Linear Thinking

1
Linear Thinking

• Chapter 9
• Solving First Degree Equations

2
In the Beginning

• 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.

3
Early Algebra Around the world

• The Babylonians in around 530 BC
• 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.

4
Rhind Papyrus

• 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)
• False position (longer and less computational)

5
False Position

• Definition posit an answer we know is incorrect,
  because it is easy to compute with, then multiply
  it to reach the correct answer
• Lets look at an example
• A quantity its fourth is added to it. It
  becomes 15.
• x1/4x15
• Assume x4 (easier to compute a fourth of four)
• Take 4 and add its fourth to it 415
• 5 does not equal 15 so multiply 5x3 to get 15
  which means multiply 4x312
• 12 is the answer

6
Double False Positions

• 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 Dabolls Schoolmasters Assistant
  published in the early 1800s
• 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 mans share of the money?

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The Brothers

• Todays method
• Old method
• First take a guess A6, then B10, C18 and D36
  but this only equals 70
• Take another guess A8, B12, C20 and D40 but
  only equals 80
• Now 6 30
• 8 20

8
Double False Positions

• Cross multiply 6x20120
• 8x30240
• Take the difference 240-120120
• Lastly, divide by difference of errors
• 30-2010 so 120/1012
• So if Adam has 12, then Benjamin has 16, Caleb
  has 24 and Daniel has 48, when added all
  together 100

9
Why does it work?

• Graph it and find out

10
Rise over Run

• Remember the form mxby
• We have to find x when y100
• 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 x12

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Restrictions

• False Positions only works with the form AxB
• 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

12
Modern way of thinking

• 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.

13
Timeline

• 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

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Resources

• Berlinghoff and Gouvea Math through the Ages
  A Gentle History for Teachers and Others
• Math Timeline http//en.wikipedia.org/wiki/Timeli
  ne_of_mathematics
• Linear Equation Article http//en.wikipedia.org/wi
  ki/Linear_equation
• Berggren, J. Lennart, M.S.Ph.C. Equations,
  Theory of
• http//www.history.com/encyclopedia.do?vendorIdF
  WNE.fw..eq051300.a
• History of Math Notes
• http//www.math.sfu.ca/histmath/math380notes/math3
  80.html

15
Thank You

• Sara and Lisa