Transcript Algebraic Symbolism - Iowa State University

Algebraic
Symbolism
Christie Epps
Abby Krueger
Maria Melby
Brett Jolly
“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
Three Stages
1. Rhetorical (1650 BCE-200 CE):
algebra was written in words without symbols.
2. Syncopated (200 CE-1500 CE):
algebra which used some shorthand or
abbreviations
3. Symbolic (1500 CE- present):
algebra which used mainly symbols
• 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 (500-300
B.C.E.) then to the Arabian Empire and India (700
C.E.) and onto Europe (1100 C.E.).
• 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.
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.
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.
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 1000-500 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
200-300 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.
1000-1200 CE: algebraic equations for geometry
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
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
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
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
“Early Renaissance” Mathematics
Transmission by 3 routes:
1.
Arabs who conquered Spain & established the first
advanced schools
1.
Arab east
1.
Turkey/Greece
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.
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
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
2 3 - powers
Stevin’s power notation
– 1, 2, … - unknowns and powers
Johannes Widman (14621498): German
“…- is the same as shortage and + is the same as
excess.” (Bashmakova)
Nicolas Chuquet (1445-1488): French
• exponential notation (12x^3 written as 12^3)
• symbolism for the zeroth power
• introduced negative numbers as exponents
16th Century: Age of Algebra
Christoff Rudolff (1499-1545): German
• Coss, first German algebra book
• current +,- signs used for first time in algebraic text
• modern symbol for square root (√ )
Michael Stifle (1487-1567)
• brought a close to the evolution of algebraic symbolism
• used (Latin) A, B, C,… to denote unknowns
• notation adopted in Germany & Italy
Robert Recorde (1510-1558):
• modern symbol for equality
Solution of the Cubic Equation
•
•
•
•
Scipione del Ferro (1456-1526)
Niccolo Tartaglia (1499-1557)
Girolamo Cardano (1501-1576)
“irreductible” case
– The form of √m with m < 0
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
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)
William Oughtred
• Born in Eton,
Buckinghamshire,
England in 1574
• Died in Albury,
Surrey, England in
1660
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)
Rene` Descartes
• Born in France, 1596
• Died in Sweden, 1650
Cartesian Graph
• Created, along with Fermat, the Cartesian graph
• Brought algebra to geometry
• Allowed circles and loops to be graphed from
algebraic equations
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)
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
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
Descartes
• Changes in Algebraic Symbolism
• Time it took
• Each person affected it in their own way
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
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 + 1156
a4 – 2a2 + 1 = 4a2 – 136a + 1156
(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 = -33 + 1
a2 -2a + 1 = -33 + 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 = 35 + 1
a2 + 2a + 1 = 35 +1
(a + 1)(a + 1) = 36
a + 1 = √36 or -√36……a = 5 or -7
Example taken from http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Harriot.html
Only change made to his work was the equals sign was different
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
Albert Girard (1595 – 1632)
• Worked with sequences, cubics, trigonometry, and
military applications
• Had different representation of algebraic formulas:
– x3 = 13x + 12 => 1 3 X 13 1 +12, 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
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
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