# Real Numbers

Welcome back to the school everyone! I hope you all had a great summer. For all those whose summer was maybe a little too great, maybe those who’ve forgotten even the basics, we’re going to take it all the way back to arithmetic a.k.a “number theory”.

A review of number theory is a perfect place to start for many levels. Calculus and a lot of what you learn in pre-calculus is based on the real number system.

When we use the word “number” we are typically referring to all real number. But how can numbers be “real”? You can’t touch the number 6 or smell 1,063. You can’t boil 1/2 or stick it in a stew. So what’s so real about real numbers? The simple answer is this: a real number is a point on a number line (1).

-2.5        -1      0       1  √2           pi
____|____|____|____|____|____|____|____|____|____

TYPES OF REAL NUMBERS:

Natural Numbers, a.k.a. “Counting Numbers”:
Natural numbers include all positive integers. One way to think about them is that they are numbers we’d use for counting. You can buy things like 1 car or 5 puppies. However, you could not have negative 5 puppies or ? of a car. Okay, while you can technically have fractions of things, like ? of a car or ½ a penny, keep in mind this is a practical term. Why would you want those things? No store would accept a penny cut in half and ? of a car simply wouldn’t work!

{1, 2, 3, 56, 100 ...}

Whole Numbers:
Whole numbers are all non-negative numbers. In other words, whole numbers are natural numbers PLUS zero (2).

{0, 1, 2, 3, 56, 100 ...}

Integers:
Integers are all whole numbers with negatives included.

{-56, -1, 0, 1, 200, 589,900, ....}

Rational Numbers:
Rational numbers include all integers PLUS fractions AND decimals . As long as the denominator isn’t zero and the fraction only contains integers, that fraction is a rational number!

{-½, -0.012, 0, 3, 10/8, 828.01, …}

All numbers have decimal representation. When the number is rational, then its decimal representation is repeating.
_                                                        ______                                                     __
1/2 = 0.50000 = 0.50               9/7 = 1.285714285714 = 1.285714            157/495 = 0.3171717 = 0.317

(The bar tells us that the particular sequence of digits beneath it repeats forever)

The Real Numbers System:

So as you’ve probably noticed, some number types belong to more than one group. That’s because all natural numbers are whole numbers and all whole numbers are integers and all integers are rational and all rational numbers are real numbers.

Real Numbers
___________________________________           _____________________
|   Rational Numbers   -1      -1/2       14.23|         |  Irrational Numbers         |
|   ____________________________        |         |                                      |
|  |   Integers   -4       -1       0      59|        |         |   √2         pi                   |
|  |    _______________________    |    0  |         |                                      |
|  |    |   Whole numbers   0    1    |   |        |         |         log(2)        sin(1°)  |
|  |    |     ________________      |   |        |         |_____________________|
|  |    |    |   Natural Numbers |  2 |   |        |
|  |    |    |     1      2      59    |     |   |  59.1|
|  |    |    |_______________ | 59|    |        |
|  |    |______________________|   |         |
|  |____________________________|        |
|___________________________________|

Irrational Numbers:
Irrational numbers are in a category all of their own. Here’s where things get a little more complicated. Some real numbers can’t be expressed as ratios of integers. In other words, as decimals they never repeat a sequence of digits or end. We call these numbers irrational numbers.

√2 = 1.41421356...        π= 3.14159265... (this is pi, in case you're having trouble seeing the symbol)

{√2, pi, sin(1°), log(2), …}

Since an irrational number could go on forever, we often need to approximate when we write them out. When we do this, the equals sign looks a little different:

π ≈ 3.14159                √2 ≈ 1.4142

Back to the number line:

-2.5        -1      0       1 √2             pi
____|____|____|____|____|____|____|____

So, as you can see, we could position all the types of numbers we've talked on a number line. Since it’s a line, it’s made up of an infinite number of points packed so closely that they look like a solid line. Thus, there’s room for all those itty-bitty little fractions and decimal places.

Every real number corresponds to a distance on the number line from zero. The number line goes forever in both directions.

End Notes:
(1) Well, this isn’t the entire truth, but this definition will suffice for high school and Freshman calculus.
(2) There's some controversy surrounding whether zero should be included as a natural number. You can, after all, have zero TVs, dogs, leaves, etc. However, a lot of textbooks like to put it elsewhere and thus the category of whole numbers was created. You should ask your teacher for his/her preference. 