Download History of the Quadratic Equation

History of the Quadratic
Equation
Sketch 10
By: Stephanie Lawrence
&
Jamie Storm
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.

Preview
 Egyptian
way of finding area
 Babylonian and Chinese method
 Pythagoras’ and Euclid’s contribution
 Brahmagupta’s Contribution
 Al-Khwarzimi’s Contribution
Egyptian’s Contribution

Their Problem

They had no equation

Tables

Solution!!!
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.
Pythagoras’ and Euclid’s
Contribution



In search of a more general method
Pythagoras hated the idea of irrational
numbers
268 years later Euclid proves him wrong
Euclid’s 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.
C
A
K
D
B
L
E
G
F
Brahmagupta’s Contribution



Indian/Hindu mathematician
Gives an almost modern solution of the
quadratic equation, allowing negatives
Brahmagupta’s formula: A  ( s  a)( s  b)( s  c)( s  d )
s=a+b+c+d
2
s=semiperimeter
Proof
Al-Khwarizmi’s Contribution



An Arab Mathematician
Lived in Baghdad; a generalist who
wrote books on mathematics
He considered single squares and
used the following formula:
x  ( 2 ) c  b 2
b
2
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
Al-Khwarizmi’s Contribution

Gave a classification of the different types
of quadratics which include:
–Squares equal to roots
–Squares equal to numbers
–Roots equal to numbers
–Squares and roots equal to numbers
–Squares and numbers equal to roots
–Roots and numbers equal to squares

His book Hisab al-jabr w-al-musqagalah
(Science of the Reunion and the
Opposition) starts out with a discussion of
the quadratic equation.
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 thirtynine?)
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
What
is the area of the missing square?
39)
-Missing square: 25

Total area?
Total area: 64
So what is the length of one of the sides of this bigger square?
-Answer: √ 64=8

Therefore how can we conclude that x=3?
Answer: Since the side of the big square is x+5, we can
conclude that x=3
Back to The Discussion

X is the unknown; the problem translates to x2+10x=39

2
Answer: x  5  39  5  25  39  5  64  5  8  5  3

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 ??
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:
x
The square is 81
32 135  3 
9  135  3 
144  3  12  3  9
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-Khwarizmi’s 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
Today
In 17th Century Rene Descartes
published La Geometrie, which
developed into modern mathematics
 General equation: ax2+bx+c=0
 Written:
2

 b  b  4ac
x
2a
Timeline

1500BC
580 BC
400 BC
300 BC

598-665AD

800AD

1145AD

1637AD



Egyptians made a table.
Pythagoras hates irrational numbers.
Babylonians solved quadratic equations.
Euclid developed a geometrical approach
and proved that irrational numbers exist.
Brahmagupta took the Babylonian method
that allowed the use of negative numbers.
Al-Khwarizmi removed the negative and
wrote a book Hisab al-jabr w-almusqagalah (Science of the Reunion and
the Opposition)
Abraham bar Hiyya Ha-Nasi
(Savasaorda)wrote the book Liber
embadorum –contained the complete
solution to the quadratic equation.
Rene Descartes published La Geometrie
containing the quadratic formula we know
today.
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 <http://jwilson.coe.uga.edu/EMT725/Class/Brooks/
Brahmagupta/Brahmagupta.htm>.
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
<http://www.mathsisgoodforyou.com/index.htm>.
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 <http://wwwgroups.dcs.st-and.ac.uk/~history/PrintHT/
Quadratic_etc_equatins.html>.
"The History Behind the Quadratic Formula." BBC homepage. 13 Oct. 2004. 11
Sept. 2006<http://www.bbc.co.uk/dna/h2g2/A2982576>.