# Real Numbers

Real numbers are any numbers that can be represented on the number line.

A number line showing the distance between -1 and 1.

The set of real numbers (also called the reals) includes both the set of rational numbers (numbers that can be written as ratios or fractions) and the set of irrational numbers (numbers that can’t be written as fractions, like pi).

There are an infinite number of reals, and there is an infinite number of real numbers between any two real numbers. For example, between 1 and 2 there are 0.1, 0.21, 0.239, and so on.

All whole numbers (1,2,3,4…) and integers (-2098, -2, 1, 2…) are also included in the set of reals.

The set of reals is denoted by the symbol ℝ.

## What Number is Not a Real Number?

Imaginary numbers of the from a+ib, with i= √(-1) and b ≠ 0, are not reals.

## Core Properties of Real Numbers.

Since every real number has its own place along the number line, reals can be ordered. Properties of real numbers include:

• The Distributive Property: a(b+c)= ab +ac for every a, b, and c in ℝ.
• The Commutative Property: ab=ba and a+b =b+a for a and b in ℝ.
• The Associative Property: (ab)c=a(bc) and (a+b)+c=a+(b+c) for all a, b, and c in ℝ
• The Closure Property: a+b, a-b, and ab are all reals if a and b are real numbers; that is to say, real numbers are closed under addition, subtraction, and multiplication.

## The Identity Properties of Real Numbers

There is one multiplicative identity, 1, and one additive identity, 0, such that

1a= a and 0+a =a
for every a in ℝ.

## The Inverse Properties of Reals

Every real number a has an unique additive inverse –a such that a + (-a) =0, with –a being a member of the set of real numbers. Every nonzero real number a also has a unique multiplicative inverse 1/a such that a(1/a)=1, where 1/a is a real number.

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