How to Classify Numbers | Portland Math Tutor, LLC

Apart from being part of our daily lives – counting money and measuring ingredients or temperature – numbers do have an identity, right? Absolutely! This post will take you on a tour through the world of number classification. The family tree of numbers helps us to understand how the different types of numbers are related. Let’s learn how to classify numbers.

Numbers are quite fascinating. They aren’t as dull as they may seem in your math textbook. They have personalities, characteristics, and categories; It’s a strange family of numbers.

Real Numbers: $\mathbb{R}$

Imaginary Numbers: $</a>\mathbb{I}$

Complex Numbers: $</a>\mathbb{C}$

Natural Numbers: $</a>\mathbb{N}$

Whole Numbers: $</a><a href="#real-numbers" title="">\</a>mathbb{W}$

Integers: $</a>\mathbb{Z}$

Rational Numbers: $</a>\mathbb{Q}$

Irrational Numbers; $</a>\mathbb{R}-\mathbb{Q}$

As we dig into the classification of numbers, our first task is to distinguish between the Real and the Imaginary Numbers.

Real Numbers: $\large\Bbb{R}$

$\text{Real Numbers: }\Bbb{R}$ . They’re the complete package and include every number that you can place on a number line.

This includes all numbers we will discuss here except for those imaginary numbers that have a whole plane of their own.

$…,\; -4,\;\sqrt{33},\;-\phi,\;0,\;e,\;\pi,\;\frac{5}{7},\;…$

$\large \...,-4,\,-3,\,-\phi,\,0,\,1,\,e,\,\pi,\,4,$

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Imaginary Numbers: $\large\text{i}$

$\text{Imaginary Numbers: }\text{I}$ . The fundamental unit of imaginary numbers is $\sqrt{-1}$ , which is represented as $\text{i}$ . Purely imaginary numbers are any multiple of the imaginary unit $[bi]$ such that $\text{b}$ is some real number.

\large[\ldots,-10i,\,-\phi i,\,0i,\,1i,\,ei,\,\pi i\,, \frac{7}{2}i, \...

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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 $\text{a+bi}$ where $\text{a}$ and $\text{b}$ 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 $\text{b}$ , 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

<span style="color: #1e1e1e; font-family: Menlo, Consolas, monaco, monospace; font-size: 15px; white-space: pre-wrap;">\large\frac{3}{4} = \frac{3}{4}+0i

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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!

\large\text{Natural Numbers; 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}

Whole Numbers: $\large\Bbb{W}$

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

\large\text{Whole Numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}

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.

\large\text{Integers: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}

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 $\Bbb{Q}$ because rational numbers are Quotients.

\large\text{Rational Numbers: }...,\,\frac{-99}{3},\,-14,-\sqrt{16},\,0,\,\frac{2}{5},\,1,\,\frac{5}{3},\,1.\bar{9},\,\sqrt{4},...

Proper Fractions

Fractions whose numerator is less than the denominator, such as: $\frac{2}{3},\,\frac{4}{15},\,\frac{9}{10},\,$

Improper Fractions

Improper fractions are fractions whose numerator is greater than the denominator, such as: $\frac{4}{3},\,\frac{13}{6},\,\frac{}{}$

Mixed Numbers

Mixed numbers are the sum of an Integer and a faction, such as; $2\frac{3}{5},\,4\frac{1}{2}$

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 $\Bbb{R} - \Bbb{Q}$ .

Transcendental

Transcendental Numbers are irrational numbers that are not the solution to any polynomial. What is a polynomial?

\large\text{Transcendental Numbers: }\pi,\,e,\,\ln{2},\,e^{\pi},\,i^{I}

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},\,...

\text{[math]}

The Family Tree of Numbers: Number Classification

So, there you have it, folks – the family tree of numbers! From the everyday natural numbers to the mysterious imaginary ones, each number has its unique place in the universe of mathematics. Mathematics is not just about calculations; it's about understanding the universe in its own unique language!

That's all for now, folks! I hope you enjoyed this fun ride through the world of number classification. Keep exploring, keep learning, and remember, in the world of mathematics, every number counts!

Mr. Laiche Owner

I am a licensed teacher in Oregon with an endorsement in Advanced Mathematics. I started Portland Math Tutor, LLC in 20216 and I have been helping students succeed in their math studies ever since.

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Real Numbers:

Imaginary Numbers:

Complex Numbers:

Natural Numbers:

Whole Numbers:

Integers:

Rational Numbers: