Number Definitions

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 ${\frac {1}{2}}$ are all rational numbers.

Integers

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 ${\frac {m}{n}}$ 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:

$i^{0}=1$

$i^{1}=i$

$i^{2}=-1$

$i^{3}=-i$

$i^{4}=1$

$i^{5}=i$

$i^{6}=-1$

$i^{7}=-i$

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 ${\sqrt {-1}}$. This is why $i^{1}=i$. Now if you multiply 2 numbers squared together, you pull the number inside the square root out, which gives you this:

$i^{2}={\sqrt {-1}}\cdot {\sqrt {-1}}=-1$

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

$i^{3}=({\sqrt {-1}}\cdot {\sqrt {-1}}=-1)\cdot {\sqrt {-1}}=-1i=-i$

Solving for $i^{4}$ now is a breeze:

$i^{4}=({\sqrt {-1}}\cdot {\sqrt {-1}}=-1)\cdot ({\sqrt {-1}}\cdot {\sqrt {-1}}=-1)=-1\cdot -1=1$

Complex Numbers

These numbers are made up of real and imaginary numbers and can take the form of $a+bi$ 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:

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