The Number System

Hi everyone. This post is about numbers and the number system in mathematics.

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Sections

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• Natural Numbers
• Non-Negative Integers
• Integers
• Rational Numbers (Fractions)
• Irrational Numbers
• Real Numbers
• Extension: Complex Numbers

Natural Numbers

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The natural numbers refer to the positive whole numbers from 1 and going up. You could see them as counting numbers starting from 1. The set of all natural numbers is denoted by a funny looking N.

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Non-Negative Integers

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The non-negative integers is just an extension of the natural numbers. Instead of counting from 1, we start counting from 0 and then go up. The set of non-negative integers is {0, 1, 2, 3, 4, ... }.

Integers

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When negative whole numbers are added to non-negative integers, we get the integers. The set of integers is denoted by a weird looking Z.

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Rational Numbers (Fractions)

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Many of us are familiar with the concepts of fractions. Fractions can also be referred to as rational numbers. A fraction/rational number contains an integer in the numerator (top) and a non-zero integer in the denominator (bottom). The set of rational numbers (fractions) is denoted by a strange looking Q.

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This is the math version of saying that a and b are integers where the bottom cannot be zero (Dividing by zero is a math crime!).

Integers are considered rational numbers as the denominator/bottom of a fraction is simply 1 (b = 1).

Irrational Numbers

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A number that is not rational (but still real) is irrational. Irrational numbers are not fractions and contain a never ending amount of decimal places. Examples include:

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(Edit On Aug 12, 9:20PM EST: Fixed the golden ratio number typo. )

Real Numbers

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All of the numbers (including the irrationals) from the previous sections belong to the real numbers. The real numbers is denoted by .

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Extension: Complex Numbers

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This particular concept may not be suitable for younger students. (You could just tell them that you cannot take the square root of a negative number as it is not a real number.)

Remember when a (high school) math teacher told you that a square root of a negative number is not real? It belongs to the set of complex numbers. The set of complex numbers is denoted by a weird looking C like this: .

A complex number contains a real number part and an imaginary number part. The form of a complex number is:

a + bi

where

a is a real number
b is a real number
i is an imaginary number with .

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Complex numbers are used in fields such as physics, electrical engineering, fluid dynamics and certain math fields. I am not an expert on complex numbers so it is best to ask others who are familiar with complex numbers.

Here is another summary image but with the complex numbers.

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