:00.850 --> 00:01:03.310
root of 36 times 2.

00:01:07.920 --> 00:01:10.690
And the square root, if you
remember from level 3

00:01:10.690 --> 00:01:13.520
exponents, square root is
the same thing as something

00:01:13.520 --> 00:01:14.920
to the 1/2 power.

00:01:14.920 --> 00:01:15.860
So let's write it that way.

00:01:15.860 --> 00:01:18.710
And I'm just writing it this
way just to show you how this

00:01:18.710 --> 00:01:20.580
radical simplification works
and that it's really

00:01:20.580 --> 00:01:22.970
not a new concept.

00:01:22.970 --> 00:01:29.750
So this is the same thing as
36 times 2 to the 1/2 power.

00:01:29.750 --> 00:01:33.210
Because it's just a square root
is the same thing as 1/2 power.

00:01:33.210 --> 00:01:35.860
And we learned from the
exponent rules that when you

00:01:35.860 --> 00:01:38.890
multiply two numbers and then
you raise that to the 1/2

00:01:38.890 --> 00:01:43.010
power, that that's the same
thing as raising each of the

00:01:43.010 --> 00:01:47.500
numbers to the 1/2 power
and then multiplying.

00:01:50.420 --> 00:01:53.330
Well that right there, that's
the same thing as saying the

00:01:53.330 --> 00:01:58.480
square root is 36 times
the square root of 2.

00:01:58.480 --> 00:02:00.780
And we already figured out what
the square root of 36 is.

00:02:00.780 --> 00:02:01.810
It's 6.

00:02:01.810 --> 00:02:07.430
So that just equals 6 times
the square root of 2.

00:02:07.430 --> 00:02:10.060
And you're probably wondering
why I went through this step

00:02:10.060 --> 00:02:12.340
of changing the radical, the
square root symbol,

00:02:12.340 --> 00:02:13.530
into the 1/2 power.

00:02:13.530 --> 00:02:16.160
And I did that just to show you
that this is just an extension

00:02:16.160 --> 00:02:17.030
of the exponent rules.

00:02:17.030 --> 00:02:22.450
It isn't really a new concept,
although, I guess sometimes

00:02:22.450 --> 00:02:24.690
it's not so obvious that
they are the same concepts.

00:02:24.690 --> 00:02:26.480
I just wanted to
point that out.

00:02:26.480 --> 00:02:28.470
So let's do another problem.

00:02:28.470 --> 00:02:30.820
I think as we do more and
more problems, these will

00:02:30.820 --> 00:02:33.260
become more obvious.

00:02:33.260 --> 00:02:37.820
The square root of 50.

00:02:37.820 --> 00:02:40.920
Well, the square root of
50 -- 50 is the same

00:02:40.920 --> 00:02:47.150
thing as 25 times 2.

00:02:47.150 --> 00:02:49.790
And we know, based on what we
just did and this is really

00:02:49.790 --> 00:02:53.670
just an exponent rule, square
root of 25 times 2 is the same

00:02:53.670 --> 00:03:01.070
thing as the square root of 25
times the square root of 2.

00:03:01.070 --> 00:03:02.580
Well we know what the
square root of 25 is.

00:03:02.580 --> 00:03:03.170
That's 5.

00:03:03.170 --> 00:03:09.700
So that just equals 5 times
the square root of 2.

00:03:09.700 --> 00:03:14.610
Now, you might be saying, "Hey,
Sal, you make it look easy, but

00:03:14.610 --> 00:03:19.210
how did you know to split 50
into 25 and 2?" Why didn't I

00:03:19.210 --> 00:03:24.010
say that 50 is equal to the
square root of 5 and 10 or that

00:03:24.010 --> 00:03:28.800
50 is equal to the square root
-- actually, I think 1 and 50?

00:03:28.800 --> 00:03:30.160
I don't know what
other factors is 50.

00:03:30.160 --> 00:03:32.570
Well, anyway, I won't go
into that right now.

00:03:32.570 --> 00:03:35.590
The reason why I picked 25 and
2 is because I wanted a factor

00:03:35.590 --> 00:03:39.640
of 50 -- I actually wanted the
largest factor of 50 that

00:03:39.640 --> 00:03:40.880
is a perfect square.

00:03:40.880 --> 00:03:42.860
And that's 25.

00:03:42.860 --> 00:03:44.970
If I had done 5 and 10, there's
really nothing I could have

00:03:44.970 --> 00:03:47.900
done with it because neither 5
nor 10 are perfect squares and

00:03:47.900 --> 00:03:50.610
same thing's with 1 and 50.

00:03:50.610 --> 00:03:52.270
So the way you should think
about it, think about the

00:03:52.270 --> 00:03:55.960
factors of the original number
and figure out if any of those

00:03:55.960 --> 00:03:57.890
factors are perfect squares.

00:03:57.890 --> 00:03:59.370
And there's no real
mechanical way.

00:03:59.370 --> 00:04:02.280
You really just have to learn
to recognize perfect squares.

00:04:02.280 --> 00:04:03.940
And you'll get familiar
with them, of course.

00:04:03.940 --> 00:04:17.150
They're 1, 4, 9, 25, 16,
25, 36, 49, 64, et cetera.

00:04:17.150 --> 00:04:19.980
And maybe by doing this module,
you'll actually learn to

00:04:19.980 --> 00:04:21.290
recognize them more readily.

00:04:21.290 --> 00:04:25.930
But if any of these numbers are
a factor of the number under

00:04:25.930 --> 00:04:27.360
the radical sign, then you'll
probably want to

00:04:27.360 --> 00:04:28.020
factor them out.

00:04:28.020 --> 00:04:30.130
And then you can take them
out of the radical sign like

00:04:30.130 --> 00:04:32.620
we did up in this problem.

00:04:32.620 --> 00:04:33.730
Let's do a couple more.

00:04:37.640 --> 00:04:43.470
What is 7 times the
square root of 27?

00:04:43.470 --> 00:04:45.420
And when I write the 7 right
next to it, that just means

00:04:45.420 --> 00:04:47.510
times the square root of 27.

00:04:47.510 --> 00:04:50.665
Well, let's think about what
other factors of 27 and

00:04:50.665 --> 00:04:52.050
whether any of them
are a perfect square.

00:04:52.050 --> 00:04:56.710
Well, 3 is a factor of 27, but
that's not a perfect square.

00:04:56.710 --> 00:04:58.260
9 is.

00:04:58.260 --> 00:05:02.800
So, we could say 7 -- that's
equal to 7 times the

00:05:02.800 --> 00:05:08.490
square root of 9 times 3.

00:05:08.490 --> 00:05:11.860
And now, based on the rules we
just learned, that's the same

00:05:11.860 --> 00:05:18.580
thing as 7 times the square
root of 9 times the

00:05:18.580 --> 00:05:21.080
square root of 3.

00:05:21.080 --> 00:05:25.230
Well that just equals 7 times 3
because the square root of 9 is

00:05:25.230 --> 00:05:29.270
3 times the square root of 3.

00:05:29.270 --> 00:05:34.670
That equals 21 times
the square root of 3.

00:05:34.670 --> 00:05:35.830
Done.

00:05:35.830 --> 00:05:37.710
Let's do another one.

00:05:37.710 --> 00:05:46.000
What is 9 times the
square root of 18?

00:05:46.000 --> 00:05:48.210
Well once again, what
are the factors of 18?

00:05:48.210 --> 00:05:50.010
Well do we have 6 and 3?

00:05:50.010 --> 00:05:52.280
1 and 18?

00:05:52.280 --> 00:05:54.550
None of the numbers I mentioned
so far are perfect squares.

00:05:54.550 --> 00:05:56.540
But we also have 2 and 9.

00:05:56.540 --> 00:05:59.010
And 9 is a perfect square.

00:05:59.010 --> 00:05:59.770
Let's write that.

00:05:59.770 --> 00:06:07.020
That's equal to 9 times the
square root of 2 times 9.

00:06:07.020 --> 00:06:11.560
Which is equal to 9 times the
square root of 2 -- that's a 2

00:06:11.560 --> 00:06:15.580
-- times the square root of 9.

00:06:15.580 --> 00:06:20.110
Which equals 9 times the square
root of 2 times 3, right?

00:06:20.110 --> 00:06:24.520
That's the square root of
9 which equals 27 times

00:06:24.520 --> 00:06:27.250
the square root of 2.

00:06:27.250 --> 00:06:28.130
There we go.

00:06:28.130 --> 00:06:30.160
Hopefully, you're starting to
get the hang of these problems.

00:06:30.160 --> 00:06:33.070
Let's do another one.

00:06:33.070 --> 00:06:39.830
What is 4 times the
square root of 25?

00:06:39.830 --> 00:06:43.000
Well 25 itself is
a perfect square.

00:06:43.000 --> 00:06:45.620
This problem is so easy, it's
a bit of a trick problem.

00:06:45.620 --> 00:06:47.130
25 itself is a perfect square.

00:06:47.130 --> 00:06:50.150
The square root is 5, so this
is just equal to 4 times

00:06:50.150 --> 00:06:52.910
5, which is equal to 20.

00:06:52.910 --> 00:06:57.020
Square root of 25 is 5.

00:06:57.020 --> 00:06:58.220
Let's do another one.

00:06:58.220 --> 00:07:01.550
What's 3 times the
square root of 29?

00:07:04.330 --> 00:07:06.190
Well 29 only has two factors.

00:07:06.190 --> 00:07:06.870
It's a prime number.

00:07:06.870 --> 00:07:09.450
It only has the
factors 1 and 29.

00:07:09.450 --> 00:07:11.750
And neither of those numbers
are perfect squares.

00:07:11.750 --> 00:07:14.220
So we really can't simplify
this one anymore.

00:07:14.220 --> 00:07:19.340
So this is already in
completely simplified form.

00:07:19.340 --> 00:07:20.480
Let's do a couple more.

00:07:23.780 --> 00:07:32.140
What about 7 times the
square root of 320?

00:07:32.140 --> 00:07:35.700
Let's think about 320.

00:07:35.700 --> 00:07:38.320
Well we could actually do
it in steps when we have

00:07:38.320 --> 00:07:39.810
larger numbers like this.

00:07:39.810 --> 00:07:43.290
I can look at it and say, well
it does look like 4 -- actually

00:07:43.290 --> 00:07:47.200
it looks like 16 would go into
this because 16 goes into 32.

00:07:47.200 --> 00:07:48.380
So let's try that.

00:07:48.380 --> 00:07:55.280
So that equals 7 times the
square root of 16 times 20.

00:07:58.540 --> 00:08:04.310
That just equals 7 times the
square root of 16 times

00:08:04.310 --> 00:08:06.960
the square root of 20.

00:08:06.960 --> 00:08:08.590
7 times the square root of 16.

00:08:08.590 --> 00:08:10.380
The square root of 16 is 4.

00:08:10.380 --> 00:08:11.630
So 7 times 4 is 28.

00:08:11.630 --> 00:08:17.110
So that's 28 times the
square root of 20.

00:08:17.110 --> 00:08:19.100
Now are we done?

00:08:19.100 --> 00:08:21.800
Well actually, I think I can
factor 20 even more because

00:08:21.800 --> 00:08:24.680
20 is equal to 4 times 5.

00:08:24.680 --> 00:08:28.650
So I can say this is equal
to 28 times the square

00:08:28.650 --> 00:08:33.570
root of 4 times 5.

00:08:33.570 --> 00:08:38.270
The square root of 4 is 2 so
that could just take the 2 out

00:08:38.270 --> 00:08:43.170
and that becomes 56 times
the square root of 5.

00:08:43.170 --> 00:08:44.450
I hope that made sense to you.

00:08:44.450 --> 00:08:45.980
And this is actually a
pretty important technique

00:08:45.980 --> 00:08:46.890
I just did here.

00:08:46.890 --> 00:08:49.060
Immediately when I look at 320.

00:08:49.060 --> 00:08:52.160
I don't know what the largest
number is that goes into 320.

00:08:52.160 --> 00:08:54.150
It actually turns
out that it's 64.

00:08:54.150 --> 00:08:56.250
But just looking at the
number, I said, well I

00:08:56.250 --> 00:08:57.610
know that 4 goes into it.

00:08:57.610 --> 00:09:00.520
So I could have just pulled out
4 and then said, "Oh, that's

00:09:00.520 --> 00:09:02.090
equal to 4 times 80." And then
I would have had

00:09:02.090 --> 00:09:03.210
to work with 80.

00:09:03.210 --> 00:09:05.760
In this case, I saw 32 and I
was like, it looks like 16 goes

00:09:05.760 --> 00:09:09.260
into it and I factored out 16
first and when I took out the

00:09:09.260 --> 00:09:11.890
square root of 16, I multiplied
the outside by 4 and

00:09:11.890 --> 00:09:13.160
that's how I got the 28.

00:09:13.160 --> 00:09:15.670
But then I reduced the number
on the inside said, "Oh, well

00:09:15.670 --> 00:09:17.430
that still is divisible
by a perfect square.

00:09:17.430 --> 00:09:20.270
It's still divisible by 4." And
then I kept doing it until I

00:09:20.270 --> 00:09:25.250
was left with essentially, a
prime number or a number that

00:09:25.250 --> 00:09:27.710
couldn't be reduced anymore
under the radical.

00:09:27.710 --> 00:09:29.950
And it actually doesn't
have to be prime.

00:09:29.950 --> 00:09:33.140
So hopefully, that gives you
a good sense of how to do

00:09:33.140 --> 00:09:34.270
radical simplification.

00:09:34.270 --> 00:09:37.020
It's really just an extension
of the exponent rules that

00:09:37.020 --> 00:09:40.180
you've already learned and
hopefully as you do the module,

00:09:40.180 --> 00:09:41.890
you'll get good at it.

00:09:41.890 --> 00:09:43.420
Have fun.


Exponent rules part 2

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:04.640
Welcome to Part 2 on the
presentation on Level

00:00:04.640 --> 00:00:06.110
1 exponent rules.

00:00:06.110 --> 00:00:07.960
So let's start off by
reviewing the rules

00:00:07.960 --> 00:00:09.340
we've learned already.

00:00:09.340 --> 00:00:21.050
If I had 2 to the tenth times 2
to the fifth, we learned that

00:00:21.050 --> 00:00:25.130
since we're multiplying
exponents with the same base,

00:00:25.130 --> 00:00:28.410
we can add the exponent, so
this equals 2 to the fifteenth.

00:00:31.320 --> 00:00:37.510
We also learned that if it was
2 to the tenth over 2 to the

00:00:37.510 --> 00:00:41.690
fifth, we would actually
subtract the exponents.

00:00:41.690 --> 00:00:46.150
So this would be 2 to
the 10 minus 5, which

00:00:46.150 --> 00:00:48.680
equals 2 to the fifth.

00:00:48.680 --> 00:00:50.460
At the end of the last
presentation, and I probably

00:00:50.460 --> 00:00:52.970
shouldn't have introduced it
so fast, I introduced

00:00:52.970 --> 00:00:54.070
a new concept.

00:00:54.070 --> 00:01:02.740
What happens if I have 2 to
the tenth to the fifth power?

00:01:02.740 --> 00:01:04.380
Well, let's think about
what that means.

00:01:04.380 --> 00:01:07.340
When I raise something to the
fifth power, that's just like

00:01:07.340 --> 00:01:14.240
saying 2 to the tenth times 2
to the tenth times 2 to the

00:01:14.240 --> 00:01:19.955
tenth times 2 to the tenth
times 2 to the tenth, right?

00:01:19.955 --> 00:01:22.465
All I did is I took 2 to
the tenth and I multiplied

00:01:22.465 --> 00:01:24.800
it by itself five times.

00:01:24.800 --> 00:01:26.370
That's the fifth power.

00:01:26.370 --> 00:01:30.550
Well, we know from this rule up
here that we can add these

00:01:30.550 --> 00:01:33.280
exponents because they're
all the same base.

00:01:33.280 --> 00:01:36.360
So if we add 10 plus 10
plus 10 plus 10 plus

00:01:36.360 --> 00:01:39.050
10, what do we get?

00:01:39.050 --> 00:01:42.480
Right, we get 2 to
the fiftieth power.

00:01:42.480 --> 00:01:44.090
So essentially, what
did we do here?

00:01:44.090 --> 00:01:50.420
All we did is we multiplied
10 times 5 to get 50.

00:01:50.420 --> 00:01:53.480
So that's our third exponent
rule, that when I raise an

00:01:53.480 --> 00:01:56.620
exponent to a power and then I
raise that whole expression to

00:01:56.620 --> 00:02:00.940
another power, I can multiply
those two exponents.

00:02:00.940 --> 00:02:02.400
So let me give you
another example.

00:02:02.400 --> 00:02:12.340
If I said 3 to the 7, and all
of that to the negative 9, once

00:02:12.340 --> 00:02:17.460
again, all I do is I multiply
the 7 and the negative 9, and

00:02:17.460 --> 00:02:23.200
I get 3 to the minus 63.

00:02:23.200 --> 00:02:28.670
So, you see, it works just as
easily with negative numbers.

00:02:28.670 --> 00:02:33.465
So now, I'm going to teach you
one final exponent property.

00:02:36.970 --> 00:02:45.740
Let's say I have 2 times 9,
and I raise that whole thing

00:02:45.740 --> 00:02:48.710
to the hundredth power.

00:02:48.710 --> 00:02:53.540
It turns out of this is equal
to 2 to the hundredth power

00:02:53.540 --> 00:02:56.890
times 9 to the hundredth power.

00:02:56.890 --> 00:02:58.180
Now let's make sure
that that makes sense.

00:02:58.180 --> 00:02:59.880
Let's do it with a
smaller example.

00:02:59.880 --> 00:03:06.190
What if it was 4 times
5 to the third power?

00:03:06.190 --> 00:03:15.440
Well, that would just be equal
to 4 times 5 times 4 times 5

00:03:15.440 --> 00:03:21.890
times 4 times 5, right, which
is the same thing as 4 times 4

00:03:21.890 --> 00:03:26.410
times 4 times 5 times
5 times 5, right?

00:03:26.410 --> 00:03:28.820
I just switched the order in
which I'm multiplying, which

00:03:28.820 --> 00:03:30.730
you can do with multiplication.

00:03:30.730 --> 00:03:33.030
Well, 4 times 4 times 4,
well, that's just equal

00:03:33.030 --> 00:03:34.640
to 4 to the third.

00:03:34.640 --> 00:03:39.220
And 5 times 5 times 5 is
equal to 5 to the third.

00:03:39.220 --> 00:03:42.110
Hope that gives you a good
intuition of why this

00:03:42.110 --> 00:03:43.350
property here is true.

00:03:43.350 --> 00:03:46.170
And actually, when I had first
learned exponent rules, I would

00:03:46.170 --> 00:03:48.480
always forget the rules, and I
would always do this proof

00:03:48.480 --> 00:03:50.750
myself, or the other proofs.

00:03:50.750 --> 00:03:52.700
And a proof is just an
explanation of why the rule

00:03:52.700 --> 00:03:56.660
works, just to make sure
that I was doing it right.

00:03:56.660 --> 00:04:00.080
So given everything that we've
learned to now-- actually, let

00:04:00.080 --> 00:04:03.810
me review all of
the rules again.

00:04:03.810 --> 00:04:10.390
If I have 2 to the seventh
times 2 to the third,

00:04:10.390 --> 00:04:13.366
well, then I can add the
exponents, 2 to the tenth.

00:04:13.366 --> 00:04:20.080
If I have 2 the seventh over
2 the third, well, here I

00:04:20.080 --> 00:04:24.580
subtract the exponents, and
I get 2 to the fourth.

00:04:24.580 --> 00:04:30.910
If I have 2 to the seventh to
the third power, well, here

00:04:30.910 --> 00:04:32.310
I multiplied the exponents.

00:04:32.310 --> 00:04:35.180
That gives you 2 to the 21.

00:04:35.180 --> 00:04:42.690
And if I had 2 times 7 to the
third power, well, that equals

00:04:42.690 --> 00:04:47.650
2 to the third times
7 to the third.

00:04:47.650 --> 00:04:52.260
Now, let's use all of these
rules we've learned to actually

00:04:52.260 --> 00:04:55.500
try to do some, what I would
call, composite problems that

00:04:55.500 --> 00:04:58.630
involve you using multiple
rules at the same time.

00:04:58.630 --> 00:05:00.630
And a good composite problem
was that problem that I had

00:05:00.630 --> 00:05:03.145
introduced you to at the
end of that last seminar.

00:05:06.600 --> 00:05:20.330
Let's say I have 3 squared
times 9 to the eighth power,

00:05:20.330 --> 00:05:25.780
and all of that I'm going to
raise to the negative 2 power.

00:05:25.780 --> 00:05:27.250
So what can I do here?

00:05:27.250 --> 00:05:32.970
Well, 3 and 9 are two separate
bases, but 9 can actually

00:05:32.970 --> 00:05:35.760
be expressed as an
exponent of 3, right?

00:05:35.760 --> 00:05:37.540
9 is the same thing as
3 squared, so let's

00:05:37.540 --> 00:05:40.210
rewrite 9 like that.

00:05:40.210 --> 00:05:44.650
That's equivalent to 3 squared
times-- 9 is the same thing as

00:05:44.650 --> 00:05:50.520
3 squared to the eighth power,
and then all of that to the

00:05:50.520 --> 00:05:52.350
negative 2 power, right?

00:05:52.350 --> 00:05:54.330
All I did is I replaced 9
with 3 squared because

00:05:54.330 --> 00:05:57.610
we know 3 times 3 is 9.

00:05:57.610 --> 00:06:00.070
Well, now we can use the
multiplication rule on

00:06:00.070 --> 00:06:01.630
this to simplify it.

00:06:01.630 --> 00:06:09.500
So this is equal to 3 squared
times 3 to the 2 times 8,

00:06:09.500 --> 00:06:15.230
which is 16, and all of
that to the negative 2.

00:06:15.230 --> 00:06:16.610
Now, we can use the first rule.

00:06:16.610 --> 00:06:18.960
We have the same base, so we
can add the exponents, and

00:06:18.960 --> 00:06:23.280
we're multiplying them, so that
equals 3 to the eighteen power,

00:06:23.280 --> 00:06:28.210
right, 2 plus 16, and all
that to the negative 2.

00:06:28.210 --> 00:06:29.150
And now we're almost done.

00:06:29.150 --> 00:06:31.870
We can once again use this
multiplication rule, and we

00:06:31.870 --> 00:06:36.580
could say 3-- this is equal to
3 to the eighteenth times

00:06:36.580 --> 00:06:42.220
negative 2, so that's
3 to the minus 36.

00:06:42.220 --> 00:06:46.040
So this problem might have
seemed pretty daunting at

00:06:46.040 --> 00:06:49.380
first, but there aren't that
many rules, and all you have to

00:06:49.380 --> 00:06:51.580
do is keep seeing, oh, wow,
that little part of the

00:06:51.580 --> 00:06:52.970
problem, I can simplify it.

00:06:52.970 --> 00:06:55.520
Then you simplify it, and
you'll see that you can keep

00:06:55.520 --> 00:06:59.215
using rules until you get
to a much simpler answer.

00:06:59.215 --> 00:07:02.290
And actually the Level 1
problems don't even involve

00:07:02.290 --> 00:07:03.340
problems this difficult.

00:07:03.340 --> 00:07:06.380
This'll be more on the
exponent rules, Level 2.

00:07:06.380 --> 00:07:07.850
But I think at this
point you're ready

00:07:07.850 --> 00:07:10.125
to try the problems.

00:07:12.860 --> 00:07:15.550
I'm kind of divided whether I
want you to memorize the rules

00:07:15.550 --> 00:07:18.800
because I think it's better to
almost forget the rules and

00:07:18.800 --> 00:07:20.740
have to prove it to yourself
over and over again to

00:07:20.740 --> 00:07:22.010
the point that you
remember the rules.

00:07:22.010 --> 00:07:25.400
Because if you just memorize
the rules, later on in life

00:07:25.400 --> 00:07:27.905
when you haven't done it for a
couple of years, you might kind

00:07:27.905 --> 00:07:29.340
of forget the rules, and
then you won't know how

00:07:29.340 --> 00:07:30.590
to get back to the rules.

00:07:30.590 --> 00:07:31.880
But it's up to you.

00:07:31.880 --> 00:07:35.160
I just hope you do understand
why these rules work, and as

00:07:35.160 --> 00:07:36.616
long as you practice and you
pay attention to the signs,

00:07:36.616 --> 00:07:40.510
you should have no problems
with the Level 1 exercises.

00:07:40.510 --> 00:07:42.010
Have fun!


Exponent rules part 1

WEBVTT
Kind: captions
Language: en

00:00:01.230 --> 00:00:05.600
Welcome to the presentation
on level one exponent rules.

00:00:05.600 --> 00:00:08.150
Let's get started
with some problems.

00:00:08.150 --> 00:00:12.870
So if I were to ask you what 2
-- that's a little fatter than

00:00:12.870 --> 00:00:15.080
I wanted it to be, but let's
keep it fat so it doesn't look

00:00:15.080 --> 00:00:20.260
strange -- 2 the third times --
and dot is another way of

00:00:20.260 --> 00:00:23.230
saying times -- if I were to
ask you what 2 to the third

00:00:23.230 --> 00:00:27.820
times 2 to the fifth is, how
would you figure that out?

00:00:27.820 --> 00:00:30.610
Actually, let me use a skinnier
pen because that does look bad.

00:00:30.610 --> 00:00:35.120
So, 2 to the third
times 2 to the fifth.

00:00:35.120 --> 00:00:37.610
Well there's one way that I
think you do know how to do it.

00:00:37.610 --> 00:00:42.150
You could figure out that
2 to the third is 9, and

00:00:42.150 --> 00:00:45.380
that 2 to the fifth is 32.

00:00:45.380 --> 00:00:46.840
And then you could
multiply them.

00:00:46.840 --> 00:00:54.010
And 8 times 32 is 240,
plus it's 256, right?

00:00:54.010 --> 00:00:55.530
You could do it that way.

00:00:55.530 --> 00:00:58.550
That's reasonable because it's
not that hard to figure out 2

00:00:58.550 --> 00:01:00.520
to the third is and what
2 to the fifth is.

00:01:00.520 --> 00:01:03.150
But if those were much larger
numbers this method might

00:01:03.150 --> 00:01:04.770
become a little difficult.

00:01:04.770 --> 00:01:08.520
So I'm going to show you using
exponent rules you can actually

00:01:08.520 --> 00:01:12.340
multiply exponentials or
exponent numbers without

00:01:12.340 --> 00:01:15.715
actually having to do as much
arithmetic or actually you

00:01:15.715 --> 00:01:18.120
could handle numbers much
larger than your normal math

00:01:18.120 --> 00:01:20.780
skills might allow you to.

00:01:20.780 --> 00:01:23.060
So let's just think what
2 to the third times

00:01:23.060 --> 00:01:24.670
2 to the fifth means.

00:01:24.670 --> 00:01:32.940
2 to the third is 2
times 2 times 2, right?

00:01:32.940 --> 00:01:35.200
And we're multiplying that
times 2 to the fifth.

00:01:35.200 --> 00:01:43.160
And that's 2 times 2
times 2 times 2 times 2.

00:01:43.160 --> 00:01:44.200
So what do we have here?

00:01:44.200 --> 00:01:47.870
We have 2 times 2 times
2, times 2 times 2 times

00:01:47.870 --> 00:01:49.780
2 times 2 times 2.

00:01:49.780 --> 00:01:52.640
Really all we're doing is we're
multiplying 2 how many times?

00:01:52.640 --> 00:01:58.920
Well, one, two, three, four,
five, six, seven, eight.

00:01:58.920 --> 00:02:03.410
So that's the same thing
as 2 to the eighth.

00:02:03.410 --> 00:02:05.050
Interesting.

00:02:05.050 --> 00:02:08.200
3 plus 5 is equal to 8.

00:02:08.200 --> 00:02:12.360
And that makes sense because 2
to the 3 is 2 multiplying by

00:02:12.360 --> 00:02:15.400
itself three times, to the
fifth is 2 multiplying by

00:02:15.400 --> 00:02:17.540
itself five times, and then
we're multiplying the two, so

00:02:17.540 --> 00:02:19.980
we're going to multiply
2 eight times.

00:02:19.980 --> 00:02:22.720
I hope I achieved my goal
of confusing you just now.

00:02:22.720 --> 00:02:23.580
Let's do another one.

00:02:26.130 --> 00:02:33.780
If I said 7 squared
times 7 to the fourth.

00:02:33.780 --> 00:02:36.550
That's a 4.

00:02:36.550 --> 00:02:42.180
Well, that equals 7 times 7,
right, that's 7 squared,

00:02:42.180 --> 00:02:44.430
times and now let's
do 7 to the fourth.

00:02:44.430 --> 00:02:50.090
7 times 7 times 7 times 7.

00:02:50.090 --> 00:02:53.780
Well now we're multiplying
7 by itself six times, so

00:02:53.780 --> 00:02:56.590
that equal 7 to the sixth.

00:02:56.590 --> 00:03:00.130
So in general, whenever I'm
multiplying exponents of the

00:03:00.130 --> 00:03:04.620
same base, that's key, I can
just add the exponents.

00:03:04.620 --> 00:03:12.520
So 7 to the hundredth power
times 7 to the fiftieth

00:03:12.520 --> 00:03:15.440
power, and notice this
is an example now.

00:03:15.440 --> 00:03:17.750
It would be very hard without
a computer to figure out what

00:03:17.750 --> 00:03:19.320
7 to the hundredth power is.

00:03:19.320 --> 00:03:22.190
And likewise, very hard without
a computer to figure out what

00:03:22.190 --> 00:03:24.050
7 to the fiftieth power is.

00:03:24.050 --> 00:03:32.730
But we could say that this is
equal to 7 to the 100 plus 50,

00:03:32.730 --> 00:03:37.790
which is equal to 7 to the 150.

00:03:37.790 --> 00:03:40.430
Now I just want to give you a
little bit of warning, make

00:03:40.430 --> 00:03:41.630
sure that you're multiplying.

00:03:41.630 --> 00:03:49.150
Because if I had 7 to the 100
plus 7 to the 50, there's

00:03:49.150 --> 00:03:50.590
actually very little
I could do here.

00:03:50.590 --> 00:03:54.440
I couldn't simplify
this number.

00:03:54.440 --> 00:03:56.710
But I'll throw out one to you.

00:03:56.710 --> 00:04:04.810
If I had 2 to the 8 times
2 to the 20, we know we

00:04:04.810 --> 00:04:06.570
can add these exponents.

00:04:06.570 --> 00:04:12.500
So that gives you 2
to the 28, right?

00:04:12.500 --> 00:04:20.820
What if I had 2 to the
8 plus 2 to the 8?

00:04:20.820 --> 00:04:22.890
This is a bit of a
trick question.

00:04:22.890 --> 00:04:25.030
Well I just said if
we're adding, we can't

00:04:25.030 --> 00:04:26.900
really do anything.

00:04:26.900 --> 00:04:28.530
We can't really simplify it.

00:04:28.530 --> 00:04:30.670
But there's a little trick
here that we actually have

00:04:30.670 --> 00:04:32.980
two 2 to the 8, right?

00:04:32.980 --> 00:04:35.080
There's 2 to the 8 times
1, 2 to the 8 times 2.

00:04:35.080 --> 00:04:41.240
So this is the same thing as 2
times 2 to the 8, isn't it?

00:04:41.240 --> 00:04:42.150
2 times 2 to the 8.

00:04:42.150 --> 00:04:44.940
That's just 2 to
the 8 plus itself.

00:04:44.940 --> 00:04:49.030
And 2 times to the 8, well
that's the same thing as 2 to

00:04:49.030 --> 00:04:53.170
the first times 2 to the 8.

00:04:53.170 --> 00:04:55.500
And 2 to the first times 2 to
the 8 by the same rule we just

00:04:55.500 --> 00:04:59.040
did is equal to 2 to the 9.

00:04:59.040 --> 00:05:01.080
So I thought I would just
throw that out to you.

00:05:01.080 --> 00:05:03.280
And it works even with
negative exponents.

00:05:03.280 --> 00:05:13.840
If I were to say 5 to the
negative 100 times 3 to the,

00:05:13.840 --> 00:05:18.370
say, 100 -- oh sorry, times
5 -- this has to be a 5.

00:05:18.370 --> 00:05:20.140
I don't know what my
brain was doing.

00:05:20.140 --> 00:05:25.150
5 to the negative 100 times
5 to the 102, that would

00:05:25.150 --> 00:05:27.890
equal 5 squared, right?

00:05:27.890 --> 00:05:30.930
I just take minus 100 plus 102.

00:05:30.930 --> 00:05:31.940
This is a 5.

00:05:31.940 --> 00:05:35.080
I'm sorry for that
brain malfunction.

00:05:35.080 --> 00:05:37.860
And of course, that equals 25.

00:05:37.860 --> 00:05:39.210
So that's the first
exponent rule.

00:05:39.210 --> 00:05:40.760
Now I'm going to show you
another one, and it kind of

00:05:40.760 --> 00:05:43.900
leads from the same thing.

00:05:43.900 --> 00:05:55.280
If I were to ask you what 2 to
the 9 over 2 to the 10 equals,

00:05:55.280 --> 00:05:56.940
that looks like that could
be a little confusing.

00:05:56.940 --> 00:06:00.720
But it actually turns out to be
the same rule, because what's

00:06:00.720 --> 00:06:03.110
another way of writing this?

00:06:03.110 --> 00:06:08.360
Well, we know that this is also
the same thing as 2 to the 9

00:06:08.360 --> 00:06:12.710
times 1 over 2 to
the 10, right?

00:06:12.710 --> 00:06:14.460
And we know 1 over 2 to the 10.

00:06:14.460 --> 00:06:18.700
Well, you could re-write right
this as 2 the 9 times 2 to

00:06:18.700 --> 00:06:20.850
the negative 10, right?

00:06:20.850 --> 00:06:25.270
All I did is I took 1 over 2 to
the 10 and I flipped it and I

00:06:25.270 --> 00:06:26.990
made the exponent negative.

00:06:26.990 --> 00:06:28.330
And I think you know
that already from

00:06:28.330 --> 00:06:30.660
level two exponents.

00:06:30.660 --> 00:06:33.090
And now, once again, we can
just add the exponents.

00:06:33.090 --> 00:06:39.300
9 plus negative 10 equals 2 to
the negative 1, or we could

00:06:39.300 --> 00:06:42.000
say that equals 1/2, right?

00:06:42.000 --> 00:06:44.730
So it's an interesting
thing here.

00:06:44.730 --> 00:06:48.110
Whatever is the bottom
exponent, you could put it in

00:06:48.110 --> 00:06:50.800
the numerator like we did here,
but turn it into a negative.

00:06:50.800 --> 00:06:53.760
So that leads us to the second
exponent rule, simplification

00:06:53.760 --> 00:06:59.860
is we could just say that this
equals 2 to the 9 minus 10,

00:06:59.860 --> 00:07:02.190
which equals 2 to
the negative 1.

00:07:02.190 --> 00:07:05.160
Let's do another
problem like that.

00:07:05.160 --> 00:07:16.400
If I said 10 to the 200 over
10 to the 50, well that

00:07:16.400 --> 00:07:23.640
just equals 10 to the 200
minus 50, which is 150.

00:07:23.640 --> 00:07:30.870
Likewise, if I had 7 to the
fortieth power over 7 to

00:07:30.870 --> 00:07:35.940
the negative fifth power,
this will equal 7 to the

00:07:35.940 --> 00:07:41.420
fortieth minus negative 5.

00:07:41.420 --> 00:07:46.230
So it equals 7 to
the forty-fifth.

00:07:46.230 --> 00:07:48.310
Now I want you to think about
that, does that make sense?

00:07:48.310 --> 00:07:54.480
Well, we could have re-written
this equation as 7 to the

00:07:54.480 --> 00:07:59.180
fortieth times 7 to
the fifth, right?

00:07:59.180 --> 00:08:02.806
We could have taken this 1 over
7 to the negative 5 and turn it

00:08:02.806 --> 00:08:06.640
into 7 to the fifth, and that
would also just be 7

00:08:06.640 --> 00:08:08.160
to the forty-five.

00:08:08.160 --> 00:08:10.810
So the second exponent rule I
just taught you actually is no

00:08:10.810 --> 00:08:12.390
different than that first one.

00:08:12.390 --> 00:08:14.810
If the exponent is in the
denominator, and of course, it

00:08:14.810 --> 00:08:18.210
has to be the same base and
you're dividing, you subtract

00:08:18.210 --> 00:08:20.570
it from the exponent
in the numerator.

00:08:20.570 --> 00:08:23.390
If they're both in the
numerator, as in this case, 7

00:08:23.390 --> 00:08:26.580
to the fortieth times 7 to the
fifth -- actually there's no

00:08:26.580 --> 00:08:29.370
numerator, but they're
essentially multiplying by each

00:08:29.370 --> 00:08:32.420
other, and of course, you have
to have the same base.

00:08:32.420 --> 00:08:35.690
Then you add the exponents.

00:08:35.690 --> 00:08:37.700
I'm going to add one variation
of this, and actually this is

00:08:37.700 --> 00:08:40.360
the same thing but it's a
little bit of a trick question.

00:08:40.360 --> 00:08:56.470
What is 2 to the 9
times 4 to the 100?

00:08:56.470 --> 00:08:58.190
Actually, maybe I shouldn't
teach this to you, you have

00:08:58.190 --> 00:08:59.480
to wait until I teach
you the next rule.

00:08:59.480 --> 00:09:01.900
But I'll give you
a little hint.

00:09:01.900 --> 00:09:09.570
This is the same thing as 2 the
9 times 2 squared to the 100.

00:09:09.570 --> 00:09:12.320
And the rule I'm going to teach
you now is that when you have

00:09:12.320 --> 00:09:15.610
something to an exponent and
then that number raised to

00:09:15.610 --> 00:09:18.930
an exponent, you actually
multiply these two exponents.

00:09:18.930 --> 00:09:24.980
So this would be 2 the
9 times 2 to the 200.

00:09:24.980 --> 00:09:27.020
And by that first rule
we learned, this would

00:09:27.020 --> 00:09:29.760
be 2 to the 209.

00:09:29.760 --> 00:09:31.140
Now in the next module
I'm going to cover

00:09:31.140 --> 00:09:31.940
this in more detail.

00:09:31.940 --> 00:09:34.650
I think I might have
just confused you.

00:09:34.650 --> 00:09:37.050
But watch the next video and
then after the next video I

00:09:37.050 --> 00:09:40.400
think you're going to be ready
to do level one exponent rules.

00:09:40.400 --> 00:09:41.930
Have fun.


Mulitplication 8: Multiplying decimals (Old video)

WEBVTT
Kind: captions
Language: en

00:00:01.290 --> 00:00:04.830
Welcome to the presentation
on multiplying decimals.

00:00:04.830 --> 00:00:07.230
Let's get started.

00:00:07.230 --> 00:00:10.120
So I think you'll find out that
multiplying decimals is not a

00:00:10.120 --> 00:00:13.220
lot more difficult than just
multiplying regular numbers.

00:00:13.220 --> 00:00:15.460
And I'll show you in a problem.

00:00:15.460 --> 00:00:16.940
Let me pick some
random numbers.

00:00:16.940 --> 00:00:25.590
Let's say I had 7,518.

00:00:25.590 --> 00:00:29.590
Actually, let's
make that's 75.18.

00:00:29.590 --> 00:00:31.700
Clearly you can tell I'm
doing this on the fly.

00:00:31.700 --> 00:00:40.010
75.18 times 0.97.

00:00:40.010 --> 00:00:41.140
So first you look at this
problem you're like,

00:00:41.140 --> 00:00:41.970
oh boy, that's tough.

00:00:41.970 --> 00:00:44.120
These decimals-- I don't even
know how to approach it.

00:00:44.120 --> 00:00:45.490
Well this is what you do.

00:00:45.490 --> 00:00:47.960
You ignore the decimals when
you start the problem and you

00:00:47.960 --> 00:00:50.640
pretend like it's just a
regular multiplication problem.

00:00:50.640 --> 00:00:53.435
And if you ignore the
decimals like I said, at

00:00:53.435 --> 00:00:57.290
the beginning, 7,518 on
top and 97 on the bottom.

00:00:57.290 --> 00:00:58.470
And if that doesn't make
sense let me just show you.

00:00:58.470 --> 00:01:00.440
I'm just going to ignore the
decimals and do this like a

00:01:00.440 --> 00:01:02.320
normal multiplication problem.

00:01:02.320 --> 00:01:03.460
So normal multiplication.

00:01:03.460 --> 00:01:05.870
I'd start at the 1's
place right here.

00:01:05.870 --> 00:01:07.430
I'd say 7 times 8.

00:01:07.430 --> 00:01:09.530
Well, 7 times 8 is 56.

00:01:09.530 --> 00:01:11.790
Carry the 5.

00:01:11.790 --> 00:01:13.890
7 times 1 is 7.

00:01:13.890 --> 00:01:17.020
Plus the 5 is 12.

00:01:17.020 --> 00:01:18.490
2 down here.

00:01:18.490 --> 00:01:20.300
Carry the 1.

00:01:20.300 --> 00:01:23.740
7 times 5 is 35.

00:01:23.740 --> 00:01:25.825
Plus the 1 is 36.

00:01:25.825 --> 00:01:27.800
Put the 6 here.

00:01:27.800 --> 00:01:29.970
Carry the 3.

00:01:29.970 --> 00:01:33.830
And then 7 times 7 is 49.

00:01:33.830 --> 00:01:36.700
Plus 2 is 52.

00:01:36.700 --> 00:01:39.370
So just put 52 here.

00:01:39.370 --> 00:01:41.430
So just like normal
multiplication we just took the

00:01:41.430 --> 00:01:43.850
1's place right here, the 7, so
it's actually not the 1's, but

00:01:43.850 --> 00:01:46.695
we're ignoring the decimals so
if there were no decimals this

00:01:46.695 --> 00:01:48.030
would be the 1's place.

00:01:48.030 --> 00:01:50.080
And we're multiplying
it by the top number.

00:01:50.080 --> 00:01:55.030
7 times 7,518 is
equal to 52,626.

00:01:55.030 --> 00:01:57.230
And like regular
multiplication, we

00:01:57.230 --> 00:01:58.050
do the 10's place.

00:01:58.050 --> 00:02:00.450
And this isn't really the 10's
place, but if you ignore

00:02:00.450 --> 00:02:02.030
the decimals it would be.

00:02:02.030 --> 00:02:05.120
And let's cross all this stuff
out since we're not using it.

00:02:05.120 --> 00:02:08.870
9 times 8 is 72.

00:02:08.870 --> 00:02:10.110
Carry the 7.

00:02:10.110 --> 00:02:12.260
9 times 1 is 9.

00:02:12.260 --> 00:02:15.450
Plus 7 is 16.

00:02:15.450 --> 00:02:16.960
Carry the 1.

00:02:16.960 --> 00:02:19.870
9 times 5 is 45.

00:02:19.870 --> 00:02:21.620
This is good practice for
me too, I haven't done

00:02:21.620 --> 00:02:24.360
my multiplication
tables in a long time.

00:02:24.360 --> 00:02:26.320
9 times 5 is 45.

00:02:26.320 --> 00:02:29.590
Plus 1 is 46.

00:02:29.590 --> 00:02:31.570
Carry the 4.

00:02:31.570 --> 00:02:35.070
9 times 7 is 63.

00:02:35.070 --> 00:02:36.710
Plus 4 is 67.

00:02:39.350 --> 00:02:40.780
Now we add.

00:02:40.780 --> 00:02:43.610
So you're probably thinking,
boy, what do decimals have

00:02:43.610 --> 00:02:44.410
to do with this at all.

00:02:44.410 --> 00:02:46.090
I'm just doing a regular
multiplication problem.

00:02:46.090 --> 00:02:46.530
And I'll show you.

00:02:46.530 --> 00:02:49.380
Actually the decimals only come
in right at the very end.

00:02:49.380 --> 00:02:52.740
So what I do is now I just add
like I do a regular level

00:02:52.740 --> 00:02:54.790
4 multiplication problem.

00:02:54.790 --> 00:02:57.880
So I say 6 plus 0 is 6.

00:02:57.880 --> 00:03:00.880
2 plus 2 is 4.

00:03:00.880 --> 00:03:03.440
6 plus 6 is 12.

00:03:03.440 --> 00:03:05.170
Carry the 1.

00:03:05.170 --> 00:03:08.640
1 plus 2 plus 6 is 9.

00:03:08.640 --> 00:03:10.990
5 plus 7 is 12.

00:03:10.990 --> 00:03:13.260
Carry the 1.

00:03:13.260 --> 00:03:15.370
1 plus 6 is 7.

00:03:15.370 --> 00:03:18.970
OK, so now here's where the
decimals come into play.

00:03:18.970 --> 00:03:21.300
And your I think you're
going to be shocked by how

00:03:21.300 --> 00:03:22.680
straightforward this is.

00:03:22.680 --> 00:03:24.770
What I do is I go back to the
original problem and now I

00:03:24.770 --> 00:03:26.430
actually pay attention
to the decimals.

00:03:26.430 --> 00:03:29.160
And I say, how many total
numbers are behind

00:03:29.160 --> 00:03:30.610
the decimal point?

00:03:30.610 --> 00:03:34.150
Well, there's 1 number behind
the decimal point, 2 numbers

00:03:34.150 --> 00:03:36.860
behind the decimal point, 3
numbers behind the decimal

00:03:36.860 --> 00:03:40.050
point, 4 numbers behind
the decimal point.

00:03:40.050 --> 00:03:41.750
1, 2, 3, 4.

00:03:41.750 --> 00:03:43.860
So there are 4 numbers behind
the decimal point in the

00:03:43.860 --> 00:03:46.330
problem I did, and
I just count here.

00:03:46.330 --> 00:03:49.810
1, 2, 3, 4.

00:03:49.810 --> 00:03:52.930
The answer will also have 4
numbers behind the decimal

00:03:52.930 --> 00:03:54.620
point, and that's the answer.

00:03:54.620 --> 00:03:59.190
72.9246.

00:03:59.190 --> 00:04:00.750
Now let me ask you a question.

00:04:00.750 --> 00:04:08.870
If I had a 0 here, would that
count as an extra number

00:04:08.870 --> 00:04:11.010
behind the decimal point?

00:04:11.010 --> 00:04:14.785
Well, it only would have been
if you actually used the

00:04:14.785 --> 00:04:16.820
0 in the multiplication.

00:04:16.820 --> 00:04:18.770
Maybe that confuses you.

00:04:18.770 --> 00:04:21.220
What I would recommend if
you have any trailing 0's

00:04:21.220 --> 00:04:22.660
with a decimal like this.

00:04:22.660 --> 00:04:25.000
you actually should just ignore
those 0's and then do the

00:04:25.000 --> 00:04:26.610
problem just the way I did it.

00:04:26.610 --> 00:04:31.140
And when remember, that's
only for trailing 0's.

00:04:31.140 --> 00:04:35.270
If this was the bottom number
then that 0 would matter

00:04:35.270 --> 00:04:38.770
because it's not a trailing 0,
it's actually part

00:04:38.770 --> 00:04:40.260
of the number.

00:04:40.260 --> 00:04:44.070
Let's do a couple more examples
and I think that'll make sense.

00:04:44.070 --> 00:04:47.830
So let's say I had 5-- and
I'm going to do a simpler

00:04:47.830 --> 00:04:49.600
example arithmetically.

00:04:49.600 --> 00:04:52.070
I think it'll help you
with some principles.

00:04:52.070 --> 00:05:00.880
If I said 5.10 times 1.09.

00:05:00.880 --> 00:05:03.020
So there's two
things we could do.

00:05:03.020 --> 00:05:05.560
We could just multiply
it the way it is.

00:05:05.560 --> 00:05:07.840
Actually let's do it both ways
and I'll show you you get the

00:05:07.840 --> 00:05:10.150
same answer whether or
not you ignore that 0.

00:05:10.150 --> 00:05:14.460
So in the first case
let's not ignore the 0.

00:05:14.460 --> 00:05:16.440
Let's use that 0, even though
that trailing 0 in the

00:05:16.440 --> 00:05:19.600
decimal-- 5.10 is the
same thing as 5.1.

00:05:19.600 --> 00:05:20.750
But let's use it.

00:05:20.750 --> 00:05:22.680
9 times 0 is 0.

00:05:22.680 --> 00:05:24.990
9 times 1 is 9.

00:05:24.990 --> 00:05:28.570
9 times 5 is 45.

00:05:28.570 --> 00:05:30.820
And in the 0's place
you put a 0 and then 0

00:05:30.820 --> 00:05:32.885
times everything is 0.

00:05:32.885 --> 00:05:36.780
0 times 0, 0 times
1, 0 times 5.

00:05:36.780 --> 00:05:37.740
Put two 0's here.

00:05:37.740 --> 00:05:40.190
And then 1 times 0 is 0.

00:05:40.190 --> 00:05:41.880
1 times 1 is 1.

00:05:41.880 --> 00:05:44.550
And 1 times 5 is 5.

00:05:44.550 --> 00:05:46.420
And now we add it all.

00:05:46.420 --> 00:05:51.750
We get 0, 9, 5, 5, 5.

00:05:51.750 --> 00:05:53.430
And like we did before, we
just count the decimals.

00:05:53.430 --> 00:05:55.650
1, 2, 3, 4.

00:05:55.650 --> 00:05:57.450
So 1, 2 3, 4.

00:05:57.450 --> 00:05:59.730
So the decimal will go here.

00:05:59.730 --> 00:06:03.470
So we got 5.5590 as the answer.

00:06:03.470 --> 00:06:05.140
Now what if we did like
I was recommending, we

00:06:05.140 --> 00:06:05.950
actually ignored the 0?

00:06:08.880 --> 00:06:15.320
And I can actually rewrite
it as 1.09 times 5.1.

00:06:15.320 --> 00:06:16.490
Because you know in
multiplication order

00:06:16.490 --> 00:06:17.510
doesn't matter.

00:06:17.510 --> 00:06:19.940
a times b is the same
thing as b times a.

00:06:19.940 --> 00:06:22.010
2 times 3 is the same
thing as 3 times 2.

00:06:22.010 --> 00:06:27.695
So 1.09 times 5.1 is the same
thing as 5.1 times 1.09.

00:06:27.695 --> 00:06:29.320
So let's just
multiply this out.

00:06:29.320 --> 00:06:30.590
And notice, these are
the same numbers.

00:06:30.590 --> 00:06:34.340
All I did is I took the 0 off.

00:06:34.340 --> 00:06:38.900
So first, I just ignore the
decimals I say 1 times 9 is 9.

00:06:38.900 --> 00:06:41.110
1 times 0 is 0.

00:06:41.110 --> 00:06:43.200
1 times 1 is 1.

00:06:43.200 --> 00:06:44.910
Put a 0 here.

00:06:44.910 --> 00:06:48.240
5 times 9 is 45.

00:06:48.240 --> 00:06:49.720
Carry the 4.

00:06:49.720 --> 00:06:51.070
5 two 0 is 0.

00:06:51.070 --> 00:06:54.060
Plus 4 is 4.

00:06:54.060 --> 00:06:57.410
5 times 1 is 5.

00:06:57.410 --> 00:06:58.830
Now I add.

00:06:58.830 --> 00:07:04.290
9, 5, 5, 5.

00:07:04.290 --> 00:07:06.890
Now I'm at the point that I
can actually pay attention

00:07:06.890 --> 00:07:09.040
to the decimal points.

00:07:09.040 --> 00:07:10.890
How many numbers are
behind the decimals?

00:07:10.890 --> 00:07:14.010
Well, there's 1, 2, 3.

00:07:14.010 --> 00:07:18.330
So I go 1, 2, 3 and put the
decimal point right here.

00:07:18.330 --> 00:07:20.520
Notice I got the
same exact answer.

00:07:20.520 --> 00:07:23.610
The only difference is that
this one had a trailing 0,

00:07:23.610 --> 00:07:25.630
which really doesn't make
a number any different.

00:07:25.630 --> 00:07:27.400
I could add a hundred 0's
here and the number's really

00:07:27.400 --> 00:07:30.490
not a different number.

00:07:30.490 --> 00:07:33.950
If you were a computer
programmer or a statistician

00:07:33.950 --> 00:07:35.250
of some kind, this could
be an important number.

00:07:35.250 --> 00:07:36.520
But ignore what I just said.

00:07:36.520 --> 00:07:41.360
And for your purposes, these
trailing 0's mean nothing.

00:07:44.250 --> 00:07:46.700
Same way a leading 0 actually
wouldn't mean nothing.

00:07:46.700 --> 00:07:47.500
No one ever does that.

00:07:50.350 --> 00:07:51.900
Well, let me see how
much time I have.

00:07:51.900 --> 00:07:52.470
I have 2 more minutes.

00:07:52.470 --> 00:07:55.520
Let me do one more problem just
to maybe hit the point home.

00:07:55.520 --> 00:08:02.770
You know, this is really
no different than level

00:08:02.770 --> 00:08:03.820
4 multiplication.

00:08:03.820 --> 00:08:06.020
And at the end you just
have to count the numbers

00:08:06.020 --> 00:08:07.620
behind the decimal point.

00:08:07.620 --> 00:08:11.040
So 5 times 5 is 25.

00:08:11.040 --> 00:08:12.390
Whoops.

00:08:12.390 --> 00:08:12.680
25.

00:08:12.680 --> 00:08:14.580
I'm already getting messy.

00:08:14.580 --> 00:08:15.910
Carry the 2.

00:08:15.910 --> 00:08:18.030
5 times 7 is 35.

00:08:18.030 --> 00:08:20.870
Plus 2 is 37.

00:08:20.870 --> 00:08:23.380
Bring down the 7, carry the 3.

00:08:23.380 --> 00:08:25.710
5 times 0 is 0.

00:08:25.710 --> 00:08:26.455
Plus 3.

00:08:26.455 --> 00:08:29.340
So it's 375, ignore that blob.

00:08:29.340 --> 00:08:30.485
I'm sorry for being so messy.

00:08:30.485 --> 00:08:32.010
And then you put a 0.

00:08:32.010 --> 00:08:34.620
1 times 5 is 5.

00:08:34.620 --> 00:08:36.870
1 times 7 is 7.

00:08:36.870 --> 00:08:38.270
Ignore that.

00:08:38.270 --> 00:08:39.114
Now we add.

00:08:39.114 --> 00:08:41.510
We say 5 plus 0 is 5.

00:08:41.510 --> 00:08:44.430
7 plus 5 is 12.

00:08:44.430 --> 00:08:47.060
1 plus 3 plus 7 is 11.

00:08:47.060 --> 00:08:48.320
So we got our answer,
now we just have to

00:08:48.320 --> 00:08:49.820
count the decimals.

00:08:49.820 --> 00:08:55.610
So here we have 1, 2,
3, 4, 5 numbers behind

00:08:55.610 --> 00:08:56.860
the decimal point.

00:08:56.860 --> 00:09:00.100
But in our answer we only have
4 digits, so how can we get 5

00:09:00.100 --> 00:09:01.610
numbers behind the
decimal point?

00:09:01.610 --> 00:09:03.090
Well, we start here.

00:09:03.090 --> 00:09:08.820
We say 1, 2, 3, 4 and we need
one more number behind the

00:09:08.820 --> 00:09:10.770
decimal point, so
we add a 0 here.

00:09:10.770 --> 00:09:13.090
And then we put the
decimal point.

00:09:13.090 --> 00:09:14.380
See what I just did.

00:09:14.380 --> 00:09:18.030
We had to have 5 numbers
behind the decimal point.

00:09:18.030 --> 00:09:20.080
And we only had 4
numbers in the answer.

00:09:20.080 --> 00:09:22.950
So I added a leading 0 and
then put the decimal point.

00:09:22.950 --> 00:09:26.990
And now we have 5 numbers
behind the decimal point.

00:09:26.990 --> 00:09:29.390
And I've shown you a very
mechanical way of doing this.

00:09:29.390 --> 00:09:32.030
Hopefully in the future I can
give you a seminar on actually

00:09:32.030 --> 00:09:35.510
why this method of counting the
numbers behind the decimal

00:09:35.510 --> 00:09:37.070
points actually works.

00:09:37.070 --> 00:09:40.340
But I think you are ready
to try some problems on

00:09:40.340 --> 00:09:41.890
multiplying decimals.

00:09:41.890 --> 00:09:43.400
Have fun.


Adding Decimals (Old)

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.660
Welcome to the presentation
on adding decimals.

00:00:04.660 --> 00:00:07.440
Let's do some problems.

00:00:07.440 --> 00:00:21.840
So let's say I had 0.008--
that's an 8-- 5 plus-- and

00:00:21.840 --> 00:00:33.260
I'm writing it side by
side on purpose-- 1.799.

00:00:33.260 --> 00:00:36.250
So at first you're like,
these decimals, they

00:00:36.250 --> 00:00:38.110
confuse me, I give up.

00:00:38.110 --> 00:00:39.750
But I'm going to show you
that it's actually very

00:00:39.750 --> 00:00:42.080
straightforward and it's
actually no more difficult

00:00:42.080 --> 00:00:44.200
than doing normal addition.

00:00:44.200 --> 00:00:46.340
And there's only one thing
you have to think about

00:00:46.340 --> 00:00:47.630
when you do decimals.

00:00:47.630 --> 00:00:50.350
You have to line up
the decimal points.

00:00:50.350 --> 00:00:52.465
So let's start with the first
number and let's rewrite it so

00:00:52.465 --> 00:00:54.750
that we can actually
do the math.

00:00:54.750 --> 00:01:01.740
0.0085.

00:01:01.740 --> 00:01:05.050
Now for the second number let's
put the decimal point right

00:01:05.050 --> 00:01:09.120
below where that first decimal
is on the top number, and

00:01:09.120 --> 00:01:10.110
then rewrite the number.

00:01:10.110 --> 00:01:17.920
So it's 1.799.

00:01:17.920 --> 00:01:21.050
So all we did is we just
rewrote both numbers with their

00:01:21.050 --> 00:01:22.230
decimal points lined up.

00:01:22.230 --> 00:01:23.320
And we could've
flipped them around.

00:01:23.320 --> 00:01:27.570
We could've written the 1.799
first and the 0.0085 below it.

00:01:27.570 --> 00:01:30.860
What's key-- and if you get
this point and you already know

00:01:30.860 --> 00:01:32.810
how to do addition, you've
already figured out how to do

00:01:32.810 --> 00:01:36.350
adding decimals-- is that you
line up the decimal points.

00:01:36.350 --> 00:01:37.810
So once we line up the
decimal points we're

00:01:37.810 --> 00:01:40.990
ready to start adding.

00:01:40.990 --> 00:01:43.440
But hey Sal, there's
something very strange

00:01:43.440 --> 00:01:44.600
here you might say.

00:01:44.600 --> 00:01:46.540
How can I add this
5 to a nothing?

00:01:46.540 --> 00:01:48.960
Well there's a very
easy solution to that.

00:01:48.960 --> 00:01:50.870
Let's just put a 0 right here.

00:01:50.870 --> 00:01:52.160
And how can I do that?

00:01:52.160 --> 00:01:56.440
How can I just randomly
augment a number with a 0?

00:01:56.440 --> 00:01:58.840
Well, I don't know if I've
already shown you in another

00:01:58.840 --> 00:02:01.930
module, but when you add
trailing 0's to decimals, it

00:02:01.930 --> 00:02:04.230
doesn't really change the
value of the decimal.

00:02:04.230 --> 00:02:05.500
And I think that
make sense to you.

00:02:05.500 --> 00:02:10.570
If I said 1.10 that that's
the same thing as 1.1.

00:02:10.570 --> 00:02:14.120
Which is the same
thing as 1.100.

00:02:14.120 --> 00:02:15.720
It's all 1 and 1/10.

00:02:15.720 --> 00:02:18.820
And these 0's just kind of add
extra digits of precision,

00:02:18.820 --> 00:02:20.570
which actually, don't
change the actual value.

00:02:20.570 --> 00:02:23.510
And maybe later when I'm
teaching computer programming

00:02:23.510 --> 00:02:25.110
and computation, the
precision might matter.

00:02:25.110 --> 00:02:26.140
But for now, it doesn't.

00:02:26.140 --> 00:02:27.210
It's just the value.

00:02:27.210 --> 00:02:29.480
So I added this 0 here and
you can always do that.

00:02:29.480 --> 00:02:31.600
You can add trailing 0's
without actually changing

00:02:31.600 --> 00:02:32.850
the value of the number.

00:02:32.850 --> 00:02:34.880
And now we're really
ready to add.

00:02:34.880 --> 00:02:38.380
5 plus 0 is 5.

00:02:38.380 --> 00:02:40.375
8 plus 9 is 17.

00:02:43.290 --> 00:02:46.730
1 plus 0 plus 9 is 10.

00:02:46.730 --> 00:02:48.020
Carry the 1.

00:02:48.020 --> 00:02:53.110
1 plus 0 plus 7 is 8.

00:02:53.110 --> 00:02:55.690
And then bring down
this 1 because there's

00:02:55.690 --> 00:02:56.830
nothing to add it to.

00:02:56.830 --> 00:02:58.350
And we could have even
added a leading zero

00:02:58.350 --> 00:02:59.630
here if you wanted to.

00:02:59.630 --> 00:03:01.680
My wife's a doctor and she says
it's key that you always add

00:03:01.680 --> 00:03:03.740
that leading 0 so that no one
gives someone the wrong

00:03:03.740 --> 00:03:04.355
amount of medicine.

00:03:04.355 --> 00:03:07.850
But anyway, so now
we are almost done.

00:03:07.850 --> 00:03:09.730
We've done the addition and now
we just have to figure out

00:03:09.730 --> 00:03:10.470
where to put the decimal point.

00:03:10.470 --> 00:03:13.496
Well, the decimal point
just drops straight down.

00:03:13.496 --> 00:03:16.020
And we're able to do that
because we already lined

00:03:16.020 --> 00:03:16.870
up the decimal points.

00:03:16.870 --> 00:03:21.160
So the answer to this
problem is 1.8075.

00:03:21.160 --> 00:03:25.150
It seems complicated, but all
you have to do, line up the

00:03:25.150 --> 00:03:29.115
decimal points, add 0's
where appropriate, and add.

00:03:29.115 --> 00:03:31.900
And if you know how to add you
already know how to do this.

00:03:31.900 --> 00:03:32.760
Let's do another problem.

00:03:36.650 --> 00:03:49.360
58.75 plus 0.028.

00:03:49.360 --> 00:03:53.240
Now if you saw a problem like
this written in this format,

00:03:53.240 --> 00:03:55.250
your temptation might be to
immediately start adding.

00:03:55.250 --> 00:03:58.550
Add the 5 to the 8, the 7
to the 2, the 8 to the 0.

00:03:58.550 --> 00:04:01.090
And then just bring down the 5
or something of that nature.

00:04:01.090 --> 00:04:03.750
And you would be wrong.

00:04:03.750 --> 00:04:05.410
And I think you realize
why you'd be wrong.

00:04:05.410 --> 00:04:06.970
Because you forgot
the first step.

00:04:06.970 --> 00:04:09.360
The first step is line
up the decimal points.

00:04:09.360 --> 00:04:12.930
When you're doing addition with
decimals, the number one step

00:04:12.930 --> 00:04:14.740
is line up the decimal points.

00:04:14.740 --> 00:04:17.220
So let's line up the
decimal points.

00:04:17.220 --> 00:04:20.010
So we can just rewrite
that top number again.

00:04:20.010 --> 00:04:23.350
58.75.

00:04:23.350 --> 00:04:25.180
And now let's rewrite the
bottom number so that the

00:04:25.180 --> 00:04:26.700
decimal points line up.

00:04:26.700 --> 00:04:29.300
So we'll put the decimal
points right below it.

00:04:29.300 --> 00:04:34.270
And it's 0.028.

00:04:34.270 --> 00:04:37.570
And now we can add.

00:04:37.570 --> 00:04:39.280
And just like before you
might say, how can I

00:04:39.280 --> 00:04:40.610
add nothing to this 8?

00:04:40.610 --> 00:04:44.150
We can add a trailing 0 to this
top number because it really

00:04:44.150 --> 00:04:46.280
doesn't change the value of
the number and now it kind of

00:04:46.280 --> 00:04:47.340
gives us a sense of security.

00:04:47.340 --> 00:04:49.840
We have something
to add to the 8.

00:04:49.840 --> 00:04:50.900
So let's do that.

00:04:50.900 --> 00:04:53.910
0 plus 8 is 8.

00:04:53.910 --> 00:04:57.370
5 plus 2 is 7.

00:04:57.370 --> 00:05:01.280
7 plus 0 is 7.

00:05:01.280 --> 00:05:04.970
8 plus blank space is 8.

00:05:04.970 --> 00:05:06.720
And we could have added a 0
there and it would've been the

00:05:06.720 --> 00:05:08.290
same thing, had the leading 0.

00:05:08.290 --> 00:05:10.550
We wouldn't have given someone
the wrong about of medicine.

00:05:10.550 --> 00:05:14.150
5 plus blank space is 5.

00:05:14.150 --> 00:05:17.990
And we just drop down that
decimal point, and we're done.

00:05:17.990 --> 00:05:19.410
It's that straightforward.

00:05:19.410 --> 00:05:23.320
You line up the decimal points,
add any 0's that might make you

00:05:23.320 --> 00:05:25.510
feel more comfortable because
it's often more comfortable to

00:05:25.510 --> 00:05:28.720
add to a 0 than to add
to a blank space.

00:05:28.720 --> 00:05:30.610
And then you do your addition
and then you drop down

00:05:30.610 --> 00:05:31.260
the decimal point.

00:05:31.260 --> 00:05:34.370
The answer is 58.778.

00:05:34.370 --> 00:05:36.920
And it's always good to kind
of do a reality check.

00:05:36.920 --> 00:05:39.060
To say, well, does my
answer make sense?

00:05:39.060 --> 00:05:44.680
I have 58.75, so
roughly 58, almost 59.

00:05:44.680 --> 00:05:45.750
58 and 3/4.

00:05:45.750 --> 00:05:48.030
And I'm adding a very
small number to it.

00:05:48.030 --> 00:05:51.140
I'm adding 0.028.

00:05:51.140 --> 00:05:55.900
So my answer shouldn't
change much I guess is the

00:05:55.900 --> 00:05:56.600
way you could view it.

00:05:56.600 --> 00:05:59.890
It should still be around 58
something because I'm only

00:05:59.890 --> 00:06:01.650
adding a very small amount.

00:06:01.650 --> 00:06:04.640
If you had done it this way and
you had just started adding

00:06:04.640 --> 00:06:08.680
immediately, you would
have gotten 59.

00:06:08.680 --> 00:06:09.140
something.

00:06:09.140 --> 00:06:12.500
And you're like, boy, but I
only added 0.028 not 0.28.

00:06:12.500 --> 00:06:14.710
And you would've been a little
bit suspicious of your answer.

00:06:14.710 --> 00:06:17.690
I hope I didn't confuse you,
but it's really healthy to

00:06:17.690 --> 00:06:19.800
always think about the
magnitude of your numbers and

00:06:19.800 --> 00:06:23.770
get a feel for just what you
are adding and not always

00:06:23.770 --> 00:06:25.810
do it purely mechanical.

00:06:25.810 --> 00:06:30.200
Although sometimes it is fun
just to chug through things.

00:06:30.200 --> 00:06:33.880
Now let's do one more problem
just for good measure.

00:06:33.880 --> 00:06:35.605
I have some time
left on the video.

00:06:38.630 --> 00:06:48.070
102.1 plus 2.56.

00:06:48.070 --> 00:06:49.360
So once again, think
about magnitude.

00:06:49.360 --> 00:06:54.020
102.1 plus 2.56, that
should be like 104.

00:06:54.020 --> 00:06:54.720
something.

00:06:54.720 --> 00:06:55.580
I won't do it in my head.

00:06:55.580 --> 00:06:57.640
We'll do it on paper.

00:06:57.640 --> 00:06:59.460
So first thing, we line
up the decimal points.

00:06:59.460 --> 00:07:08.630
102.1 plus-- line up the
decimal points-- 2.56.

00:07:08.630 --> 00:07:11.170
Now we could add a trailing 0
here and we're ready to add.

00:07:11.170 --> 00:07:13.510
0 plus 6 is 6.

00:07:13.510 --> 00:07:15.760
1 plus 5 is 6.

00:07:15.760 --> 00:07:18.260
2 plus 2 is 4.

00:07:18.260 --> 00:07:20.000
0 plus nothing is 0.

00:07:20.000 --> 00:07:22.380
1 plus nothing is 1.

00:07:22.380 --> 00:07:24.120
Bring down that decimal point.

00:07:24.120 --> 00:07:26.330
104.66.

00:07:26.330 --> 00:07:27.400
It was that easy.

00:07:27.400 --> 00:07:28.050
Hopefully it's easy.

00:07:28.050 --> 00:07:30.480
I don't want to make you think
that it's easy if you're

00:07:30.480 --> 00:07:32.360
finding it hard because
it actually is hard the

00:07:32.360 --> 00:07:33.510
first time you do it.

00:07:33.510 --> 00:07:37.390
But anyway, I think you're
ready now to try the module

00:07:37.390 --> 00:07:38.500
on adding decimals.

00:07:38.500 --> 00:07:40.470
I hope you have fun.


Dividing decimals with hundredths example 3

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.560
Welcome to the presentation
on dividing decimals.

00:00:03.560 --> 00:00:05.620
Let's get started
with a problem.

00:00:05.620 --> 00:00:23.925
If I were to say how many times
does 0.28 go into 23.828?

00:00:23.925 --> 00:00:26.320
So you're going to see that
these dividing decimal problems

00:00:26.320 --> 00:00:28.480
are actually just like the
level 4 division problems.

00:00:28.480 --> 00:00:30.730
You just have to figure out
where to put the decimal.

00:00:30.730 --> 00:00:34.060
So what you do is you always
want to take this decimal and

00:00:34.060 --> 00:00:37.180
move it over as many spaces
as necessary to turn this

00:00:37.180 --> 00:00:39.380
number into a whole number.

00:00:39.380 --> 00:00:42.670
So in this case, we have to
move it one space, two spaces

00:00:42.670 --> 00:00:44.680
over to put the decimal here.

00:00:44.680 --> 00:00:47.230
Well, if I did that with
this number I have to do it

00:00:47.230 --> 00:00:48.350
with this number as well.

00:00:48.350 --> 00:00:50.595
So I moved it over two spaces
to the right, so I have

00:00:50.595 --> 00:00:54.370
to move this decimal two
spaces to the right-- 1, 2.

00:00:54.370 --> 00:00:56.950
Decimal goes here, and I
put the decimal right

00:00:56.950 --> 00:00:58.450
above, right there.

00:00:58.450 --> 00:01:03.570
Now I can treat this
28 as a whole number.

00:01:03.570 --> 00:01:07.460
And if I want to, let me see if
I could-- well, I want to erase

00:01:07.460 --> 00:01:09.210
the old decimal because if you
were doing it with a pen you

00:01:09.210 --> 00:01:11.540
would kind of have the
same problem I have.

00:01:11.540 --> 00:01:14.450
So now we do it just like a
level 4 division problem.

00:01:14.450 --> 00:01:17.950
So we say, how many times
does 28 go into 2?

00:01:17.950 --> 00:01:19.260
Well, no times.

00:01:19.260 --> 00:01:21.100
2 is smaller than 28.

00:01:21.100 --> 00:01:23.780
How many times does
28 go into 23?

00:01:23.780 --> 00:01:27.600
Once again, still, it goes
into it zero times because

00:01:27.600 --> 00:01:29.820
23 is smaller than 28.

00:01:29.820 --> 00:01:30.110
How Many.

00:01:30.110 --> 00:01:34.570
Times does 28 go into 238?

00:01:34.570 --> 00:01:36.680
So let's think about that.

00:01:36.680 --> 00:01:39.320
28 is almost 30.

00:01:39.320 --> 00:01:44.170
238 is almost 240.

00:01:44.170 --> 00:01:47.940
So 30 goes into 240 eight
times Because 3 goes

00:01:47.940 --> 00:01:49.630
into 24 eight times.

00:01:49.630 --> 00:01:54.480
So I'm going to guess that 28
goes into 238 eight times.

00:01:54.480 --> 00:01:55.690
And it literally it a guess.

00:01:55.690 --> 00:01:58.380
You have to try out some
numbers sometimes.

00:01:58.380 --> 00:02:00.250
8 times 8 is 64.

00:02:03.200 --> 00:02:05.620
8 times 2 is 16.

00:02:05.620 --> 00:02:07.260
Plus 2 is 22.

00:02:10.470 --> 00:02:11.590
Subtract.

00:02:11.590 --> 00:02:15.390
I get 14.

00:02:15.390 --> 00:02:19.410
I guessed right because the
remainder when I divide 28 into

00:02:19.410 --> 00:02:23.030
238 and I say it goes into
it eight times is 14,

00:02:23.030 --> 00:02:24.480
which is less than 28.

00:02:24.480 --> 00:02:28.740
So 8 was the largest number of
times that the 28 could go into

00:02:28.740 --> 00:02:31.120
238 without being larger.

00:02:31.120 --> 00:02:32.600
So now I bring down this 2.

00:02:32.600 --> 00:02:35.170
Once again, you recognize this
is just purely a level 2

00:02:35.170 --> 00:02:39.110
division problem-- a level
4 division problem.

00:02:39.110 --> 00:02:42.180
So now I say, how many
times does 28 go into 142?

00:02:42.180 --> 00:02:43.980
Well, once again, I'm
going to approximate.

00:02:43.980 --> 00:02:47.360
28, it's almost 30.

00:02:47.360 --> 00:02:50.900
Let's see, 30 times 4 is 120.

00:02:50.900 --> 00:02:53.400
So yeah, I'll take a guess
and I'll say let's say it

00:02:53.400 --> 00:02:54.490
goes into it four times.

00:02:54.490 --> 00:02:58.770
I could be wrong, but let's
see if it works out.

00:02:58.770 --> 00:02:59.980
Let me get rid of this old 6.

00:02:59.980 --> 00:03:03.710
4 times 8 is 32.

00:03:03.710 --> 00:03:06.960
And 4 times 2 is 8.

00:03:06.960 --> 00:03:09.440
Plus 3 is 11.

00:03:12.790 --> 00:03:14.480
2 minus 2 is 0.

00:03:14.480 --> 00:03:16.830
4 minus 1 is 3.

00:03:16.830 --> 00:03:17.410
Huh.

00:03:17.410 --> 00:03:18.600
Interesting.

00:03:18.600 --> 00:03:22.640
So it turns out that my
remainder here is larger than

00:03:22.640 --> 00:03:27.410
28, so I actually could have
divided 28 into 142

00:03:27.410 --> 00:03:28.860
one more time.

00:03:28.860 --> 00:03:31.450
So let me go back
and change that.

00:03:31.450 --> 00:03:34.130
See, it's not a
mechanical thing.

00:03:34.130 --> 00:03:37.330
And if you feel unsure
sometimes, you just have to try

00:03:37.330 --> 00:03:38.780
numbers and see if they work.

00:03:38.780 --> 00:03:41.360
And otherwise, you raise or
lower the number accordingly.

00:03:41.360 --> 00:03:43.410
So let me erase that 4.

00:03:46.480 --> 00:03:47.640
I'm going to try
not to mess up.

00:03:52.510 --> 00:03:54.130
Erase all this stuff down here.

00:03:56.790 --> 00:03:59.420
I probably should have tried it
out on the side first before

00:03:59.420 --> 00:04:01.230
doing all this and I wouldn't
have had to go back

00:04:01.230 --> 00:04:03.270
and erase it.

00:04:03.270 --> 00:04:06.510
And then let me get back
to what I was doing.

00:04:06.510 --> 00:04:08.800
So when I went into it four
times the remainder was too

00:04:08.800 --> 00:04:10.710
large, so let me try five now.

00:04:13.350 --> 00:04:17.590
5 times 8 is 40.

00:04:17.590 --> 00:04:18.890
5 times 2 is 10.

00:04:18.890 --> 00:04:19.910
Plus 4 is 14.

00:04:22.890 --> 00:04:25.530
142 minus 140 is 2.

00:04:25.530 --> 00:04:25.950
Good.

00:04:25.950 --> 00:04:27.900
2 is less than 28.

00:04:27.900 --> 00:04:29.200
This 5 is correct.

00:04:29.200 --> 00:04:32.910
Now I just bring down the 8.

00:04:32.910 --> 00:04:36.500
28 goes into 28
exactly one time.

00:04:36.500 --> 00:04:40.154
1 times 28 is 28.

00:04:40.154 --> 00:04:41.320
Remainder of 0.

00:04:41.320 --> 00:04:42.340
Done.

00:04:42.340 --> 00:04:49.610
So 28 goes into
2,382.8 85.1 times.

00:04:49.610 --> 00:05:03.550
Or you could say, 0.28 goes
into 23.828 85.1 times.

00:05:03.550 --> 00:05:05.300
That's the answer
we had gotten.

00:05:05.300 --> 00:05:06.540
And that makes sense.

00:05:06.540 --> 00:05:08.740
It's always good to do a
reality check because if I took

00:05:08.740 --> 00:05:14.070
85.1 and I multiplied it by
0.28, it makes sense that

00:05:14.070 --> 00:05:16.110
I'd get a number around 23.

00:05:16.110 --> 00:05:18.770
0.28 is almost 1/3.

00:05:18.770 --> 00:05:22.060
So 23 is almost 1/3 of 85.

00:05:22.060 --> 00:05:23.820
So at least it makes
sense in rough numbers.

00:05:23.820 --> 00:05:27.600
When you're doing decimals, if
I had gotten 800 here instead

00:05:27.600 --> 00:05:30.530
of 85, I'd be like, oh,
well, 0.28 times 800?

00:05:30.530 --> 00:05:32.820
I don't know if that equals 23.

00:05:32.820 --> 00:05:35.270
So it's always good to just do
a reality check and get a sense

00:05:35.270 --> 00:05:39.710
for at least the magnitude of
what your answer should be.

00:05:39.710 --> 00:05:40.660
Let's do another problem.

00:05:43.740 --> 00:05:58.130
Let's do 3.3 goes into 43.23.

00:05:58.130 --> 00:05:59.490
That's a 3.

00:05:59.490 --> 00:06:01.340
So first thing we want to
do is move the decimal.

00:06:01.340 --> 00:06:03.230
We just have to move it one
space here, so we move it

00:06:03.230 --> 00:06:04.870
once space here as well.

00:06:04.870 --> 00:06:06.300
Put the decimal right up here.

00:06:06.300 --> 00:06:09.200
And now it's just a level
4 division problem.

00:06:09.200 --> 00:06:13.480
33 goes into 4 zero times.

00:06:13.480 --> 00:06:16.880
33 goes into 43 one time.

00:06:16.880 --> 00:06:17.830
That's easy.

00:06:17.830 --> 00:06:21.680
1 times 33 is 33.

00:06:21.680 --> 00:06:22.540
Do the subtraction.

00:06:22.540 --> 00:06:25.400
43 minus 33 is 10.

00:06:25.400 --> 00:06:27.830
Bring down this 2.

00:06:27.830 --> 00:06:31.180
33 goes into 102?

00:06:31.180 --> 00:06:33.300
You could eyeball that one
and say, about three times

00:06:33.300 --> 00:06:34.740
because 3 times 33 is 99.

00:06:37.520 --> 00:06:41.240
3 times 33 is 99.

00:06:41.240 --> 00:06:42.570
102 minus 99?

00:06:42.570 --> 00:06:43.380
Well, that's easy.

00:06:43.380 --> 00:06:44.760
That's 3.

00:06:44.760 --> 00:06:48.440
We just bring down this 3.

00:06:48.440 --> 00:06:52.500
33 goes into 33 one time.

00:06:52.500 --> 00:06:53.720
1 times 33 is 33.

00:06:57.360 --> 00:06:58.330
0.

00:06:58.330 --> 00:07:06.010
So 3.3 goes into
43.23 13.1 times.

00:07:06.010 --> 00:07:08.470
Or, if you move the decimal
over, and when you move the

00:07:08.470 --> 00:07:11.230
decimal over to the right one
spot, all you're doing is

00:07:11.230 --> 00:07:16.940
you're multiplying both the
divisor and the dividend by 10.

00:07:16.940 --> 00:07:20.030
Which is fine as long as you
multiply both of them by 10.

00:07:20.030 --> 00:07:27.830
It's also like saying 33
goes into 432.3 13.1 times.

00:07:27.830 --> 00:07:28.830
Let's do one more problem.

00:07:28.830 --> 00:07:30.390
I think I have time.

00:07:30.390 --> 00:07:32.290
YouTube puts a limit
on this stuff.

00:07:32.290 --> 00:07:44.310
so let's say 2.5 goes into
0.3350 how many times?

00:07:44.310 --> 00:07:47.860
Well once again, let's move the
decimal point over one here.

00:07:47.860 --> 00:07:49.780
So we move the decimal
point over one here.

00:07:49.780 --> 00:07:51.660
Put the decimal here.

00:07:51.660 --> 00:07:54.140
So how many times
does 25 go into 3?

00:07:54.140 --> 00:07:55.580
Well 0.

00:07:55.580 --> 00:07:58.790
So you could put a 0 here
just for fun if you want.

00:07:58.790 --> 00:08:01.470
How many times does
25 go into 33?

00:08:01.470 --> 00:08:02.990
Well, it goes into it one time.

00:08:02.990 --> 00:08:06.950
1 times 25 is 25.

00:08:06.950 --> 00:08:09.120
33 minus 25 is 8.

00:08:09.120 --> 00:08:11.490
Bring down the five.

00:08:11.490 --> 00:08:12.730
25 goes into 85?

00:08:12.730 --> 00:08:16.440
Well, we know 25 times 3 is 75.

00:08:16.440 --> 00:08:18.920
So it'll go into
it three times.

00:08:18.920 --> 00:08:20.080
3 times 25.

00:08:20.080 --> 00:08:23.000
We know that's 75.

00:08:23.000 --> 00:08:26.040
85 minus 75 is 10.

00:08:26.040 --> 00:08:27.890
Bring down the 0.

00:08:27.890 --> 00:08:30.220
Up here we had brought
down the 5 before.

00:08:30.220 --> 00:08:33.300
And 25 goes into
100 four times.

00:08:33.300 --> 00:08:42.920
So our answer is 2.5 goes
into 0.3350 0.134 times.

00:08:42.920 --> 00:08:45.680
So as you see, the only
difference step between what

00:08:45.680 --> 00:08:49.360
we're doing when we're dividing
decimals and when we're doing

00:08:49.360 --> 00:08:51.510
level 4 division is we just
have to make sure we get the

00:08:51.510 --> 00:08:54.070
decimal in the right place.

00:08:54.070 --> 00:08:57.620
You shift the decimal here
enough so that this becomes a

00:08:57.620 --> 00:09:00.450
whole number and you just have
to shift the decimal here

00:09:00.450 --> 00:09:01.910
the same number of times.

00:09:01.910 --> 00:09:04.600
And once you do that it
just becomes a level

00:09:04.600 --> 00:09:05.950
4 division problem.

00:09:05.950 --> 00:09:09.080
And the whole trick with level
4 division is always be willing

00:09:09.080 --> 00:09:12.330
to try numbers, and if
the numbers don't work,

00:09:12.330 --> 00:09:13.240
adjust them accordingly.

00:09:13.240 --> 00:09:15.450
Don't feel that there should be
a way that you can just always

00:09:15.450 --> 00:09:16.380
power through these problems.

00:09:16.380 --> 00:09:18.320
You have to do a little bit of
trial and error and maybe use

00:09:18.320 --> 00:09:21.070
your eraser or do some work
on side every now and then.

00:09:21.070 --> 00:09:24.140
But anyway, I think you're
ready to do some dividing

00:09:24.140 --> 00:09:25.960
decimals problems.

00:09:25.960 --> 00:09:28.010
I hope you have some fun.


Level 4 division

WEBVTT
Kind: captions
Language: en

00:00:01.250 --> 00:00:05.640
Welcome to the presentation
on level 4 division.

00:00:05.640 --> 00:00:09.540
So what makes level 4 division
harder than level 3 division is

00:00:09.540 --> 00:00:13.690
instead of having a one-digit
number being divided into a

00:00:13.690 --> 00:00:15.700
multi-digit number, we're now
going to have a two-addition

00:00:15.700 --> 00:00:17.890
number divided into a
multi-digit number.

00:00:17.890 --> 00:00:21.540
So let's get started with
some practice problems.

00:00:21.540 --> 00:00:25.140
So let's start with what I
would say is a relatively

00:00:25.140 --> 00:00:26.210
straightforward example.

00:00:26.210 --> 00:00:27.595
The level 4 problems
you'll see are actually a

00:00:27.595 --> 00:00:28.300
little harder than this.

00:00:28.300 --> 00:00:40.320
But let's say I had
25 goes into 6,250.

00:00:40.320 --> 00:00:46.660
So the best way to think about
this is you say, OK, I have 25.

00:00:46.660 --> 00:00:48.520
Does 25 go into 6?

00:00:48.520 --> 00:00:49.300
Well, no.

00:00:49.300 --> 00:00:52.900
Clearly 6 is smaller than 25,
so 25 does not go into 6.

00:00:52.900 --> 00:00:55.990
So then ask yourself, well,
then if 25 doesn't go into

00:00:55.990 --> 00:00:58.990
6, does 25 go into 62?

00:00:58.990 --> 00:00:59.740
Well, sure.

00:00:59.740 --> 00:01:03.940
62 is larger than 25,
so 25 will go into 62?

00:01:03.940 --> 00:01:05.230
Well, let's think about it.

00:01:05.230 --> 00:01:07.370
25 times 1 is 25.

00:01:07.370 --> 00:01:10.800
25 times 2 is 50.

00:01:10.800 --> 00:01:13.180
So it goes into 62
at least two times.

00:01:13.180 --> 00:01:15.700
And 25 times 3 is 75.

00:01:15.700 --> 00:01:16.900
So that's too much.

00:01:16.900 --> 00:01:21.300
So 25 goes into 62 two times.

00:01:21.300 --> 00:01:23.820
And there's really no
mechanical way to go

00:01:23.820 --> 00:01:25.030
about figuring this out.

00:01:25.030 --> 00:01:27.170
You have to kind of think
about, OK, how many times do

00:01:27.170 --> 00:01:28.375
I think 25 will go into 62?

00:01:28.375 --> 00:01:29.440
And sometimes you get it wrong.

00:01:29.440 --> 00:01:30.560
Sometimes you'll
put a number here.

00:01:30.560 --> 00:01:33.180
Say if I didn't know, I
would've put a 3 up here and

00:01:33.180 --> 00:01:34.530
then I would've said 3 times
25 and I would've

00:01:34.530 --> 00:01:35.930
gotten a 75 here.

00:01:35.930 --> 00:01:37.640
And then that would have been
too large of a number, so

00:01:37.640 --> 00:01:39.840
I would have gone back
and changed it to a 2.

00:01:39.840 --> 00:01:44.870
Likewise, if I had done a 1 and
I had done 1 tmes 25, when I

00:01:44.870 --> 00:01:46.580
subtracted it out, the
difference I would've gotten

00:01:46.580 --> 00:01:48.290
would be larger than 25.

00:01:48.290 --> 00:01:50.150
And then I would know
that, OK, 1 is too small.

00:01:50.150 --> 00:01:51.970
I have to increase it to 2.

00:01:51.970 --> 00:01:53.490
I hope I didn't
confuse you too much.

00:01:53.490 --> 00:01:56.060
I just want you to know that
you shouldn't get nervous if

00:01:56.060 --> 00:01:58.520
you're like, boy, every time I
go through the step it's kind

00:01:58.520 --> 00:02:01.170
of like- I kind of have to
guess what the numbers is as

00:02:01.170 --> 00:02:02.800
opposed to kind of a method.

00:02:02.800 --> 00:02:04.540
And that's true; everyone
has to do that.

00:02:04.540 --> 00:02:08.650
So anyway, so 25 goes
into 62 two times.

00:02:08.650 --> 00:02:10.480
Now let's multiply 2 times 25.

00:02:10.480 --> 00:02:13.840
Well, 2 times 5 is 10.

00:02:13.840 --> 00:02:18.740
And then 2 times 2 plus 1 is 5.

00:02:18.740 --> 00:02:21.860
And we know that 25
times 2 is 50 anyway.

00:02:21.860 --> 00:02:23.020
Then we subtract.

00:02:23.020 --> 00:02:24.970
2 minus 0 is 2.

00:02:24.970 --> 00:02:27.630
6 minus 5 is 1.

00:02:27.630 --> 00:02:31.000
And now we bring down the 5.

00:02:31.000 --> 00:02:33.400
So the rest of the mechanics
are pretty much just like a

00:02:33.400 --> 00:02:35.540
level 3 division problem.

00:02:35.540 --> 00:02:41.250
Now we ask ourselves, how many
times does 25 go into 125?

00:02:41.250 --> 00:02:44.720
Well, the way I think about it
is 25-- it goes into 100 about

00:02:44.720 --> 00:02:47.750
four times, so it will go
into 125 one more time.

00:02:47.750 --> 00:02:49.560
It goes into it five times.

00:02:49.560 --> 00:02:51.650
If you weren't sure you could
try 4 and then you would see

00:02:51.650 --> 00:02:53.110
that you would have
too much left over.

00:02:53.110 --> 00:02:57.540
Or if you tried 6 you would see
that you would actually get 6

00:02:57.540 --> 00:03:00.020
times 25 is a number
larger than 125.

00:03:00.020 --> 00:03:02.380
So you can't use 6.

00:03:02.380 --> 00:03:07.520
So if we say 25 goes into 125
five times then we just

00:03:07.520 --> 00:03:10.270
multiply 5 times 5 is 25.

00:03:13.630 --> 00:03:18.090
5 times 2 is 10 plus 2, 125.

00:03:18.090 --> 00:03:19.400
So it goes in exact.

00:03:19.400 --> 00:03:21.922
So 125 minus 125 is clearly 0.

00:03:21.922 --> 00:03:25.150
Then we bring down this 0.

00:03:25.150 --> 00:03:28.270
And 25 goes into 0 zero times.

00:03:28.270 --> 00:03:29.430
0 times 25 is 0.

00:03:29.430 --> 00:03:30.990
Remainder is 0.

00:03:30.990 --> 00:03:39.500
So we see that 25 goes into
6,250 exactly 250 times.

00:03:39.500 --> 00:03:40.460
Let's do another problem.

00:03:45.820 --> 00:03:50.820
Let's say I had-- let me
pick an interesting number.

00:03:50.820 --> 00:03:56.170
Let's say I had 15 and I
want to know how many

00:03:56.170 --> 00:04:05.710
times it goes into 2,265.

00:04:05.710 --> 00:04:07.330
Well, we just do the same
thing we did before.

00:04:07.330 --> 00:04:09.780
We say OK, does 15 go into 2?

00:04:09.780 --> 00:04:10.710
No.

00:04:10.710 --> 00:04:12.910
So does 15 go into 22?

00:04:12.910 --> 00:04:13.430
Sure.

00:04:13.430 --> 00:04:16.250
15 goes into 22 one time.

00:04:16.250 --> 00:04:18.220
Notice we wrote the
1 above the 22.

00:04:18.220 --> 00:04:20.800
If it go had gone into 2 we
would've written the 1 here.

00:04:20.800 --> 00:04:23.260
But 15 goes into 22 one time.

00:04:23.260 --> 00:04:25.430
1 times 15 is 15.

00:04:28.070 --> 00:04:35.000
22 minus 15-- we could do the
whole carrying thing-- 1, 12.

00:04:35.000 --> 00:04:36.380
12 minus 5 is 7.

00:04:36.380 --> 00:04:37.850
1 minus 1 is 0.

00:04:37.850 --> 00:04:40.120
22 minus 15 is 7.

00:04:40.120 --> 00:04:43.190
Bring down the 6.

00:04:43.190 --> 00:04:46.980
OK, now how many times
does 15 go into 76?

00:04:46.980 --> 00:04:50.330
Once again, there isn't a real
easy mechanical way to do it.

00:04:50.330 --> 00:04:53.210
You can kind of eyeball
it and estimate.

00:04:53.210 --> 00:04:55.530
Well, 15 times 2 is 30.

00:04:55.530 --> 00:04:58.490
15 times 4 is 60.

00:04:58.490 --> 00:05:01.510
15 times 5 is 75.

00:05:01.510 --> 00:05:05.500
That's pretty close, so let's
say 15 goes into 76 five times.

00:05:08.380 --> 00:05:11.820
So 5 times 5 once again, I
already figured it out in

00:05:11.820 --> 00:05:13.760
my head, but I'll
just do it again.

00:05:13.760 --> 00:05:15.200
5 times 1 is 5.

00:05:15.200 --> 00:05:17.920
Plus 7.

00:05:17.920 --> 00:05:20.740
Oh, sorry.

00:05:20.740 --> 00:05:22.210
5 times 5 is 25.

00:05:25.420 --> 00:05:26.620
5 times 1 is 5.

00:05:26.620 --> 00:05:27.425
Plus 2 is 7.

00:05:30.030 --> 00:05:32.040
Now we just subtract.

00:05:32.040 --> 00:05:34.640
76 minus 75 is clearly 1.

00:05:34.640 --> 00:05:37.770
Bring down that 5.

00:05:37.770 --> 00:05:42.090
Well, 15 goes into 15
exactly one time.

00:05:42.090 --> 00:05:44.330
1 times 15 is 15.

00:05:47.600 --> 00:05:50.340
Subtract it and we get
a remainder of 0.

00:05:50.340 --> 00:05:57.410
So 15 goes into 2,265
exactly 151 times.

00:05:57.410 --> 00:06:01.370
So just think about what we're
doing here and why it's a

00:06:01.370 --> 00:06:04.100
little bit harder than when you
have a one-digit number here.

00:06:04.100 --> 00:06:05.950
Is that you have to kind of
think about, well, how many

00:06:05.950 --> 00:06:08.580
times does this two-digit
number go into this

00:06:08.580 --> 00:06:09.580
larger number?

00:06:09.580 --> 00:06:13.480
And since you don't know
two-digit multiplication

00:06:13.480 --> 00:06:15.770
tables-- very few people do--
you have to do a little

00:06:15.770 --> 00:06:16.440
bit of guesswork.

00:06:16.440 --> 00:06:18.500
Sometimes you can look at this
first digit and look at the

00:06:18.500 --> 00:06:20.600
first digit here and
make an estimate.

00:06:20.600 --> 00:06:21.800
But sometimes it's
trial and error.

00:06:21.800 --> 00:06:23.990
You'll try and when you
multiply it out you might get

00:06:23.990 --> 00:06:25.510
it wrong on the first try.

00:06:25.510 --> 00:06:27.690
Let's do another problem.

00:06:27.690 --> 00:06:29.600
And actually, I'm going to pick
numbers at random, so it might

00:06:29.600 --> 00:06:31.020
not have an easy remainder.

00:06:31.020 --> 00:06:32.300
But I think you'll
get the point.

00:06:32.300 --> 00:06:34.800
I won't teach you decimals
now, so I'll just leave the

00:06:34.800 --> 00:06:36.590
remainder if there is one.

00:06:36.590 --> 00:06:49.580
Let's say I had 67
going into 5,978.

00:06:49.580 --> 00:06:52.220
So I just picked these numbers
randomly out of my head, so

00:06:52.220 --> 00:06:55.790
I'll show you that I also
sometimes have to do a little

00:06:55.790 --> 00:06:57.680
bit of guesswork to figure out
how many times one of these

00:06:57.680 --> 00:07:00.350
two-digit numbers go
into a larger number.

00:07:00.350 --> 00:07:03.560
So 67 goes into 5 zero times.

00:07:03.560 --> 00:07:06.620
67 goes into 59 zero times.

00:07:06.620 --> 00:07:13.440
67 goes into 597--
so let's see.

00:07:13.440 --> 00:07:19.880
67 is almost 70 and
597 is almost 600.

00:07:19.880 --> 00:07:27.850
So if it was 70 goes
into-- 70 times 9 to 630.

00:07:27.850 --> 00:07:29.820
Because 7 times 9 is 63.

00:07:29.820 --> 00:07:33.110
So I'm going to just
eyball approximate.

00:07:33.110 --> 00:07:35.060
I'm going to say that it
goes into it eight times.

00:07:35.060 --> 00:07:35.950
I might be wrong.

00:07:38.550 --> 00:07:40.980
And you can always check, but
well, we're going to actually

00:07:40.980 --> 00:07:42.930
check in this step essentially.

00:07:42.930 --> 00:07:44.620
8 times 7-- well that's 56.

00:07:47.820 --> 00:07:51.160
And then 8 times 6 is 48.

00:07:51.160 --> 00:07:52.900
Plus 2 is 53.

00:07:57.330 --> 00:07:59.310
7 minus 6 is 1.

00:07:59.310 --> 00:08:01.740
9 minus 9 is 6.

00:08:01.740 --> 00:08:03.560
5 minus 5 is 0.

00:08:03.560 --> 00:08:04.650
61.

00:08:04.650 --> 00:08:05.170
So good.

00:08:05.170 --> 00:08:08.240
I got it right because if I got
a number here that was larger

00:08:08.240 --> 00:08:13.560
than-- 67 or larger, than that
means that this number up

00:08:13.560 --> 00:08:15.050
here wasn't large enough.

00:08:15.050 --> 00:08:17.770
But here, I got a number
that's positive because

00:08:17.770 --> 00:08:20.530
536 is less than 597.

00:08:20.530 --> 00:08:24.070
And it's less than 67, so
I did that step right.

00:08:24.070 --> 00:08:28.250
So now we bring down this 8.

00:08:28.250 --> 00:08:30.760
Now this one might be a little
bit trickier this time.

00:08:30.760 --> 00:08:35.500
Once again, we have almost 70
and here we have almost 630.

00:08:35.500 --> 00:08:37.990
So maybe it will go
into it 9 times.

00:08:37.990 --> 00:08:40.610
Well, let's give it a
try and see if it does.

00:08:43.780 --> 00:08:46.215
9 times 7 is 63.

00:08:49.340 --> 00:08:51.720
9 times 6 is 54.

00:08:51.720 --> 00:08:53.870
Plus 6 is 60.

00:08:53.870 --> 00:08:54.700
Good.

00:08:54.700 --> 00:08:56.610
So it did actually go into
it nine times because

00:08:56.610 --> 00:08:59.480
603 is less than 618.

00:08:59.480 --> 00:09:01.730
8 minus 3 is 5.

00:09:01.730 --> 00:09:04.580
1 minus 0 is 1.

00:09:04.580 --> 00:09:06.970
And 6 minus 6 is 0.

00:09:10.420 --> 00:09:13.480
We have a remainder of 15,
which is smaller than 67.

00:09:13.480 --> 00:09:15.420
So I'm not going to teach you
decimals right now, so we can

00:09:15.420 --> 00:09:16.970
just leave that remainder.

00:09:16.970 --> 00:09:23.690
So what we could say is that
67 goes into 5,978 89 times.

00:09:23.690 --> 00:09:25.980
And when it goes into it 89
times, you're left with

00:09:25.980 --> 00:09:28.670
a remainder of 15.

00:09:28.670 --> 00:09:31.850
hopefully you're ready
now to try some level

00:09:31.850 --> 00:09:33.560
4 division problems.

00:09:33.560 --> 00:09:35.130
Have fun.