An equation is a mathematical expression with an equals sign in it.

Other possible elements of an equation are:

- numbers
- variables
- binary operators (+, -, * or ÷)
- unary operators (+ or -)
- brackets
- powers (x
^{y}) - functions (sin(),log() , etc.)

These are explained below:

## Numbers

There are different types of numbers, for instance:

**Z**- Integers - whole numbers: ... -3, -2, -1, 0, 1, 2, 3 ... both positive, zero and negative.- - Real numbers - numbers which are continuous such as when we are representing points along a line - On this site I will sometimes use the term 'Scalar' to mean 'Real' numbers although strictly the term should be used when scaling a vector - In computer programs real numbers have a finite length and may have decimal point and/or exponent this allows us to approximate most real numbers but it is only an approximation.
**Q**- Rational numbers - Integers and fractions where numerator and denominator are integers.- Radical Integers - The integers plus any combination of addition, subtraction, multiplication, division and root extraction.
**Q**^{alg}- The root of a**Z**-polynomial - A complex number made up more than just radical integers although it is closed under sum, difference, product, quotient and n^{th}root.- Modulo 'n' numbers
**C**- Complex Numbers - numbers with real and imaginary parts.**H**- Quaternion - Complex number whose elements are complex numbers.**O**- Octonion - Quaternion whose elements are complex numbers.- Vectors - one dimensional arrays of numbers.
- Matrices - two dimensional arrays of numbers.

and these numbers may be coded in different ways:

- binary
- octal
- decimal
- hexadecimal

Unless otherwise specified we usually assume that numbers are decimal.

## Variables

Sometimes an equation contains a number but we don't yet know its value, or we may want to apply the equation to a range of values.

An example of the first is using x as the unknown, for example,

x + 1 = 3

An example of the second might be an equation of a line:

y = 2 * x + 3

In general we use:

x, y, z for unknowns.

a, b, c for values which are not yet specified.

## Binary Operators

The operators:

- + add
- - subtract
- * multiply (was x but for computers has to be changed to distinguish it from 24th letter of the alphabet)
- ÷ or / divide

take the two numbers on either side and replace it by a single number.

### Addition

In the following plot the height is given by a+b, this gives a flat plane at 45° to both a and b:

Of course the plane is infinite in all directions but here we have only shown a section of the plane.

### Multiplication

In the following plot the height is given by a*b, this gives a surface made up of straight lines along a or b directions (bi-linear). If we take a direction at 45° to both a and b then we get a parabola.

## Unary operators

The operators:

- + plus
- - minus

apply to the number to the right. '-' inverts the number (subtracts from 0) '+' says the number to the right is positive (the default).

## Brackets

When we mix + and * then the answer we get depends on the order that we apply them.

For example

2 + 1 * 3

To clarify this we can put brackets around the operation to be applied first:

(2 + 1) * 3 = 9

2 + (1 * 3) = 5

If we don't specify which has precedence by using bracket then by default * and ÷ have precedence over + and -. So,

2 + 1 * 3 = 5

## Functions

A function takes one number and uses it to generate another number. For example the function sin() takes an angle as input and returns the ratio of opposite and hypotenuse in a right angled triangle.