A Strange Family: Number Classification

Complex Numbers: \large\Bbb{C}

Real and Imaginary Numbers are subsets of an even larger set. They form what we call \text{Complex Numbers: }\Bbb{C}. When we say “complex numbers” we don’t mean difficult to understand. The “complex” refers to a combination of two different types of numbers, as in, “a complex of numbers.” Complex Numbers are of the form a+bi �+��[][]where a and b �[]and �  are real numbers and i�[][] is the imaginary unit.

We know from above that zero is part of the \text{Real Numbers}, so it is possible for the imaginary component b, a \text{Real Number}, to be zero. Therefore, we can see that all \text{Real Numbers} are considered Complex. For example, 5=5+0i.

Complex Numbers whose imaginary component is nonzero, exist apart from the Real number line. To graph a complex number we first locate the real component on the horizontal (Real) number line and then move up or down the appropriate number of units equal to the b value.

\large5=5+0i

\large\pi=\pi+0i

\large\frac{3}{4}=\frac{3}{4}+0i

Natural Numbers: \large\Bbb{N}

\text{Natural Numbers: }\Bbb{N} are a subset of the real numbers. You’ve known these guys since kindergarten – they’re all the counting numbers starting from 1, and they go on forever! 

Whole Numbers: \large\Bbb{W}

And when you throw a 0 into this mix, boom, you get a new set called the \text{Whole Numbers}.

Integers: \large\Bbb{Z}

Integers are composed of negative integers and whole numbers. Let’s say we have 3 apples, and we give away 5. How many do we have left? Integers are the saviors here, introducing the concept of negative numbers. So, integers are whole numbers, but they also include their negative counterparts. The use of a \Bbb{Z} to represent the set of integers comes from the German word Zahlen; Zahlen relates to “numbers”.

It is common for people to refer to any negative round number, or negative decimal number as an integer (-2.5 is not an integer, it is a rational number). However, integers are all the whole numbers (0, 1, 2, 3, …) and their opposites (…, -3, -2, -1). This is a technical distinction, that often gets overlooked in practice, so don’t feel compelled to correct your teacher if they refer to -1.25 as an integer.

Rational Numbers: \large\Bbb{Q}

Next up, we have \text{Rational Numbers}. If you can write a number as a fraction where both the numerator and the denominator are integers (remember, the denominator cannot be zero), then you’re looking at a rational number; the quotient of two integers. This set includes all integers because you can write any integer as a fraction with 1 in the denominator, right? Rational Numbers include all Natural Numbers, Whole Numbers, Integers, and of course all the decimals and  fractions.

We use a stylized capital \Bbb{Q} to represent the set of \text{Rational Numbers}. We use a Q because rational numbers are Quotients.

Irrational Numbers: \large\Bbb{R} - \Bbb{Q}

But the world of numbers doesn’t end with \text{Rational Numbers: }\Bbb{Q}. Have you heard about the number \pi (pi)? It’s a non-repeating, non-terminating decimal that can’t be expressed as a simple fraction. Such numbers are called \text{Irrational Numbers}. We don’t have a set symbol to represent these numbers because they are customarily defined as the set of \text{Real Numbers} minus the set of \text{Rational Numbers} or \Bbb{R} - \Bbb{Q}.

SURDS

There is an infinite set of irrational roots. Any root that does not equal a integer, is irrational. For instance, all the square roots between the perfect squares that you’ve likely already learned about, are all irrational.

\large\text{Irrational Square Roots; }\sqrt{2},\,\sqrt{3},\,\sqrt{5},\,\sqrt{6},\,\sqrt{7},\,\sqrt{8},\,...

\large\text{Other Irrational Roots; }\sqrt[3]{4},\,\sqrt[7]{8},\,\sqrt[4]{5},\,\sqrt[9]{64},\,\sqrt[4]{7},\,\sqrt[21]{8},\,...