Subject: Trigonometry

Real Numbers

What are Real Numbers?

To understand real numbers, you first need to understand the following categories of numbers:

  • Natural Numbers (positive integers) are all positive whole numbers above zero on a number line. This definition is is generally used in Number Theory and basic mathematics (i.e. Trig).

1,2,3,4,5,6...

  • Natural Numbers (non-negative integers) are all non-negative whole numbers on a number line. Note that this would include zero. The definition is generally used in set theory and computer science (more advanced mathematical concepts).

0,1,2,3,4,5,6...

  • Whole Numbers are all positive whole numbers including zero on a number line. Whole numbers are also sometimes interchanged with "integers"; see below for additional information.

0,1,2,3,4,5,6...

  • Integers are all positive, zero and negative whole numbers on a number line:

...-3, -2, -1, 0, 1, 2, 3, 4...

  • Rational Numbers are made of Natural, Whole and Integer Numbers, however they're described or interchanged. These can also be explained as numbers that could be represented as fractions.

\frac{2}{3}, 23, -493, 0.583333, \frac{5}{25}...

  • Irrational Numbers are numbers that will never repeat, will never terminate and are not the quotient (result of a division) of two integers.

\pi , \sqrt{7}, -\sqrt{2}, e

Finally, we can use these definitions to define real numbers:

  • Real Numbers are defined by the whole set of rational and irrational numbers.

Real numbers represent any number you would find on a number line. Real numbers are often denoted with a large R in mathematical notations and equations. This R represents the set of all real numbers.

Let's try an Example

Given the following set of real numbers, define which belong to a) natural numbers, b) whole numbers, C) integers, d) rational numbers and e) irrational numbers:

0.25, -358, 27, 0, \pi , \sqrt{5}, -\sqrt{25}

The answers are as follows:

a) 27
b) 27, 0
c) -358, 27, 0, -\sqrt{25}
d) 0.25, -358, 27, 0, -\sqrt{25}
e) \pi, \sqrt{5}