# Algebra

### Algebra

Algebra deals with letters and symbols and rules to manipulate those symbols used to represent numbers and quantities within formulae and equations.

Algebraic expressions commonly include a mixture of letters, numbers and other symbols. E.g. x + 5 = y. To represent multiplication you don’t need to use a multiplication symbol, 5 multiplied by x can simply be represented by 5x.

The Distributive Property is a mathematical law which means we can distribute the same operation to terms with parenthesis in the same equation.

The order of operations is used in both mathematics and computer programming to define which procedures are performed first in a complex formula. PEMDAS is an acronym to help remember the order:
Parentheses
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right))

Linear equations are equations between two variables that give a straight line when plotted on a chart. While we can’t determine definite values of either variable, we can solve the equation for how to calculate one of the variables with respect to the other.
Example: 2y-4x=2
Firstly isolate the y term: 2y=2+4x
Divide both sides by two: y=1+2x
Now we know the value of y for any given value of x. All the possible values of x and y could be plotted on a chart to visualise the linear growth of the equation.

Equations that can be rearranged into the form y = mx + c will produce what is known as a straight line graph.
Example: x + y = 3 can be rearranged into y = 3 – x

Quadratic equations in algebra follow the form ax^2 + bx + c = 0 where x represents an unknown variable but a, b and c have known values. The value of a must be >0 for the equation to be quadratic, otherwise it is linear.

Polynomials can include many different terms: numeric values, variables (like x or y) and exponents (like squaring numbers). They can be combined by addition, subtraction, multiplication and division but not by division within a variable. E.g. 5xy^2 = 3x + 4y is a polynomial but 3xy^-2 is not.

An exponent is a quantity representing the power to which a given value is to be raised. For example 2^4 is equivalent to 2*2*2*2. Logarithms explain how many times a value needs to be multiplied to achieve another value. Using the same example of 2^4 = 16, this could also be represented as the log of 16 equalling 4: log(16)=4.

A linear scale can be visualised similar to a ruler or a tape measure, where the distance between all numbers are the same. This can also be applicable to minus numbers on a linear scale. A logarithmic scale, however, is based on multiplication of numbers rather than addition. When you move distance between numbers on a logarithmic number line you are multiplying by the previous number. E.g. a linear scale may add 10 to each value (0, 10, 20, 30, 40, 50) whereas a logarithmic scale which is scaling by 10 would multiply each value by 10 (1, 10, 100, 1000, 10000, 100000).

Linear growth is when a constant amount is fixed as the increase between each value e.g. 2, 5, 8, 11 which indicates a linear relationship. Exponential growth multiplies between values e.g. 1, 3, 9, 27 multiplying each by 3 and indicating an exponential relationship.

Vectors are quantities which explain the position of a point in space relative to another, defined by directional co-ordinates.

A matrix is an array of numbers that can be arranged into a tabular format with rows and columns. You can add or subtract matrices of the same size and structure and you can also transpose matrices to switch the row and column orientation.