Linear Thinking
Chapter 9:
Solving First Degree
Equations
In the Beginning
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Earliest mathematicians developed ways to find
solutions to equations of the first degree.
- Babylonians
- Egyptians
- Chinese
- Greeks
As we already know, the Egyptians documented all
their “findings” in the “Rhind Papyrus” and the
Chinese wrote in the “Nine Chapters of the
Mathematical Art.”
Early Algebra Around the world
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The Babylonians in around 530 BC
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There were no symbols
No negative numbers
Every number was rounded to a whole number
The Egyptians had much less knowledge of Algebra
than the Babylonians. Their “Rhind Papyrus” only
consisted of linear equations.
In Greece, their concept of algebra took a more
graphical approach. They borrowed a lot of their
knowledge from Asia Minor and Babylonia.
Rhind Papyrus
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Collection of problems used to train young scribes
“A quantity; its half and its third are added to it. It
becomes 10.”
Contained two different versions of problems
–
Modern method (shorter and more algebraic)
1
1
1
10  x  x  x
2
3
4
–
False position (longer and less computational)
False Position
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Definition: posit an answer we know is incorrect,
because it is easy to compute with, then multiply it to
reach the correct answer
Let’s look at an example:
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A quantity; it’s fourth is added to it. It becomes 15.
x+1/4x=15
Assume x=4 (easier to compute a fourth of four)
Take 4 and add its fourth to it: 4+1=5
5 does not equal 15 so multiply 5x3 to get 15 which means
multiply 4x3=12
12 is the answer
Double False Positions
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Double false positions is a way to solve for first degree equations
without any manipulation
So effective it was used after the introduction of algebraic functions
Ex: From Daboll’s Schoolmaster’s Assistant published in the early
1800’s
A purse of 100 dollars is to be divided
among four brothers Adam, Benjamin,
Caleb, and Daniel, so that Benjamin may
have four dollars more than Adam, and Caleb
eight dollars more that Benjamin, and
Daniel twice as many as Caleb; what is
each man’s share of the money?
The Brothers
Today’s method:
x  x  4  x 12  2 x  12  100
Old method:
First take a guess: A=6, then B=10, C=18 and D=36
but this only equals 70
Take another guess: A=8, B=12, C=20 and D=40 but
only equals 80
Now 6
30
8
20
Double False Positions
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Cross multiply: 6x20=120
8x30=240
Take the difference: 240-120=120
Lastly, divide by difference of errors:
30-20=10 so 120/10=12
So if Adam has $12, then Benjamin has $16,
Caleb has $24 and Daniel has $48, when
added all together = $100
Why does it work?
Graph it and find out:
100
75
50
25
3
6
9
12
Rise over Run
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Remember the form mx+b=y
We have to find x when y=100
So our two points are our two guess that we
made earlier (6,70) and (8,80) we want
(x,100)
100-70 = 100-80
x-6
x-8
cross multiply and solve
and the answer is the same x=12
Restrictions
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False Positions only works with the form
Ax=B
Double Positions only works when the errors
are either both an underestimate, or both an
overestimate
If the errors are of different types,
sum of the products
sum of the errors
Modern way of thinking
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Thinking of equations as lines is only as recent as
the 17th century
Before the 17th century linear thinking was limited
because negative and complex numbers were still
difficult concepts to grasp. The only one to
recognize negatives was Brahmagupta.
The concept of linear thinking is based around the
change of output proportional to the change in input.
This is something that the Egyptians, Babylonians,
or Chinese did understand.
Timeline
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1750 BCE – Babylonians solve linear and quadratic
equations
1650 BCE – Rhind Papyrus first knowledge of solved
linear equations
500s – obtains solutions by methods similar to
modern methods
628 – Brahmagupta methods of solving linear
equations
750 – Al-Khawarizmi systemization of the theory of
linear equations
Resources
Berlinghoff and Gouvea: “Math through the Ages: A
Gentle History for Teachers and Others”
Math Timeline:
http://en.wikipedia.org/wiki/Timeline_of_mathematics
Linear Equation Article
http://en.wikipedia.org/wiki/Linear_equation
Berggren, J. Lennart, M.S.Ph.C. “Equations, Theory of”
http://www.history.com/encyclopedia.do?vendorId=F
WNE.fw..eq051300.a
History of Math Notes:
http://www.math.sfu.ca/histmath/math380notes/math38
0.html
Thank You
Sara and Lisa