# Beginning Mathematics/Algebra

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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

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

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

• The set of integers is closed under addition, multiplication, and exponentiation, but not division.
• 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

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

• ${\displaystyle Z_{4}}$ is the set of integers modulo 4. The elements of ${\displaystyle Z_{4}}$ are ${\displaystyle \{0,1,2,3\}}$

## Groups

A group is an ordered pair ${\displaystyle (G,\cdot )}$ where G is a set closed under the operation ${\displaystyle \cdot }$. The integers modulo any integer form a group.