1. Christie Epps Abby Krueger Maria Melby Brett Jolly
Algebraic Symbolism
Christie Epps
Abby Krueger
Maria Melby
Brett Jolly

2. “Every meaningful mathematical statement can also be expressed in plain language. Many plain language statements of mathematical expressions would fill several pages, while to express them in mathematical notation might take as little as one line. One of the ways to achieve this remarkable compression is to use symbols to stand for statements, instructions and so on.”
Lancelot Hogben

3. Three Stages Rhetorical (1650 BCE-200 CE):
algebra was written in words without symbols.
Syncopated (200 CE-1500 CE):
algebra which used some shorthand or abbreviations
Symbolic (1500 CE- present):
algebra which used mainly symbols

4. Historically algebra developed in Egypt and Babylonia around 1650 B. C
Historically algebra developed in Egypt and Babylonia around 1650 B.C.E.
Developed in response to practical needs in agriculture, business, and industry.
Egyptian algebra was less sophisticated possibly because of their number system
Babylonian influence spread to Greece ( B.C.E.) then to the Arabian Empire and India (700 C.E.) and onto Europe (1100 C.E.).

5. Two factors played a large role in standardizing mathematical symbols:
Invention of the printing press
Strong economies who encouraged the traveling of scholars resulting in the transmission of ideas
Still today there are differences in the use of notation:
Log and ln
In Europe they use a comma
where Americans use a period
(i.e. 3,14 for 3.14).
Printing press 1445 C.E.

6. Rhetorical Algebra 1650 BCE-200 CE no abbreviations or symbols
Early Babylonian and Egyptian algebras were both rhetorical
In Greece, the wording was more geometric but was still rhetorical.
The Chinese also started with rhetorical algebra and used it longer.

7. Greek Contributions Three periods:
1. Hellenic (6th Century BCE): Pythagoras, Plato, Aristotle
Pythagorean Theorem
2. Golden Age (5th Century BCE): Hippocrates, Eudoxus
Translation of arithmetical operations into geometric language
3. Hellenistic (4th Century BCE): School of Alexandria, Euclid, Archimedes, Apollonius, Ptolemy, Pappus
Euclid’s Elements, conic sections, cubic equations.

8. Chinese History
Decline of learning in the West after the 3rd century BCE but development of math continued in the East.
The first true evidence of mathematical activity in China can be found in numeration symbols on tortoise shells and flat cattle bones (14th century B.C.E.).
About the same time the magic square was founded and led to the development of the dualistic theory of Yin and Yang. Yin represents even numbers and Yang represents odd numbers.
Between BCE the Chinese discovered the equivalent of the Pythagorean Theorem.
300 BCE to the turn of the century: square and cube roots, systems of linear equations, circles, volume of a pyramid
CE we see Liu Hui and his approximation of pi
By 600 CE there was translation of some Indian math works in China
700 CE: The Chinese are credited with the concept of 0.
CE: algebraic equations for geometry

9. Syncopated Algebra 200 CE-1500 CE some shorthand or abbreviations
Started with Diophantus and lasted until 17th Century BCE.
However, in most parts of the world other than Greece and India, rhetorical algebra persisted for a longer period (in W. Europe until 15th Century CE).
The revival of the Alexandrian school was accompanied by a fundamental change of orientation of math research.
Geometry was the foundation of math, now the number was the foundation which resulted in the independent evolution of Algebra

10. Diophantus
This independence of algebra is attributed to Diophantus who used syncopated algebra in his Arithmetica (250 CE).
He defined a number as a collection of units
Introduced negative numbers but used them only in indeterminate computations and sought only positive solutions
Introduced signs for an unknown and its powers
Had a symbol for equality and an indeterminate square

11. Aryabhata and Brahmagupta
Ist century CE from India
Developed a syncopated algebra
Ya stood for the main unknown and their words for colors stood for other unknowns

12. Symbolic Algebra mainly symbols
Began to develop around 1500 but did not fully replace rhetorical and syncopated algebra until the 17th century
Symbols evolved many times as mathematicians strived for compact and efficient notation
Over time the symbols became more useable and standardized

13. “Early Renaissance” Mathematics
Transmission by 3 routes:
Arabs who conquered Spain & established the first advanced schools
Arab east
Turkey/Greece

14. Jordanus Nemorarius
Picked letters in alphabetical order to stand for concrete numbers with no distinction between knowns and unknowns.
He used Roman numerals and did not have signs for equality and algebraic operations.

15. 14th Century
Italian mathematicians translated Arab words into Latin for the unknown and its powers.
co – x (thing)
ce – x2
cu – x3
ce-ce – x4
R – square root
q.p0 – y
Pui – addition
Meno – subtraction

16. 15th Century: Revival of Algebraic Investigations
Luca Pacioli (1494)
Had symbol for the constant and was the first to show symbols for the first 29 powers of the unknown.
Symbol for a second unknown
Symbols for addition and subtraction
Bombelli:
3√2+√-3
R.c.L2puidimeno
di menoR.q.3
1 – unknown
powers
Stevin’s power notation
1, 2, … - unknowns and powers

17. Johannes Widman (1462-1498): German
“…- is the same as shortage and + is the same as excess.” (Bashmakova)
Nicolas Chuquet ( ): French
exponential notation (12x^3 written as 12^3)
symbolism for the zeroth power
introduced negative numbers as exponents

18. 16th Century: Age of Algebra
Christoff Rudolff ( ): German
Coss, first German algebra book
current +,- signs used for first time in algebraic text
modern symbol for square root (√ )
Michael Stifle ( )
brought a close to the evolution of algebraic symbolism
used (Latin) A, B, C,… to denote unknowns
notation adopted in Germany & Italy
Robert Recorde ( ):
modern symbol for equality

19. Solution of the Cubic Equation
Scipione del Ferro ( )
Niccolo Tartaglia ( )
Girolamo Cardano ( )
“irreductible” case
The form of √m with m < 0

20. Rafael Bombelli (1526-1573): Italy
introduced complex numbers and used them to solve algebraic equations
introduced successive integral powers of rational numbers
explains “irreductible” case

21. Francios Viete (1540-1603): France
“An Introduction to the Art of Analysis”
introduced the language of formulas into math
IMPORTANT STEP: use of literal notation for knowns and unknowns
allowed writing equations and identities in general form
“The end of the 16th century marked a crucial turning point in the evolution of algebra, for the first time it found its own language, namely the literal calculus.” (Bashmakova)

22. William Oughtred Born in Eton, Buckinghamshire, England in 1574
Died in Albury, Surrey, England in 1660

23. William Oughtred Wrote Clavis Mathematicae in 1631
Described Hindu-Arabic notation and decimal fractions
Created new symbols
Multiplication x
Proportion ::
Pi for circumference  (not for ratio of circumference to diameter)

24. Rene` Descartes
Born in France, 1596
Died in Sweden, 1650

25. Cartesian Graph Created, along with Fermat, the Cartesian graph
Brought algebra to geometry
Allowed circles and loops to be graphed from algebraic equations

26. Imaginary Roots Created the name imaginary for imaginary roots
Descartes says “one can ‘imagine’ for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.”
(J.J. O’ Conner and E. F. Robertson)

27. Polynomial Roots
Stated a polynomial that disappears at y has a root x-y.
Reason why solving for the roots using the factor theorem form: (x-y)*(x-z)=r

28. Variables Descartes was also known for today’s variables
Changed unknowns from Viete’s (a e i o u) to (u v w x y z) –end of alphabet.
Created knowns from consonants to (a b c d) –beginning of alphabet

29. Descartes Changes in Algebraic Symbolism Time it took
Each person affected it in their own way

30. Thomas Harriot (1560 – 1621) Known best for his work in algebra
Introduced a simplified notation for algebra
Debate as to who was first, Viete or Harriot
Ahead of his time in his theory of equations and notation simplification
Accepted real and imaginary roots
Worked with cubics
If a, b, c are the roots of a cubic then the cubic equation is (x-a)(x-b)(x-c)=0

31. Reproduction of his solution to an equation of degree four:
Harriot’s Notation
Our Notation
aaaa-6aa+136a=1155
a4 – 6a2 +136a = 1155
aaaa – 2aa + 1 = 4aa – 136a
a4 – 2a2 + 1 = 4a2 – 136a
(aa – 1)(aa – 1) = 2(2a – 34)(a – 17)
(a2 – 1)2 = 2(2a – 34)(a – 17)
aa – 1 = 2a -34
a2 – 1 = 2a -34
aa – 2a = -33
a2 -2a = -33
aa -2a + 1 =
a2 -2a + 1 =
(a – 1)(a – 1) = -32
(a – 1) 2 = -32
a -1 = √-32 or -√-32
a = 1 + √-32 or a = 1 - √-32 Complex Roots
aa – 1 = 34 – 2a
a2 -1 = 34 -2a
aa + 2a = 35
a2 + 2a = 35
aa + 2a + 1 =
a2 + 2a + 1 = 35 +1
(a + 1)(a + 1) = 36
a + 1 = √36 or -√36……a = 5 or -7
Example taken from
Only change made to his work was the equals sign was different

32. Harriot cont.
He never published any of his findings, circulated amongst his peers
His works were published after his death (Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (1631))
were badly edited
< > controversy
He was also an explorer, navigational
expert, scientist and astronomer
Worked with Sir Walter Raleigh ~1583
did not discuss negative solutions

33. Albert Girard (1595 – 1632)
Worked with sequences, cubics, trigonometry, and military applications
Had different representation of algebraic formulas:
x3 = 13x => 1 3 X , with a circle around the 3 and 1 superscripted
1626-publishes an essay on trigonometry
first to use negative numbers in geometry
introduces sin, cos, and tan
also included formulas for area of a spherical triangle

34. Girard cont.
1629- Invention nouvelle en l'algebre (New Discoveries in Algebra) is published
writes the beginnings of the Fundamental Theorem of Algebra
talks about relationship between roots and coeffiecients
allowing negative and imaginary roots to equations
his understanding of negative solutions lead the way toward the number line
“laid off in the direction opposite that of the positive”
introduced the idea of a fractional exponent
numerator = power, denominator = root
introduced the modern notation for higher roots
3√9 instead of 91/3

35. Girard cont.
1634- Formulates the inductive definition fn+2= fn+1+ fn for the Fibonacci Sequence
Interested in the military applications of mathematics
This was a time of discovery and conquering
The “New World” was being explored….America is being colonized