The Real Number System
The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers. 1, 2, 3, 4, 5, . . The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 (an integer), 4/3 (a rational number that is not an integer), 8.6 (a rational number given by a finite decimal representation), √2 (the square root of two, an irrational number) and π (3.1415926535..., a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation (such as that of π above), where the consecutive digits indicate into which tenth of an interval given by the previous digits the real number belongs to. The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include real numbers as a special case.
The real number system is a very complicated . It states that there is a lot of numbers, like a farmer has two sheep so he has +2 sheep but, if he owed someone 2 sheep he would have -2 sheep. There are different sections of the number system.Natural Numbers
Natural Numbers or “counting numbers” are regular numbers. These kind of numbers go on forever. 0 or zero is a actual natural number, because if you owe someone one dollar and you give it to them you would owe them 0 dollars .Whole Numbers
Whole numbers are Natural numbers with zero. But this is a story on the number zero . Is it a number or not.About the Number Zero
What is zero? Is it a number? How can the number of nothing be a number? Is zero nothing, or is it something?Well, before this starts to sound like a Zen koan, let’s look at how we use the numeral “0.” Arab and Indian scholars were the first to use zero to develop the place-value number system that we use today. When we write a number, we use only the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.These numerals can stand for ones, tens, hundreds, or whatever depending on their position in the number. In order for this to work, we have to have a way to mark an empty place in a number, or the place values won’t come out right. This is what the numeral “0″ does. Think of it as an empty container, signifying that that place is empty. For example, the number 302 has 3 hundreds, no tens, and 2 ones.
So is zero a number? Well, that is a matter of definition, but in mathematics we tend to call it a duck if it acts like a duck, or at least if it’s behavior is mostly duck-like. The number zero obeys most of the same rules of arithmetic that ordinary numbers do, so we call it a number. It is a rather special number, though, because it doesn’t quite obey all the same laws as other numbers-you can’t divide by zero, for example.
Note for math purists: In the strict axiomatic field development of the real numbers, both 0 and 1 are singled out for special treatment. Zero is the additive identity, because adding zero to a number does not change the number. Similarly, 1 is the multiplicative identity because multiplying a number by 1 does not change it.
Integers
The next generalization that we can make is to include the idea of fractions. While it is unlikely that a farmer owns a fractional number of sheep, many other things in real life are measured in fractions, like a half-cup of sugar. If we add fractions to the set of integers,we get the set of rational numbers.
Rational Numbers
Rational numbers include what we usually call fractions. Notice that the word “rational” contains the word “ratio,” which should remind you of fractions. The bottom contains the denominator and the top contains the numerator.Irrational numbers
These and rational numbers are almost the same only irrational number never stop rational do.The sets of numbers:
| Names | Sets | Notes and examples |
| natural numbers | {1, 2, 3, . . .} | 1. See note 1. on set notation below: |
| whole numbers | {0, 1, 2 , 3, . . .} | |
| integers | {0, 1, 2, 3,. . .} | |
| rational | { p/q | p and q are integers, q0} | 2.  |
| real | {x | x can be written as a decimal} | 3. |
| irrational | {x | x is a nonrepeating and nonterminating decimal} | 3.14159. . , e 2.71828, 2 |
