Exploring the Real Number Line

The Real Number Line

In this lecture, we will see the first example of an infinite set. It is one that pervades
all of data science, the idea of the real number line. In this lecture, we will review
fundamental concepts such as positive numbers and negative numbers, nonnegative
numbers and nonpositive numbers, and absolute value.
The Real Number Line is the concept of a line that has infinite points on it. You may
have heard the term real number before.

Real numbers are numbers that can be
expressed as a decimal (or fraction). For example, 5, 6, 7 and -8 are all real
numbers. But O or m are not real numbers because they cannot be expressed as
decimals or fractions. The Real Number Line is simply a collection of all the possible
points in this infinitely long continuous line.
The real numbers are represented by the number line. When we graph them, we plot
them on an extension of the number line that extends from positive infinity on the
right to negative infinity on the left.

We define the set of real numbers, R. as the
infinite set containing all rational and irrational numbers. It is said that every real
number can be represented as an infinite sequence of digits, each digit being either
O or 1. Now, we will mark some numbers we know with an equal sign. The equal sign
means that the sequence of digits continues infinitely in both directions. So here are
some integers: 0, 1, 2, 3, 4, 5
The real number line includes all numbers between and including the integers, but
not every number on the real number line is an integer.

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In fact, most aren't. If we
blow up this little stick between one and two, we can see this more clearly.
The only thing you need to know is that every single real number can be obtained by
adding together an integer and any number of integers that follow it. For example,
1.1 is a real number. So are the numbers 1.4 and 1.1538. And anything else that you
can create by continuing the sequence of integers forever is also a real number. This
iS true for any subinterval of the real numbers because no matter how small or large
a subinterval may be, if you blow it up enough to see all of its possible values, you
will be able to see every single real number within it somewhere.
[Graph]
As you already may know, there are real numbers such as pi. Pi is approximately
3.14, and it goes on forever without any pattern or repetition. There are also
numbers that do repeat, like 2.353199258, which is called a repeating decimal. We
will not discuss that type of number here because it is not relevant to the main point
of this paper. The take-home message is that a real number is any number on the
real line; there are an infinite amount of them, with some being positive and some
being negative.
So an example of a positive real number might be 5.3 might be 0.001. For example
the negative real number right here might be negative 11.7 if we include zero. So the
positive reals but including zero we often write the non negative reals- and if we
include 0 on the other side we go from right, non positive reals.
Let us draw the real number line again, using increasingly straight lines. It is
important to note that numbers come in pairs: a positive number and its negative
counterpart. For example, here we have 0: the positive version of this number is +0.
And here we have 7.1; its negative counterpart is -7.1. Likewise, 10 has a friend on
the other side called -10. Now notice 7.1 is not equal to -7.1. 10 is not equal to -10.
However, 7.1 and -7.1 have one important thing in common, which is that they have
the same distance to zero. The distance from here to here is about 7.1, is exactly
7.1. And the distance from here to here is 7.1. There's a concept called absolute
value. Let's define that.
The absolute value of a real number is the distance from 0 to that number, which we
denote by X. Notice that the absolute value symbol looks like the definition of
cardinality for sets, which is unfortunate.
[Graph]
We will notice over here that the absolute value of 7.1 is 7.1 and the absolute value
of negative 7.1 is 7.1, which by the way is the same thing as saying that negative 7.1
is positive 7.1, or -7.1 is +7.1. That's not just a huge little formula that we all know:
negative times negative equals positive. But it allows us to make a general definition
through what is known as a definition by cases: For any real number in R (that is,
any x in R), the following is true: The absolute value of an x can be one of two things:
It can be equal to plain old x if x is nonnegative (that's O or greater than 0). But it can
also be equal to negative x if x is negative (that's less than 0).
Let's check if that's true, and while we check this, this'll sort of show us how to parse
the definition by cases.
So let's compute the absolute value of 8.7. So according to this definition, 8.7 is our
It's either going to be 8.7 or negative 8.7. But which case happens? 8.7 is
non-negative so we're in this case up here. So this is just equal to 8.7. And that's
true.
The distance between -1 and -10 is 10. Let's check another example: the distance
between 8.7 and -1. If x = 8.7, then by our definition d(x) = -1. The distance between
8.7 and -1 is negative 10, because this formula tells us to take the negative of any
number whose absolute value is greater than 0, which means it's greater than or
equal to 0. In this case, we want to take the absolute value of 8.7, which is 9. This
gives us a result of 9; therefore, the distance between 8.7 and -1 is 9 units away
from zero on the number line.
General Rule-for any XєR
|x|={x=if x is non-negative; -x=if is negative}
Check 18,71=8,7
1-101=(-10)=10
[Graph]