Number Definitions

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There are lots of different meanings for each kind of number, to sort this out, I'm going to list all the different kinds of numbers and their definitions here as a reference.

Real Numbers

Includes all rational and irrational numbers. May be positive, negative or zero.

Rational Numbers

Any number that can be named, for instance five, six or three-hundred sixty-two, five point four or one-fourths. It can be any whole number, fraction or decimal with it's negative pair. 1, -1, 1.1 and are all rational numbers.


Integers are numbers without decimal or fractional parts. An example set of some integers: 0, 1, -1, 2, 54, 99, 1000, -1000, 13, 453, -234. All integers would exist along an infinitely long number line with 0 at the center going from the negative integers to the positive integers.

Natural Numbers

Any positive integer. Sometimes when the 0 is included in this set, the numbers are referred to as whole numbers or counting numbers.

Irrational Numbers

A number that cannot be expressed as a fraction where m and n are integers and n is non-zero. These numbers cannot be represented as simple or infinitely repeating decimals. These numbers are all the Real numbers that are not considered rational.

Imaginary Numbers

A number taking the form bi where b is a real number, and i is the square root of minus one. The imaginary numbers repeat in a constant pattern as shown here:

Calculating the imaginary number for x number of powers for i is a matter of knowing a few things, first the imaginary unit which is . This is why . Now if you multiply 2 numbers squared together, you pull the number inside the square root out, which gives you this:

Knowing how two squared numbers interact with each other helps us solve the 3rd power like so:

Solving for now is a breeze:

Complex Numbers

These numbers are made up of real and imaginary numbers and can take the form of where a and b are real numbers, and i is imaginary.

Algebraic Numbers

A complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients.

Transcendental Numbers

A number (possibly a complex number) that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients. Some example transcendental numbers:

= pi, sin(a), cos(a), tan(a) along with csc(a), sec(a), cot(a).