Algebra

Introduction & Useful Websites

Origin of Algebra

Many say that the Babylonians first developed systems of quadratic equations. This calls for over simplification, because the Babylonians had no concept of an equation. Also, all solutions to Babylonian problems were positive because they were solutions to problems involving lengths.

The knowledge of algebra comes mainly from two Mathematicians: Al-Khwarizmi & Diophantus

Algebra

The word ‘algebra’comes from a 9th century Mathematics book written by a famous Arabian mathematician Al-Khwarizmi .

Al-Khwarizmi Mohammed ibn-Musa al-Khwarizmi was born sometime before 800 A.D. and died after 847 A.D..He flourished as a mathematician and astronomer who was a faculty member at the "House of Wisdom" established in Baghdad by Al-Mamun.

A picture of al-Khwarizmi

Al-Khwarizmi and AlgorithmAl-Khwarizmi wrote numerous books that played important roles in arithmetic and algebra.

Al-Khwarizmi & arithmetic

De numero indorum About the Hindu-Arabic numerals and place values. Because of this book with the Latin translations made a

false inquiry that our system of numeration is Arabic in origin.

The new numeral notation came to be known as that of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm.

Al-Khwarizmi & Algebra

From Al-Khwarizmi’s most important book, Al-jabr wa'l muqabalah , comes the word algebra.

Al-gabr - muqabalahThe word al-jabr presumably meant something like "restoration" or "completion“It seems to refer to the transposition of subtracted terms to the other side of an equation;

X + 12 = 54X + 12 –12 = 54 - 12

Al-gabr - Muqabalah

the word muqabalah is said to refer to "reduction" or "balancing"It refers to the cancellation of like terms on opposite sides of the equations or the simplification of the resulting expression.

5x +12 = 50 + 125x = 50

Al-gabr - Muqabalah

The book deals with solution of equations and the methods of which involve transposing and cancelling terms.

More about Al-Khwarizmi

http://www.ms.uky.edu/~carl/ma330/project2/al-khwa21.html

Diophantus of Alexandria

Born in about 200 A.D. and died in about 284 A.D. sometimes called "the father of Algebra”best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers

Diophantus’ book “Arithmetica”

More about Diophantus

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Diophantus.html

Basics of Algebra

Algebra is a division of mathematics designed to help solve certain types of problems quicker and easier. Algebra operates on the idea that an equation represents a scale such as the one shown on the right. Instead of keeping the scale balanced with weights, we use numbers, or constants.

Variable / unknown & Constant

Coefficients

In arithmetic, we work with numbers:

5 + 8 = 13

In algebra, we often use letters to stand for

numbers:a + b = c

Can you raise some examples of using letters to represent numbers in our daily life?

Numerical expressions

54 (34 – 54) 7

Algebraic expressions

3n2 –13

x + 3y –12

An example of algebraic expressions

Pick up 9 adjacent days from the following calendar and add the numbers.Express these days as algebraic expressionsCan you generalize a pattern to calculate the sum of days?

Example: January 2001The sum of the numbers is:

Sun Mon Tue Wed Thu Fri Sat

1 2 3 4 5

6 7 8 9 10 11 12

13 14 15 16 17 18 19

20 21 22 23 24 25 26

27 28 29 30

Example: the algebraic expressions

X-8 X-7 X-6

X-1 x X+7

X+6 X+7 X+8

The sum of the expressions is:

Algebraic expressions can be used to represent numbers and solve problems

Advantages of algebraic expressions

Simple and directEasy to read and understandAn international language

Some exercises in equations

http://www.algebrahelp.com/worksheets/

The End