Oxford Essays: Algebra

Algebra

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ALGEBRA

Algebra is a part of mathematics (often called math in the United States and maths in the United Kingdom). It uses variables to represent a value that is not yet known. When an equals sign (=) is used, this is called an equation. A very simple equation using a variable is: 2 + 3 = x In this example, x = 5, or it could also be said, "x is five". This is called solving for x.
Besides equations, there are inequalities (less than and greater than). A special type of equation is called the function. This is often used in making graphs.

Algebra can be used to solve real problems because the rules of algebra work in real life and numbers can be used to represent the values of real things. Physics, engineering and computer programming are areas that use algebra all the time. It is also useful to know in surveying, construction and business, especially accounting.

People who do algebra need to know the rules of numbers and mathematic operations used on numbers, starting with adding, subtracting, multiplying, and dividing. More advanced operations involve exponents, starting with squares and square roots. Many of these rules can also be used on the variables, and this is where it starts to get interesting.

Algebra was first used to solve equations and inequalities. Two examples are linear equations (the equation of a line, y=mx+b) and quadratic equations, which has variables that are squared (power of two, a number that is multiplied by itself, for example: 2*2, 3*3, x*x). How to factor polynomials is needed for quadratic equations.

History

Early forms of algebra were developed by the Babylonians and the Greeks. However the word "algebra" is a Latin form of the Arabic word Al-Jabr ("casting") and comes from a mathematics book Al-Maqala fi Hisab-al Jabr wa-al-Muqabilah, ("Essay on the Computation of Casting and Equation") written in the 9th century by a famous Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, who was a Muslim born in Khwarizm in Uzbekistan. He flourished under Al-Ma'moun in Baghdad, Iraq through 813-833 AD, and died around 840 AD. The book was brought into Europe and translated into Latin in the 12th century. The book was then given the name 'Algebra'. (The ending of the mathematician's name, al-Khwarizmi, was changed into a word easier to say in Latin, and became the English word algorithm.)

RULES

In algebra, there are a few rules that can be used for further understanding of equations. These are called the rules of algebra. While these rules may seem senseless or obvious, it is wise to understand that these properties do not hold throughout all branches of mathematics. Therefore, it will be useful to know how these axiomatic rules are declared, before taking them for granted. Before going on to the rules, reflect on two definitions that will be given.

Opposite - the opposite of a is -a.
Reciprocal - the reciprocal of a is \frac{1}{a}.

Rules

Commutative property of addition
'Commutative' means that a function has the same result if the numbers are swapped around. In other words, the order of the terms in an equation do not matter. When the operator of two terms is an addition, the 'commutative property of addition' is applicable. In algebraic terms, this gives a + b = b + a.

Note that this does not apply for subtraction! (i.e. a - b \ne b - a)

Commutative property of multiplication
When the operator of two terms is an multiplication, the 'commutative property of multiplication' is applicable. In algebraic terms, this gives a \cdot b = b \cdot a.

Note that this does not apply for division! (i.e. \frac{a}{b} \ne \frac{b}{a}, when a \neq b )

Associative property of addition
'Associative' refers to the grouping of numbers. The associative property of addition implies that, when adding three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives a + (b + c) = (a + b) + c. Note that this does not hold for subtraction, e.g. 1 = 0 - (0 - 1) \neq (0 - 0) - 1 = -1 (see the distributive property).

Associative property of multiplication
The associative property of multiplication implies that, when multiplying three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives a \cdot (b \cdot c) = (a \cdot b) \cdot c. Note that this does not hold for division, e.g. 2 = 1/(1/2) \neq (1/1)/2 = 1/2.

Distributive property
The distributive property states that the multiplication of a number by another term can be distributed. For instance: a \cdot (b + c) = ab + ac. (Do not confuse this with the associative properties! For instance, a \cdot (b + c) \ne (a \cdot b) + c.)

Additive identity property
'Identity' refers to the property of a number that it is equal to itself. In other words, there exists an operation of two numbers so that it equals the variable of the sum. The additive identity property states that the sum of any number and 0 is that number: a + 0 = a. This also holds for subtraction: a - 0 = a.

Multiplicative identity property
The multiplicative identity property states that the product of any number and 1 is that number: a \cdot 1 = a. This also holds for division: \frac{a}{1} = a.

Additive inverse property
The additive inverse property is somewhat like the opposite of the additive identity property. When an operation is the sum of a number and its opposite, and it equals 0, that operation is a valid algebraic operation. Algebraically, it states the following: a - a = 0. Additive inverse of 1 is (-1).

Multiplicative inverse property
The multiplicative inverse property entails that when an operation is the product of a number and its reciprocal, and it equals 1, that operation is a valid algebraic operation. Algebraically, it states the following: \frac{a}{a} = 1. Multiplicative inverse of 2 is 1/2.