Algebra

Algebra is an ancient form of mathematical analytical methodology and is one of the most fundamental in our modern practice of analysis.

Its origins can be traced back to ancient civilizations such as the Babylonians and Greek, who developed rudimentary algebraic techniques to solve practical problems in areas like geometry and astronomy. Over time, algebra has evolved and expanded, becoming a powerful tool for solving complex equations and understanding abstract mathematical structures.

One of the key concepts in algebra is the idea of variables, which represent unknown quantities that can be manipulated using mathematical operations. By using variables, mathematicians are able to generalize patterns and relationships, making it possible to solve a wide range of problems efficiently.

In addition to its practical applications, algebra plays a crucial role in the development of mathematical reasoning and problem-solving skills. By studying algebra, students learn to think logically, analyze problems methodically, and communicate their solutions effectively. This foundational knowledge is essential for success in fields such as engineering, computer science, and economics.

## Learning projects

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### Pre-University Level Courses

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* Pre-Algebra
* Algebra I
* Algebra II
* College Algebra

### Undergraduate Level Courses

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* Introduction to Linear Algebra
* Introduction to Logic and Proofs
* Abstract Linear Algebra
* Introduction to Group Theory

### Graduate Level Courses

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* Higher Algebra
* Group Theory
* Ring Theory
* Virasoro Algebra

## Algebra Resources

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### Wikiversity

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* Continuum mechanics/Tensor algebra identities
* Portal:Mathematics
* Powers

### Wikibooks

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* Algebra
* Linear algebra
* Abstract algebra

### Wikipedia

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* Algebra
* Geometry
* Pre-algebra

## Digits

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Numbers 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

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The following laws are true for all  a , b , c {\displaystyle a,b,c} whether these are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.

### Addition

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* Commutative Law:  a + b = b + a {\displaystyle a+b=b+a} .
* Associative Law:  ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} .
* Additive Identity:  a + 0 = a {\displaystyle a+0=a} .
* Additive Inverse:  a + ( − a ) = 0 {\displaystyle a+(-a)=0} .

### Subtraction

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* Definition:  a − b = a + ( − b ) {\displaystyle a-b=a+(-b)} .

### Multiplication

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* Commutative law:  a × b = b × a {\displaystyle a\times b=b\times a} .
* Associative law:  ( a × b ) × c = a × ( b × c ) {\displaystyle (a\times b)\times c=a\times (b\times c)} .
* Multiplicative identity:  a × 1 = a {\displaystyle a\times 1=a} .
* Multiplicative inverse:  a × 1 a = 1 {\displaystyle a\times {\frac {1}{a}}=1} , whenever  a ≠ 0 {\displaystyle a\neq 0}
* Distributive law:  a × ( b + c ) = ( a × b ) + ( a × c ) {\displaystyle a\times (b+c)=(a\times b)+(a\times c)} .

### Division

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* Definition:  a b = a × 1 b {\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}} , whenever  b ≠ 0 {\displaystyle b\neq 0} .

Let's look at an example to see how these rules are used in practice.

( x + 2 ) ( x + 3 ) x + 3 {\displaystyle {\frac {(x+2)(x+3)}{x+3}}} |  = [ ( x + 2 ) × ( x + 3 ) ] × ( 1 x + 3 ) {\displaystyle =\left[(x+2)\times (x+3)\right]\times \left({\frac {1}{x+3}}\right)} (from the definition of division)

|

|  = ( x + 2 ) × [ ( x + 3 ) × ( 1 x + 3 ) ] {\displaystyle =(x+2)\times \left[(x+3)\times \left({\frac {1}{x+3}}\right)\right]} (from the associative law of multiplication)
|  = ( ( x + 2 ) × ( 1 ) ) , x ≠ − 3 {\displaystyle =((x+2)\times (1)),\qquad x\neq -3} (from multiplicative inverse)
|  = x + 2 , x ≠ − 3 {\displaystyle =x+2,\qquad x\neq -3} (from multiplicative identity)

Of course, the above is much longer than simply cancelling  x + 3 {\displaystyle x+3} 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:

2 × ( x + 2 ) 2 = 2 2 × x + 2 2 = 1 × x + 2 2 = x + 2 2 {\displaystyle {\frac {2\times (x+2)}{2}}={\frac {2}{2}}\times {\frac {x+2}{2}}=1\times {\frac {x+2}{2}}={\frac {x+2}{2}}} .

The correct simplification is

2 × ( x + 2 ) 2 = ( 2 × 1 2 ) × ( x + 2 ) = 1 × ( x + 2 ) = x + 2 {\displaystyle {\frac {2\times (x+2)}{2}}=\left(2\times {\frac {1}{2}}\right)\times (x+2)=1\times (x+2)=x+2} ,

where the number  2 {\displaystyle 2} cancels out in both the numerator and the denominator.