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Algebra

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.

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Algebra

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  1. Algebra Introduction & Useful Websites

  2. 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.

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

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

  5. 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.

  6. A picture of al-Khwarizmi

  7. Al-Khwarizmi and Algorithm • Al-Khwarizmi wrote numerous books that played important roles in arithmetic and algebra.

  8. 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.

  9. Al-Khwarizmi & Algebra • From Al-Khwarizmi’s most important book, Al-jabrwa'l muqabalah, comes the word algebra.

  10. Al-gabr - muqabalah • The 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 = 54 X + 12 –12 = 54 - 12

  11. Al-gabr - Muqabalah • the word muqabalahis 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 + 12 5x = 50

  12. Al-gabr - Muqabalah • The book deals with solution of equations and the methods of which involve transposing and cancelling terms.

  13. More about Al-Khwarizmi http://www.ms.uky.edu/~carl/ma330/project2/al-khwa21.html

  14. 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

  15. Diophantus’ book “Arithmetica”

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

  17. 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.

  18. Variable / unknown & Constant

  19. Coefficients

  20. In arithmetic, we work with numbers:5 + 8 = 13

  21. In algebra, we often use letters to stand for numbers:a + b = c

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

  23. Numerical expressions 54  (34 – 54)  7

  24. Algebraic expressions 3n2–13 x + 3y –12

  25. An example of algebraic expressions • Pick up 9 adjacent days from the following calendar and add the numbers. • Express these days as algebraic expressions • Can you generalize a pattern to calculate the sum of days?

  26. Example: January 2001 The sum of the numbers is:

  27. Example: the algebraic expressions The sum of the expressions is:

  28. Algebraic expressions can be used to represent numbers and solve problems

  29. Advantages of algebraic expressions • Simple and direct • Easy to read and understand • An international language

  30. Some exercises in equations http://www.algebrahelp.com/worksheets/

  31. The End

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