Welcome again guys! To the second video on
our Prep For Your University Test and today
we will continue with the review of the Fundamental
Concepts of Algebra. Remmember, if you have
any doubts, please write them down in the
comment section.
We will start with the Real Number Line but
for that, first we may know what is a number
line and the different ones that exist. A
Number Line is a horizontal line which describes
a set by containing its ordered members and
this set must be a number set. At the left
we usually find the negative members and the
line extends up to negative infinity. At the
right we find the positive members and it
extends up to infinity. At the middle is the
zero or also called, the origin.
For example, let us build the Natural Number
Line. First we must know which are the members
that conform this set. If we remember the
Natural Set's notation, we see that only the
whole positive numbers conform this set, along
with the zero most of the cases; for know
let´s just assume the zero is always part
of the Natural Numbers. Its number line will
just have the members implied in the notation
and to identify this we put a circle on them,
as if we were filling the line.
For the Integer Number Line we revisit its
notation and we see that it is conformed by
the positive and negative whole numbers, along
with the zero. We continue with the same process
we did with the Naturals and as you can see
there are more members in the Integer Numbers
Set than in the Natural Numbers Set and we
didn´t need to write a lot of numbers. Later
in this video you will see that there is no
need to write all the numbers but you must
always take into account the order, because
it is a fundamental part for you to understand
the inequalities.
For the Rational Number Line we see that it
gets denser due to the fact that all the available
quotients of two integers other than zero
are filling the line even more. Though, little
holes are found along this number line, where
the numbers that cannot be described as the
quotient of two integers are. These are...yes!
the irrational numbers. Before we move to
the Irrational Numbers Line you must notice
that with the Rational Number Line we have
somehow proven that there are more rational
numbers than Integer Numbers because between
two consecutive Integers there are infinite
Rational numbers. Why? Well, if we imagine
the space between these two integers as a
cake then we theoretically can cut it into
very thin pieces; as thin as one millionth
wide and even less. As a note, remember that
the bigger the denominator of a quotient is,
the smaller the number it represents. It takes
some time for some people to wrap their heads
around all of this but stay with me.
For the Irrational Number Line things are
kind of different; we know that the irrational
numbers will fill the holes that we mentioned
in the Rational Number Line, the trick is
knowing were are those holes. For now, you
should know that between any two rational
numbers there are an infinite number of irrational
numbers, and to prove this you need more advanced
knowledge about Maths. The latter proves that
there are more irrational than rational numbers;
this means that the holes we are looking for
are so tiny and dense that it would be kind
of difficult to graph the Irrational Numbers
Line because at first glance you would thing
is the whole line itself. So let´s just graph
the most prominent Irrational Numbers.
From 0 to 4 we could say that the most prominent
irrationals are the square root of 2, Phi,
square root of 3,5,6,7, Euler's constant,
square root of 8; Pi square root 10,11,12,13,14,15.
And their respective negatives as well.
Finally the Real Number Line is the easiest
to graph. We should recall that the Real Numbers
Set contains all the previously discussed
Number Sets, thus, if we were to graph the
line it would be the line itself. Infinite
and continuous from the origin to both sides.
Anonther important property of real numbers
is that they are ordered.
Now let´s disccuss inequalities, which are
the relations made between two numbers or
mathematical expressions that makes a non-equal
comparison. Earlier I said that the order
is an important part of inequalities and let's
see why. We have two numbers, a and b which
are located on the real number line like this.
If you notice a it is at the left of b that
means that it is less than b and it can be
denoted by the inequality a<b. On the Real
Number line all the numbers, such as n1 that
are at the left of some other number lets
say n0 will be less than it, despite wherever
n0 is located. And all the numbers at the
right will be greater than it.
For example, if we have the number 4 all the
numbers at its left are smaller; 1 is less
than 4, negative 5 is less than 4 and all
the numbers at the right of 4 are bigger;
7 is greater than 4 as well as 64. So as you
see, the order is very important to understand
inequalities.There are different types of
inequalities, there are the strict and non-strict
ones; the strict ones are the ones we have
already practiced; the symbols involved are
the "greater-than sign" and the "less-than
sign". These are called strict because if
we say that b>a and let's say "a" equals to
12 then b can take the value of every real
number at the right of 12 but not 12. On the
other hand we have the non-strict inequalities
where the symbols involved are the "greater-than-or-equal-to
sign" and "less-than-or-equal-to sign". Now,
if we have the inequality b<=a and again "a"
takes the value of 12 then it means that b
can take the value of every real number at
the left of 12 including the 12 itself.
Another inequality we may found is the one
that uses the "different-to sign" or "non-equal
to sign" and it is a strict one because it
would mean that a and b may take whichever
value we want except the same, this also implies
that it is the only type of inequality where
the order of the numbers is not important.
The inequalities can be simple, double, mixed
double and compound. All the ones we have
written before are simple. Don't get scared
with all these variations, you just need to
look at the signs that are being used and
remmember how the numbers would be ordered
on the Real Number Line with each one of them.
Anyway, don´t worry; soon, we'll get deeper
into the inequalities due to being a very
important part of pre-calculus. An example
of a double inequality may be 2<x<7 and it
means that 2 is less than the variable x and
at the same time x is less than 7 so we can
assume that x may take any value between 2
and 7 but those two numbers. The double inequalities
are used to describe intervals of numbers
which may be subsets of a bigger set of numbers
such as the Real Numbers Set. As an extra
and last commentary on inequalities, for now;
is this, the Law of Trichotomy tells us that
for any two real numbers a and b precisely
one of three relations are possible, "a" could
be equal or greater or less than "b". You
don't have to remmember the name but believe
me, sometimes taking these little laws into
account, come in handle from time to time.
All of this, clear us the path to talk about
intervals. We know that a set of numbers may
be infinite as it is the case of all the sets
we talked about last video, but it also may
be not infinite; for example the next set
A has only 5 members. Now, the intervals have
uncountable members but can be of definite
length; for example (-5,3) contains all the
real numbers between -5 and 3 but not those
two numbers.
An interval is a set of real numbers that
contains all real numbers between any two
numbers of the set which naturally are the
endpoints of the interval. There are different
types of intervals.
We have different types of intervals. First,
we have the Bounded ones, this means that
these intervals are of definite length and
both endpoints are open or closed. The Closed
ones are denoted with brackets and this means
that "x" can take the value of "a" and "b"
and any Real Number between those two values.
The double inequality that describes this
type of interval is written as this, with
the non-strict types of inequalities. And
finally we have the opened ones that are noted
with parentheses or with inverted brackets,
which is a French notation and this type of
interval means that x can take the value of
any Real Number between "a" and "b" but not
those two values. Then, we have a special
type of intervals called the Mixed ones, which
can be Right-Closed or Left-Closed. First
the right-closed which means the endpoint
at the right is denoted with bracket and at
the left is denoted with parentheses, then
"x" can take the value of "b" but not of "a"
and also may take the value of any Real Number
between those two. We must say that the Mixed
Bounded Intervals are of definite length but
one endpoint is closed and the other open
or viceversa. The Left-Closed is denoted like
this andthis means that "x" can take the value
of "a" but not of "b", and may take any value
between "a" and "b".
Ok, let's see the Unbounded intervals, which
are those of indefinite length, or of a single
endpoint. There are the Left-Bounded, are
denoted as follows; they can be left-closed
or left-open, if they are left-closed then
we use a bracket next to the "a" and this
means that x can take any value at the right
of "a" and "a" also. Then wehave the left-open
which are denoted with the parentheses and
this means that x can take any value at the
right of a but not "a". The other type of
Unbounded intervals are the Right-Unbounded
ones which can be written like these ones;
they can be right-closed and right-open also.
The Right Closed means that the right endpoint
is denoted with brackets and this means that
x can take the value of "b" and every value
at the left of "b", then, the right-open are
denoted with parentheses on both sides and
this means that x cannot take the value of
"b" and can take any value at the left of
b. At last but not least, let's talk about
the other type of intervals. We will write
them on interval notation set notation and
inequality notation, we will take into account
again that "a" is less than "b" so "a" is
located at the left of "b". For the empty
set which has no member we have differrent
notations on interval notation, the set notation
is that type of zero or the empty braces,
and there is no inequality notation.
Then, we have the degenerate interval, which
contains juyst one member, which in this case
is "a" so x can only take the value of "a",
so we have the equality x equals "a".
Finally we have the Unbounded at both ends,
which is described by that interval, from
negative infinity up to infinity, this means
can take any value from right to left. This
is how we call the real number set, unbounded
at both ends interval.
The last topic for this video is the Absolute
Value, its properties and Distance. First,
you should know that the absolute value of
a real number is its magnitude or distance
between the origin and the point representing
said real number on the real number line.
To understand this better let´s draw the
real number line and choose 5 as our desired
real number, which its located here, so this
is the point representing the number on the
line; now, what is the distance between the
origin an it? Well it is 5 and if we chose
-5 then the distance from the origin would
still be 5, so for any positive number "a"
its absolute value would be "a" and if "a"
were negative its absolute value but be -a
so when we substitute the value of a it turns
out to be positive.
Using the example of -5 we take its absolute
value, due to -5 is less than 0 then we apply
the absolute value definition being -(-5)
and finally we get 5. So now we know that
"For every number a that belongs to the REal
Numbers the absolute value of said number
is Real positive but never zero unless it
is 0 which is the ony real number that its
absolute value is 0."
Some properties of the absolute value are
the following, we have that |a|>=0 always,
so |-a| is equivalent to |a|, this second
property is called evenness. The third property
is that |ab| is as if we took the absolute
value of both numbers individually and then
multiply them. It happens the same with the
absolute value of a division which is our
fourth and last property.
Let's end this video with the method to get
the distance between two points on the real
number line which is denoted by the next equation
an it reads as "the distance between a and
b is given by the absolute value of b minus
"a" or "a" minus b, at the end it doesn't
matter if b is greater or less than "a" because
the absolute value will always be positive
as it is the distance.
Then, something you must take into accountis
that the notation d(a,b) is specifically referring
to a distance between a pair of ordered points
"a" and "b". Don't confuse with the notation
for a Bounded Open Interval.
Ok guys! that was everything for this video.
Yes, I know I said we would talk about the
algebraic expressions, though this topics
extended more than I intended to and it would
get very tedious for you. On the next video
we will talk about them and the Basic Rules
of Algebra, then we will talk about exponent
and radicals before entering into Polynomials
and Factoring on the fourth video. I should
remind you that this series of videos won´t
only consist of theory but of Practice and
Excercises as well; I will be telling you
the order you should watch both so you have
the bases to understand and correctly answer
the excercises. Even though I will be explaining
them.
Don´t forget to like the video, subscribe,
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you don't miss any video, never forget that
even in the most unexpected of times, doubts
will be cleared and knowledge that may now
seem complex will be easily acquired. Until
the next time!!!
