The Real Numbers
In this book you will learn about The Real Numbers. This system is made up of many different types of numbers:
- The Natural Numbers
- The Whole Numbers
- The Integers
- The Rational Numbers
- The Irrational Numbers
Have you ever played hide and seek?
If you have, you probably closed your eyes, and counted like this: "1, 2, 3, 4, 5, ... "
Anytime you count this way, you are using the Natural Numbers.
These numbers are called "natural" because they are the first ones we learn to count when we are little.
Now based on what you know about the Natural Numbers, what do you think about the number zero? Could the number zero be classified as a Natural Number?
Suppose you and a friend ate an entire pie. How much pie is left in the tin?
The answer is: Nothing.
When humans were first beginning to use numbers they used them to represent things in their daily lives. Unfortunately, the Natural Numbers did not provide a way for them to represent "nothing" - so they had to create a new number.
This number is called zero. Now, these early humans could count, starting at zero: "0, 1, 2, 3, 4, 5, ..."
When we count in this way, we are using the Whole Numbers. You can remember this name easily because zero looks like a number with a hole in it.
Any time you pay a bill or purchase an item, you are spending money. This means you have less money than you started with - people consider this a "negative transaction".
Unfortunately, the Natural Numbers and the Whole Numbers cannot be used to represent these "negative transactions" - so humans had to create a new type of number. We call them "negative numbers."
The "negative numbers" along with the Whole Numbers form a new set of numbers called the Integers.
Integers allow us to count like this: "...-3, -2, -1, 0, 1, 2, 3, ..."
What happens if we have a fraction? Based on what you have just learned, can the number 1/2 be classified as an Integer?
Let us briefly summarize what we have learned so far:
The Natural Numbers
- Count beginning at one: "1, 2, 3, 4, 5..."
The Whole Numbers
- Are formed by combining the number "zero" and the Natural Numbers.
- Count beginning at zero: "0, 1, 2, 3, 4, 5, ..."
The Integers
- Are formed by combining the "negative numbers" and the Whole Numbers.
- Represent a loss or a gain of an entire item.
- Count like this: "...-3, -2, -1, 0, 1, 2, 3, ..."
Now suppose we have a fraction? Can the number 1/2 be classified as a Natural Number, Whole Number, or Integer?
Would you say that the glass above is about half-full or about half-empty?
Regardless of how you look at the glass, we can all agree that it is not entirely filled. In other words, we only have a "fractional unit."
Unfortunately, the Natural Numbers, Whole Numbers, and the Integers can not be used to represent these "fractional units" - so we need to introduce a new type of number - the Rational Numbers.
Rational Numbers are simply any Integer that can be written as a fraction.
This means that we can have both positive and negative Rational Numbers.
We count Rational Numbers like this: "..., -1/2, -1/4, 0, 1/4, 1/2, ..."
Based on what you have just learned, can the number 2 be classified as a Rational Number?
Before we proceed any further, we must understand that the Natural Numbers, Whole Numbers, and the Integers can all be classified as Rational Numbers.
The Natural Numbers:
- Each number is an Integer.
- Each number can be written as a fraction.
- "1/1, 2/1, 3/1, 4/1, 5/1, ...."
The Whole Numbers
- Each number is an Integer.
- Each number can be written as a fraction.
- "0/1, 1/1, 2/1, 3/1, 4/1, 5/1, ..."
The Integers
- Each Integer can be written as a fraction.
- "...-2/1, -1/1, 0/1, 1/1, 2/1, ..."
If we were recording how much a plant grows in a week, we might say it grows about six inches. Others might measure more precisely and say about 6.4 inches. Others still might measure and say about 6.47 inches.
Unfortunately, when we measure objects, the precision of our instrument is not always very finely accurate. Even if we had the best measuring device in the world our measurements would be only approximations - the actual amount that the plants grow in a week would be represented by a decimal that never ends.
For decimals that never end, or for decimals that do not ever have any pattern, we have to create a new type of number - the Irrational Numbers.
The Irrational Numbers do not share any of the properties of the Natural Numbers, Whole Numbers, Integers, or Rational Numbers.
Some famous examples of Irrational numbers are the number "e", the number "pi", and the square-root of two.
If we placed all of the Rational and Irrational numbers in order in a horizontal row, we create what is called a Number Line.
Number Lines are a good representation of all of the numbers that we have learned about in this book - it extends out to negative-infinity (on the left) and positive-infinity (on the right).
The further left you are on the Number Line, the more negative a number is becomming.
Likewise, the further right you are on the Number Line, the more positive a number is becomming.
Theorem
- Between any two Rational Numbers on the Number Line, there will always be at least one Irrational Number.
To think about why the theorem is true, suppose you have two bricks (each one represents a Rational Number). Then no matter how close you get these two bricks together, you can always get something very thin between them, like a peice of hair (the peice of hair represents an Irrational Number).
This theorem is very important, because if it wasn't true, then we could have holes in our Number Line.
As we just learned, a Number Line is a representation of the number system that is formed when we combine the Rational Numbers and the Irrational Numbers.
This number system that is formed is called the Real Numbers. It includes all of the numbers that you will work with until you get to about Algebra 2.
At that point, we will need to introduce a new number system, the Complex Numbers... But until then, I encourage you to continue to study and if you have any further questions about the Real Numbers, ask your teacher or search the Internet.