# Beginning Mathematics/Algebra

Arithmetic is the study and use of numbers and their relationships, whilst **Algebra** uses letters as a preliminary substitute for numbers.

Unknown numbers are often given the letters x, y and z, while the temporary substitutes for known numbers may be a, A, b, B, c, A1, FF, etc. Algebra also comes up with various simplifications.

## An example[edit | edit source]

How to calculate wages to be paid:

- Employee A worked 50 hours at 10 per hour, total is 50 times 10 = 500
- Employee B worked 45 hours at 10 per hour, total is 45 times 10 = 450, etc. - That is arithmetic.

In algebraic terms, this same table could be express as such: Let h = the number of hours worked, and r = the rate of pay per hour.

- Then t = the total to be paid = h times r
- Note that this covers the calculation for ALL employees, just
__substitute__the correct numbers for each of the letters, as the case may be. - For Employee A: h = 50, r = 10, and t = h times r, therefore t= 500
- For Employee B: h = 45, r = 10, and t = h times r, therefore t = 450

## Closure[edit | edit source]

**Closure** is a term which describes a relationship between an operation and a set. A set is said to be **closed** under a certain operation if any application of the given operation applied between members of the set yields another member of the set.

### Examples[edit | edit source]

- The set of integers is closed under addition, multiplication, and subtraction, but not division and exponentiation.
- The set of complex numbers is closed under addition, multiplication, exponentiation, and division.
- The set of odd integers is not closed under addition.

## Modular Arithmetic[edit | edit source]

Modular arithmetic acts like regular arithmetic, except that it is dealing only with remainders with respect to a certain number. We call the number the modulus and refer to arithmetic modulo the given number.

### Example[edit | edit source]

- is the set of integers modulo 4. The elements of are

## Groups[edit | edit source]

A group is an ordered pair where **G** is a set closed under the operation . The integers modulo any integer form a group.