An Introduction to Algebra

The origins of algebra can be traced to the cultures of the ancient Egyptians who were actually Black Niolitic Peoples of the lower Nile, or Kush and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.

Around 300 BC Greek mathematician Euclid in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.

Around 100 BC Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.

Around 150 AD Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.

Around 200 AD Greek mathematician Diophantus , often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.

The word algebra itself is derived from the name of the treatise first written by Persian mathematician Al-Khwarizmi in 820 AD titled: Kitab al-mukhtasar fi Hisab Al-Jabr wa-al-Moghabalah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr (from which algebra is derived) means "reunion", "connection" or "completion".

Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci in 1202. Variables A variable is a symbol that represents a number. Usually we use letters such as n, t, or x for variables. For example, we might say that s stands for the side-length of a square. We now treat s as if it were a number we could use. The perimeter of the square is given by 4 × s. The area of the square is given by s × s.

When working with variables, it can be helpful to use a letter that will remind you of what the variable stands for: let n be the number of people in a movie theater; let t be the time it takes to travel somewhere; let d be the distance from my house to the park.

Expressions An expression is a mathematical statement that may use numbers, variables, or both. Example: The following are examples of expressions: 2 x 3 7 2 × y 5 2 6 × (4 - 2) z 3 × (8 - z)

Example: Roland weighs 70 kilograms, and Mark weighs k kilograms. Write an expression for their combined weight. The combined weight in kilograms of these two people is the sum of their weights, which is 70 k.

Example: A car travels down the freeway at 55 kilometers per hour. Write an expression for the distance the car will have traveled after h hours. Distance equals rate times time, so the distance traveled is equal to 55 × h..

Example: There are 2000 liters of water in a swimming pool. Water is filling the pool at the rate of 100 liters per minute. Write an expression for the amount of water, in liters, in the swimming pool after m minutes. The amount of water added to the pool after m minutes will be 100 liters per minute times m, or 100 × m. Since we started with 2000 liters of water in the pool, we add this to the amount of water added to the pool to get the expression 100 × m 2000.

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To evaluate an expression at some number means we replace a variable in an expression with the number, and simplify the expression.

Example: Evaluate the expression 4 × z 12 when z = 15. We replace each occurrence of z with the number 15, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction. 4 × z 12 becomes 4 × 15 12 = 60 12 = 72

Example: Evaluate the expression (1 z) × 2 12 ÷ 3 - z when z = 4. We replace each occurrence of z with the number 4, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction. (1 z) × 2 12 ÷ 3 - z becomes (1 4) × 2 12 ÷ 3 - 4 = 5 × 2 12 ÷ 3 - 4 = 10 4 - 4 =

Equations An equation is a statement that two numbers or expressions are equal. Equations are useful for relating variables and numbers. Many word problems can easily be written down as equations with a little practice. Many simple rules exist for simplifying equations. Example:

The following are examples of equations: 2 = 2 17 = 2 15 x = 7 7 = x t 3 = 8 3 × n 12 = 100 w 4 = 12 - w y - 1 - 2 - 9.3 = 34 3 × (d 4) - 11 = 321 - 23

Example: Translate the following word problem into an equation: My age in years y plus 20 is equal to four times my age, minus 10. The first expression stands for "my age in years plus 20", which is y 20.

This is equal to the second expression for "four times my age, minus 10", which is 4 × y - 10.

Setting these two expressions equal to one another gives us the equation: y 20 = 4 × y - 10

Solution of an Equation When an equation has a variable, the solution to the equation is the number that makes the equation true when we replace the variable with its value.

Example: We say y = 3 is a solution to the equation 4 × y 7 = 19, for replacing each occurrence of y with 3 gives us 4 × 3 7 = 19 ==> 12 7 = 19 ==> 19 = 19 which is true.

Examples: x = 100 is a solution to the equation x ÷ 2 - 40 = 10 z = 12 is a solution to the equation 5 × (z - 6) = 30

Counterexample: y = 10 is NOT a solution to the equation 4 × y 7 = 19. When we replace each y with 10, we get 4 × 10 7 = 19 ==> 40 7 = 19 ==> 47 = 19 not true!

Counterexamples: x = 200 is NOT a solution to the equation x ÷ 2 - 40 = 10 z = 20 is NOT a solution to the equation 5 × (z - 6) = 30

Simplifying Equations To find a solution for an equation, we can use the basic rules of simplifying equations. These are as follows:

1) You may evaluate any parentheses, exponents, multiplications, divisions, additions, and subtractions in the usual order of operations. When evaluating expressions, be careful to use the associative and distributive properties properly.

2) You may combine like terms. This means adding or subtracting variables of the same kind. The expression 2x 4x simplifies to 6x. The expression 13 - 7 3 simplifies to 9.

3) You may add any value to both sides of the equation.

4) You may subtract any value from both sides of the equation. This is best done by adding a negative value to each side of the equation.

5) You may multiply both sides of the equation by any number except 0.

6) You may divide both sides of the equation by any number except 0.