Number Systems

Set notation can be used to define each of the number systems. All numbers are built up from the natural numbers. Since these numbers can be counted on one’s fingers, they are also called the counting numbers.

 

The natural numbers are defined as: N= {1,2,3...}

 

The use of the ... shows that the natural numbers are never ending, or an infinite set.

By adding the element 0 to the natural numbers, the whole numbers defined as: W= {0,1,2,3,...}.

Adding a sequence of negative natural numbers to the set W defines the integers, defined as:

J= {..., -2,-1,0,1,2...}.

 

To define the next number system, a new type of set notation is needed. Set-builder notation provides the rules and constraints for constructing, or building, each element of the set. The rational numbers do not follow a sequence as the previous number systems did.

Using set-builder notation, the rational numbers are defined as:

In words, this is read "the set of all p over q such that p and q are integers with q not equal to zero."

The rational numbers can also be described as all numbers whose decimal representation is either terminating

(like $\frac{1}{2}=.5$ ) or repeating (like $\frac{1}{3}=.333$... )

 

The irrational numbers are the next number system, and they do not contain any elements of the previously defined sets of numbers.

The irrational numbers are defined as those numbers whose decimal representation is both non-terminating and non-repeating. This set is represented by the letter H. Examples of irrational numbers are $\pi$ and $\sqrt{2}$ .

The final number system covered in this course is the real numbers, represented by the letter R.

The real numbers are defined as the union of the set of all rational numbers with the set of all irrational numbers.

That is, R= Q ∪ H. Unless otherwise specified, the work in this course will involve the real numbers.

The real numbers can be pictured as follows:

Reals
Rationals			Irrationals
..., $\frac{1}{2}$, $\frac{3}{4}$ , $\frac{-7}{6}$ , ... Integers Negative   Positive ..., -2, -1, 0, 1, 2, ...			..., $\sqrt{2}$, $\sqrt[3]{5}$, $\pi$, ...

 

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