Download MERIT Number and Algebra

MERIT Number and Algebra
Algebra T polynomials in one variable
This session:
• Linear and quadratic polynomials
Next sessions:
Algebra
• It is a branch of mathematics which uses
symbols to represent numbers and
variable quantities. It is used to express
relationships between variable quantities.
• More on the algebra of
polynomials
• Polynomials and abstract algebra
The History of Algebra
• Algebra may divided into "classical
algebra" (equation solving or "find the
unknown number" problems) and "abstract
algebra", also called "modern algebra" (the
study of groups, rings, and fields).
• Classical algebra has been developed
over a period of 4000 years. Abstract
algebra has only appeared in the last 200
years.
• Understanding Algebra: The process
involved in learning algebra parallels the
historic development of algebra as a
symbol system.
• Rhetorical Stage: (3000 B.C TO A.D. 250)
era in which problems and solutions were
written out in words.
Egyptian Algebra
• Syncopated Stage (250 to 1591) era in
which letters are used to represent
unknown quantities and abbrevations were
created for operations. Letters represented
unknowns for which there was one unique
solution.
• Symbolic Stage: (1591-> present) letters
used to represent both the given and
unknown quantities.
MORE INFO
• http://www.math.buffalo.edu/mad/AncientAfrica/mad_ancient_egypt_algebra.html#rhind28
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• Much of our knowledge of ancient Egyptian
mathematics, including algebra, is based on
the Rhind papyrus, which was written about
1650 B.C. and is thought to represent the
state of Egyptian mathematics of about 1850
B.C.
• They could solve problems equivalent to a
linear equation in one unknown.
• Their method was what is now called the
"method of false position."
• Their algebra was rhetorical, that is, it used
no symbols. Problems were stated and
solved verbally.
• The mathematics of the Old Babylonian
Period (1800 - 1600 B.C.) was more
advanced than that of Egypt.
• Their excellent sexagesimal [base 60]
led to a highly developed algebra.
• They had a general procedure
equivalent to solving quadratic
equations, although they recognized
only one root and that root had to be
positive.
• They worked out a calendar for the year
along with developed units of
measurement. As early as 2500 B.C.
the Babylonians had established
standards for length, weight, and
volume.
• Mathematical and astronomical texts
show that the Babylonians came up with
the 360 degree circle and also the 60
minute hour. The Babylonians could
predict eclipses of the sun and moon.
Babylonian Algebra
MORE INFO:
• http://www.math.tamu.edu/~don.allen/history/babylon/babylo
n.html
• http://www.angelfire.com/il2/babylonianmath/
• In effect, they had the quadratic formula.
• They also dealt with the equivalent of
systems of two equations in two
unknowns.
• They considered some problems involving
more than two unknowns and a few
equivalent to solving equations of higher
degree
• The Babylonian system also had units.
They did not have a symbol for zero, but
the idea of zero was used.
• When Babylonians wanted to express
zero they just left a blank space in the
number they were writing.
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Maya Mathematics
MORE INFO:
•http://www.michielb.nl/maya/math.html
•http://www.hanksville.org/yucatan/mayamath.html
• The Mayans devised a counting system that was
able to represent very large numbers by using
only 3 symbols, a dot, a bar, and a symbol for
zero, or completion, usually a shell.
• Like our numbering system, they used place
values to expand this system to allow the
statement of very large values.
• Their system has two significant differences from
the system we use: 1) the place values are
arranged vertically, and 2) they use a base 20,
or vigesimal, system.
• For example,the number in the third place has a
value of 202, or 400, times the value of the
numeral (whereas the number in the third place
in our number system represents 100 (102)).
Roman Algebra
MORE INFO:
• http://home.hiwaay.net/~lkseitz/math/roman/
• Instead of ten digits like we have today, the
Maya used a base number of 20. (Base 20 is
vigesimal.)
• Because the base of the number system was 20,
larger numbers were written down in powers of
20.
• We do that in our decimal system too: for
example 32 is 3*10+2.
• In the Maya system, this would be 1*20+12,
because they used 20 as base.
• It was very easy to add and subtract using this
number system, but they did not use fractions.
Other Achievements
• With crude instruments (by today’s standards) the Maya were
able to calculate the length of the year to be 365.242 days
(the modern value is 365.242198 days).
• Two further remarkable calculations are of the length of the
lunar month. At Copán (now on the border between
Honduras and Guatemala) the Mayan astronomers found
that 149 lunar months lasted 4400 days, which gives 29.5302
days as the length of the lunar month.
• At Palenque in Tabasco they calculated that 81 lunar months
lasted 2392 days. This gives 29.5308 days as the length of
the lunar month.
• The modern value for the lunar month is 29.53059 days.
• Addition and subtraction were made easy with the
concepts of place holders and this was the first real
system in base 10.
• I – represents one
• II – means “one plus one” (two), followed by III (three)
• IV – means “five minus one” (four).
• V – represents five.
• VI – The V followed by a 'I' means “five plus one” (six)
followed by VII (seven), VIII (eight).
• IX – means “ten minus one” (nine).
• X – represents ten
• L – represents fifty
• C – represents one hundred
• D – represents five hundred
• M – represents one thousand
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Example
Hindu Algebra
• 982
= 900 + 80 + 2
• 900 = one thousand minus one hundred = CM
• 80 = fifty plus ten plus ten plus ten = LXXX
• 2 = one plus one = II
= CM + LXXX + II
= CMLXXXII.
MORE INFO:
• http://www.geocities.com/dipalsarvesh/mathematics.html
• The Hindu civilization dates back to at least 2000
B.C.
• Their record in mathematics dates from about 800
B.C.
• Most Hindu mathematics was motivated by
astronomy and astrology.
• A base ten, positional notation system was standard
by 600 A.D.
• They treated zero as a number and discussed
operations involving this number.
• The Hindus introduced negative numbers to
represent debts.
• The Hindus recognized that quadratic equations
have two roots, and included negative as well as
irrational roots.
Abu Ja'far Muhammad ibn
Musa Al- Khwarizmi
• Born: about 780 in Baghdad (now in
Iraq)
Died: about 850
• Known as the father of
Algebra
• He introduced the natural
numbers
Arabic Algebra
MORE INFO
• http://members.aol.com/bbyars1/algebra.html
• http://www.ms.uky.edu/~carl/ma330/project2/al-khwa21.html
He first reduced an equation (linear or quadratic) to one of
six standard forms:
1. Squares equal to roots.
x2 = 2x
2. Squares equal to numbers. x2 = 2
3. Roots equal to numbers.
10x = 2
4. Squares and roots equal to numbers;
x2 + 10 x = 39
5. Squares and numbers equal to roots;
x2 + 21 = 10 x.
6. Roots and numbers equal to squares;
3 x + 4 = x2.
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Example: x2 + 10 x = 39
•Symbolic Stage: (1591->
present) letters used to
represent both the given
and unknown quantities.
Al- Khwarizmi solved this problem as follows:
... a square and 10 roots are equal to 39 units. The
question therefore in this type of equation is about as
follows: what is the square which combined with ten of
its roots will give a sum total of 39? The manner of
solving this type of equation is to take one-half of the
roots just mentioned. Now the roots in the problem
before us are 10. Therefore take 5, which multiplied by
itself gives 25, an amount which you add to 39 giving 64.
Having taken then the square root of this which is 8,
subtract from it half the roots, 5 leaving 3. The number
three therefore represents one root of this square, which
itself, of course is 9. Nine therefore gives the square.
European Algebra after 1500
• At the beginning of this period, zero had
been accepted as a number
• Irrationals were used freely although
people still worried about whether they
were really numbers.
• Negative numbers were known but were
not fully accepted.
MORE INFO
• http://members.aol.com/jeff570/mathsym.html
• Complex numbers were as yet
unimagined. Full acceptance of all
components of our familiar number system
did not come until the 19th century.
• Algebra in 1500 was still largely rhetorical.
Renaissance mathematics was to be
characterized by the rise of algebra.
• In the 16th century there were great
advances in technique, notably the
solution of the cubic and quadratic
equations - achievements called by Boyer
“perhaps the greatest contribution to
algebra since the Babylonians learned to
solve quadratic equations almost four
millennia earlier.”
• There were also at this time many
important improvements in symbolism
which made possible a science of algebra.
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• The landmark advance in symbolism was
made by Viète (French, 1540-1603) who
used letters to represent known constants
(parameters).
• This advance freed algebra from the
consideration of particular equations and
thus allowed a great increase in generality
and opened the possibility for studying the
relationship between the coefficients of an
equation and the roots of the equation
("theory of equations")
Polynomial Function
Two branches of studying polynomial
functions such as:
f(x) = a0 x0 + a1x1+ a2x2 + … + anxn
• Symbolic algebra reached full maturity
with the publication of Descartes' La
Géométrie in 1637.
• This work also gave the world the
wonderfully fruitful marriage of algebra and
geometry that we know today as analytic
geometry (developed independently by
Fermat and Descartes).
Solving Polynomial EquationsAlgebraic Method
• Modern Algebra view - theory of solving
polynomials by factoring
• Modeling view – solving real world
problems modeled by polynomial functions
which almost never factor
• Set Polynomial Equation equal to zero.
• Factor the resulting polynomial expression
into a product of linear expressions.
• Reduce the equation to a series of linear
equations.
This is a classic example of analytic
reasoning – reducing a more complex
problem to one we already know how to
solve.
Example
Zero Product Rule
f ( x) = x 2 + x − 10
• When does f(x) = 20?
• If a • b = 0, then a=0 or b=0
• Question: What property of Modern
Algebra that we have studied supports this
statement?
• Question: What does this indicate about
the replacement set or domain set?
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Linear equation a *x = b
a+x = b
ax = b
a*x=b
http://www.math.wvu.edu/~mays/AVdemo/Labs/Lab01/Lab01-02.htm
Linear equation ax+b = cx+d
• Two operations implicit—need a Field
Theorem
If a, b, c, and d are elements of a field F and
a≠c, then the equation ax+b=cx+d has
exactly one solution in F. If a=c then…
Babylonian problem
Find two numbers whose sum is 10 and
whose product is 18.
http://www.math.wvu.edu/~mays/AVdemo/deployed/Diophantine.html
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Quadratic formula
ax 2 + bx + c = 0
x=
− b ± b 2 − 4ac
2a
http://www.math.wvu.edu/~mays/AVdemo/126/Quadratic/QuadraticFunctions01.htm
http://www.math.wvu.edu/~mays/AVdemo/deployed/QuadraticComplexRoots.html
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