Transcript - Who Invented Algebra?

Algebra
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Many advanced ideas in
Algebra make use of the
idea of symmetry.
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Who did this to us and why?
How an algebra problem was
solved a thousand years ago
If some one say: "You divide ten into two parts: multiply
the one by itself; it will be equal to the other taken eightyone times." Computation: You say, ten less thing,
multiplied by itself, is a hundred plus a square less twenty
things, and this is equal to eighty-one things. Separate the
twenty things from a hundred and a square, and add them
to eighty-one. It will then be a hundred plus a square,
which is equal to a hundred and one roots. Halve the
roots; the moiety is fifty and a half. Multiply this by itself,
it is two thousand five hundred and fifty and a quarter.
Subtract from this one hundred; the remainder is two
thousand four hundred and fifty and a quarter. Extract the
root from this; it is forty-nine and a half. Subtract this
from the moiety of the roots, which is fifty and a half.
There remains one, and this is one of the two parts
Solving the same problem today!
10 − 𝑥 2 = 81𝑥
𝑥 2 − 20𝑥 + 100 = 81𝑥
𝑥 2 − 101𝑥 + 100 = 0
𝑥 − 1 𝑥 − 100 = 0
The solutions are x=1 and x=100
Algebra traded the words in for symbols,
making problems and solutions easier to
understand once you learn the language!
Counting, Subtotals and the Abacus
• The Romans kept the accounts for
their businesses using counting
boards, hand-abaci, and Roman
Numerals.
• The Roman empire’s engineers
and architects used the same
methods for their calculations.
• The abacus may have spread East
to China along the Silk Road.
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Commerce on the Mediterranean
Between the 12th
and 14th Centuries,
the city of Pisa
operated a large
trading empire
throughout the
Mediterranean.
• Around 1200, Leonardo Pisano
Fibonacci, from Pisa, learned from
Arab scholars about doing arithmetic
with pen and paper and the HinduArabic numerals, 9, 8 7,6,5,4,3,2,1, 0.
• Over the next 300 years, the new
system of arithmetic became
important. More than a thousand
hand-written arithmetic texts were
written.
• Fibonacci used this new system to
calculate wages,
calculate profits,
compute interest, exchange money
• By 1500, Hindu-Arabic arithmetic
had replaced the Roman system of
arithmetic throughout Europe.
Harnessing wind and water power
In 16th Century Europe, craftsmen skilled in the
mechanical arts worked on complex problems
such as using wind and water power to drive
machinery for milling and weaving.
In the 16th Century, complex, powerful
machines were built from carefully fitted
wedges, levers, wheels, and pulleys
Careful measurement and calculation were
required to make sure the machines actually
worked
But, multiplication and division were regarded as time consuming and error prone.
In response to this need, new notations and instruments were developed for calculation
These tools could be mastered by craftsmen not well schooled in mathematics
Calculation in the late 16th Century
Mathematical Instruments
First Modern Equations
• Many mathematical instruments
were invented to help avoid errors
• Robert Recorde was the first to use
=, +, and – to write out calculations
as equations
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Sector
Napier Bones
Slide rules
Tables of functions
• With equations, a craftsman could
plan out and double check all the
steps in a long calculation
Variables invented in the 17th Century
• Because of Viète’s discoveries,
we can write the quadratic
formula as
𝑥=
• Around 1600, François Viète
invented the use of variables
to represent unknown
quantities in equations
• Viète used different parts of
the alphabet to distinguish
parameters from genuine
unknowns.
−𝑏± 𝑏2 −4𝑎𝑐
2𝑎
• During the rest of the 17th
Century other mathematicians
and scientists made important
use of the idea of an equation
describing the relationship
between two unknown
variables.
Graphing on the Coordinate Plane
• Pierre de Fermat in 1693 was first
to use a coordinate plane to draw
graphs representing the set of
possible solutions to an equation
in two or three variables.
• Fermat observed that the shape
of the graph, whether curved or
straight, could be predicted by
analyzing properties of the
equation
• Fermat’s idea of graphing
equations gave engineers an
important new tool for
studying physical systems.
• Others improved on Fermat’s
graphs by using both positive
and negative coordinates.
Functions in Algebra
• Leonhard Euler in 1734 introduced the
function notation. He wrote equations in
the form f(t)=2t+5.
• Euler found that for many functions
describing physical systems, the rate of
change itself was not constant
• Euler used functions to describe how
measurable quantities changed as time
passed.
• Studying these complex algebraic
equations lead to advanced mathematics
such as calculus and differential
equations.
Scientists learned to model many
classes of physical systems as
algebraic functions. A few of these
important classes of physical
principles are listed on the right.
18th and 19th Century Scientific Principles
• Laws of Motion
• Laws of Mechanics
• Laws of Optics
• Laws of Fluid Mechanics
• Laws of Chemistry
• Laws of Electricity
• Laws of Thermodynamics
Algebra in the Industrial Revolution
• The industrial revolution began in
the early 18th Century with the
construction, in England, of the first
factories driven by stream power.
• As the industrial revolution gained
steam, engineers and craftsmen
were using physical principles
described as algebraic functions
when inventing new things
• Respected mathematicians of the
era wrote popular textbooks on
Algebra.
• During the 18th and 19th Centuries,
Algebra became widely used
18th and 19th Century Inventions
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Steam Engine
Electric Telegraph
Power Loom
Self-winding Clock
Seed Drill
Threshing Machine
Cotton Gin
Battery
Wireless Communication
Tungsten Steel
Thermometer
Train Locomotive
Telephone
Gasoline Engine
Electric Light
Electric Generator
Electric Motor
Automobile
Air Glider
Spectrograph
Computers in Algebra
As Algebra became more widely used,
calculations depended on mathematical tables,
often the size of books, which recorded the
values of a function. Produced by hand, these
tables were often full of errors.
Babbage’s Difference Engine
In an effort to eliminate these errors, Charles
Babbage, in the mid 19th Century, designed but
failed to build mechanical calculating engines.
Other 19th Century mathematicians tried to
reduce the amount of repetitive hand calculation
needed to solve large systems of linear equations
and equation of degree five or more.
However, it was not until the electronic
computers of the 1940s that these repetitive
tasks could be automated.
ENIAC: first electronic computer
What’s next for Algebra?
Today, computer algebra systems can
automate all of the calculation in an algebra
problem, allowing users to focus on
discovery activities such as
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exploring deeper mathematical ideas
connected with algebra
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using the tools of algebra to investigate
realistic, practical problems.
In the spirit of those who invented Algebra
over the last 1500 years, we must not regret
that computers can solve an equation for an
unknown faster than we can.
Instead, we must embrace our new freedom
by using algebra to explore, understand, and
invent the world around us.