History of the Quadratic Equation Sketch 10

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History of the Quadratic EquationSketch 10

• By Stephanie Lawrence
• Jamie Storm

2
Introduction

• Around 2000 BC Egyptian, Chinese, and Babylonian
  engineers acquired a problem.
• When given a specific area, they were unable to
  calculate the length of the sides of certain
  shapes.
• Without these lengths, they were unable to design
  a floor plan for their customers.

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Preview

• Egyptian way of finding area
• Babylonian and Chinese method
• Pythagoras and Euclids contribution
• Brahmaguptas Contribution
• Al-Khwarzimis Contribution

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Egyptians Contribution

• Their Problem

• They had no equation

• Tables

• Solution!!!

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Babylonian and Chinese Contribution

• Started a method known as completing the square
  and used it to solve basic problems involving
  area.
• Babylonians had the base 60 system while the
  Chinese used an abacus. These systems enabled
  them to double check their results.

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Pythagoras and Euclids Contribution

• In search of a more general method

• Pythagoras hated the idea of irrational numbers

• 268 years later Euclid proves him wrong

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Euclids Contribution

• Using strictly a geometric approach.
• -If a straight line be cut into equal and unequal
  segments, the rectangle contained by the unequal
  segments of the whole together with the square on
  the straight line between the points of section
  is equal to the square on the half.

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Brahmaguptas Contribution

• Indian/Hindu mathematician
• Gives an almost modern solution of the quadratic
  equation, allowing negatives
• Brahmaguptas formula
• sabcd
• 2
• ssemiperimeter

Proof
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Al-Khwarizmis Contribution

• An Arab Mathematician

• Lived in Baghdad a generalist who
• wrote books on mathematics

• He considered single squares and
• used the following formula

This part of the quadratic formula was brought to
Europe by the Jewish mathematician Abraham bar
Hiyya (Savasaorda), who then wrote a book
containing the complete solution to the
quadratic equation in 1145 called Liber
Embadorum
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Al-Khwarizmis Contribution

• Gave a classification of the different types of
  quadratics which include

• His book Hisab al-jabr w-al-musqagalah (Science
  of the Reunion and the Opposition) starts out
  with a discussion of the quadratic equation.

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The Discussion

• Ex One square and ten roots of the same are
  equal to thirty-nine dirhems. (i.e. What must be
  the square that when increased by ten of its own
  roots, amounts to thirty-nine?)

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Can you Show this Geometrically?

• We draw a square with side x and add a 10 by x
  rectangle.

-The area is 39

• To determine x cut the number of roots in half

• Move one of these halves to the bottom of the
  square (total area is still 39)

• What is the area of the missing square? Total
  area?

-Missing square 25 Total area 64

• So what is the length of one of the sides of
  this bigger square?

-Answer v 648

• Therefore how can we conclude that x3?

Answer Since the side of the big square is x5,
we can conclude that x3
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Back to The Discussion

• X is the unknown the problem translates to
  x210x39

• Proof You halve the number of roots, which in
  the present instance yields five. This you
  multiply by itself the product is twenty-five.
  Add this to thirty-nine the sum is sixty-four.
  Now take the root of this, which is eight, and
  subtract from it half the number of the roots
  which is five the remainder is three. This is
  the root of the square which you sought for the
  square itself is nine.

?? Does this remind you of anything ??
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Try One

• One square and 6 roots of the same are equal to
  135 dirhems. (i.e. What must be the square which,
  when increased by 6 of its own roots amounts to
  135?)

• Answer The square is 81

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Extra Information

• Methods and justifications became more
  sophisticated over time
• From the 9th Century to the 16th Century, almost
  all algebra books started their discussions of
  quadratic equations with Al-Khwarizmis example
• In the 17th Century European mathematicians began
  representing numbers with letters
• Finally Thomas Harriot and Rene Descartes
  realized that it is much easier to write all
  equations as something 0

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Today

• In 17th Century Rene Descartes published La
  Geometrie, which developed into modern
  mathematics
• General equation ax2bxc0
• Written

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Timeline

• 1500BC Egyptians made a table.
• 580 BC Pythagoras hates irrational numbers.
• 400 BC Babylonians solved quadratic
  equations.
• 300 BC Euclid developed a geometrical
  approach and proved that irrational
  numbers exist.
• 598-665AD Brahmagupta took the Babylonian
  method that allowed the use of negative
  numbers.
• 800AD Al-Khwarizmi removed the negative and
  wrote a book Hisab al-jabr w-al-
  musqagalah (Science of the Reunion and
  the Opposition)
• 1145AD Abraham bar Hiyya Ha-Nasi
  (Savasaorda)wrote the book Liber
  embadorum contained the complete
  solution to the quadratic equation.
• 1637AD Rene Descartes published La
  Geometrie containing the quadratic formula
  we know
• today.

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References

• Artmann, Benno (1999). Euclid The Creation of
  Mathematics. New York, NY Springer-Verlag.
• Fishbein, Kala, Brooks, Tammy. Brahmagupta's
  Formula. The University of Georgia. 16 September
  2006 lthttp//jwilson.coe.uga.edu/EMT725/Class/Broo
  ks/ Brahmagupta/Brahmagupta.htmgt.
• Katz, Victor J. (2004). A History of Mathematics.
  New York, NY Pearson Addison Wesley.
• Lawrence, Dr. Dnezana. Math is Good for You! 17
  September 2006 lthttp//www.mathsisgoodforyou.com/i
  ndex.htmgt.
• Merlinghoff, W, Fernando, G (2002). Math
  Through the Ages A Gentle History for Teachers
  and Others.Farminton, ME Oxton House Publishers.
  105-108.
• O'Conner, J. J., E. F. Robertson. "History
  topic Quadratic, cubic, and quartic equations."
  Quadratic etc equations. Feb. 1997. 4 Sept. 2006
  lthttp//www-groups.dcs.st-and.ac.uk/history/Print
  HT/Quadratic_etc_equatins.htmlgt.
• "The History Behind the Quadratic Formula." BBC
  homepage. 13 Oct. 2004. 11 Sept.
  2006lthttp//www.bbc.co.uk/dna/h2g2/A2982576gt.