Linear Thinking Chapter 9: Solving First Degree Equations

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.”

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.

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)

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 Let’s look at an example: – 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 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: 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

Double False Positions Cross multiply: 6x20=120 8x30=240 Take the difference: =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:

Rise over Run 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) = x-6 x-8cross multiply and solve and the answer is the same x=12

Restrictions 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 Thinking of equations as lines is only as recent as the 17 th century Before the 17 th 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 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: Linear Equation Article Berggren, J. Lennart, M.S.Ph.C. “Equations, Theory of” WNE.fw..eq a History of Math Notes: 0.html

Thank You Sara and Lisa