Algebra
Algebra is an ancient form of mathematical analytical methodology and is one of the most fundamental in our modern practice of analysis.
Learning projects
editPre-University Level Courses
editUndergraduate Level Courses
edit- Introduction to Linear Algebra
- Introduction to Logic and Proofs
- Abstract Linear Algebra
- Introduction to Group Theory
Graduate Level Courses
editAlgebra Resources
editWikiversity
editWikibooks
editWikipedia
editDigits
editNumbers are made of digits. Here are their names:
0 - zero 1 - one 2 - two 3 - three 4 - four 5 - five 6 - six 7 - seven 8 - eight 9 - nine
Rules of arithmetic and algebra
editThe following laws are true for all whether these are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.
Addition
edit- Commutative Law: .
- Associative Law: .
- Additive Identity: .
- Additive Inverse: .
Subtraction
edit- Definition: .
Multiplication
edit- Commutative law: .
- Associative law: .
- Multiplicative identity: .
- Multiplicative inverse: , whenever
- Distributive law: .
Division
edit- Definition: , whenever .
Let's look at an example to see how these rules are used in practice.
(from the definition of division) | |
(from the associative law of multiplication) | |
(from multiplicative inverse) | |
(from multiplicative identity) |
Of course, the above is much longer than simply cancelling out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:
- .
The correct simplification is
- ,
where the number cancels out in both the numerator and the denominator.