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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 <u>substitute</u> 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===
* <math>Z_4</math> is the set of integers modulo 4.  The elements of <math>Z_4</math> are <math>\{0,1,2,3\}</math>

==Groups==
A group is an ordered pair <math>(G,\cdot)</math> where '''G''' is a set closed under the operation <math>\cdot</math>.  The integers modulo any integer form a group.

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