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

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 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

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

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

Groups

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