A Little History of Some
Familiar Mathematics
Gary Talsma
Mathematics and Statistics Department
Calvin College
Grand Rapids, MI 49546
([email protected])
First-Year Seminar
November 30, 2011
1
I.
The Quadratic Formula(s)
A.
Muhammad ibn Musa al-Khwarizmi (ca. 780-850)
1.
Persian mathematician
2.
Worked at the Bayt al-Hikma (House of Wisdom) in Baghdad
3.
Books
a)
Algoritmi de numero Indorum (Al-Khwarizmi on the Hindu Art of
Reckoning)
• Explained the basics of the decimal number system from India
• This Latin translation was later very influential in popularizing HinduArabic numeration in Europe
• Our word algorithm originated with al-Khwarizmi’s name in the title
b)
al-Kitab al-mukhtasar fi hisab al-jabr w'al-muqabala (The Compendious
Book on Calculation by Restoring and Comparing)
• Al-jabr: moving a subtracted quantity to the other side of an equation
• Al-muqabala: applying operation to both sides of an equation
• Translated into Latin in 1140 as Liber algebrae et almucabala, from
which our word algebra comes
B.
al-Khwarizmi’s classification of quadratic equations
1.
Involved quantities he called “squares”, “roots”, and “numbers”
2.
Negative quantities were not considered; all coefficients and roots had to be
positive (that’s where al-jabr was useful)
3.
Approach was purely “rhetorical” – no algebraic symbolism, everything (even the
numbers!) written out in words
4.
Six cases
Case 1: “squares equal to roots” (ax2 = bx)
Case 2: “squares equal to number” (ax2 = c)
Case 3: “roots equal to number” (bx = c)
Case 4: “squares and roots equal to number” (ax2 + bx = c)
Case 5: “squares and number equal to roots” (ax2 + c = bx)
Case 6: “squares equal to roots and number” (ax2 = bx + c)
al-Khwarizmi presented a general method for each case, and a specific example; he also included a
geometric “proof” of each method. al-Khwarizmi’s “claim to fame” was his set of “quadratic formulas”
to solve the last 3 cases.
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Al-Khwarizmi’s Quadratic Formulas
In modern symbolic notation, al-Khwarizmi’s 3 cases that include all three relevant quantities (squares,
roots, and numbers) are:
Case 4: square and roots equal number:
x2 + bx = c;
Case 5: square and number equal roots:
x2 + c = bx; and
Case 6: square equals roots and number:
x2 = bx + c
Note that al-Khwarizmi prescribes that we always start by making the coefficient of the squared term 1:
“Whether there are many or few squares, they will have to be reduced in the same manner to the form of
one square… However many squares are proposed all are to be reduced to one square. Similarly also
you may reduce whatever numbers or roots accompany them in the same way in which you have
reduced the squares.” That step has already been taken for the equations shown at the top of this page,
compared to the corresponding cases on the previous page.
Below are 3 sets of instructions al-Khwarizmi gives for solving these cases (translated into English for
those of you who, like me, don’t read Arabic). Translate each rule into an algebraic formula involving
the coefficients b and c from the equations above, and match the formula to the appropriate case.
a) Halve the number of roots; multiply this by itself; subtract from this the number, and take the square
root of the difference; subtract this from, or add it to, half the number of roots.
Formula: x =
Equation: Case __
b) Halve the number of roots; multiply this by itself; add this to the number, and take the square root of
the sum; subtract from this half the number of roots.
Formula: x =
Equation: Case __
c) Halve the number of roots; multiply this by itself; add this to the number, and take the square root of
the sum; add this to half the number of roots.
Formula: x =
Equation: Case __
3
Al-Khwarizmi’s Geometric Proofs of His Quadratic Formulas
4
5
II.
The Pythagorean Theorem
The “Babylonian Box”
We’ve just seen how this was used to justify one of
al-Khwarizmi’s quadratic formulas.
We’ve probably all used it as a visual representation
of
(a + b)2 = a2 + 2ab + b2 .
How can it be used to prove the Pythagorean
Theorem? There’s not a right triangle in sight!
Does this seem more promising?
The square has the same dimensions as the one
above. Let’s say each side has length a + b.
This diagram, along with the algebra above, is often
used to prove the Pythagorean Theorem.
Can you see how to use the two diagrams together
to produce an “algebra-free” proof?
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If we take the previous diagram and “fold in” the
corner triangles, we get this diagram. The 12th century
Indian mathematician Bhaskara used this diagram to
prove the Pythagorean Theorem. One can use an
algebraic analysis of the diagram:
c2 = 4(ab/2) + (a – b)2 = 2ab + a2 – 2ab + b2 = a2 + b2
But I like the geometric, or “dissection”, approach:
slide the upper right triangle to the lower left, and slide
the upper left triangle to the lower right to produce this
diagram. Can you see a square of side a and a square
of side b sitting there?
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How did the Pythagoreans prove the
theorem?
(Please label the intersection of AB and CJ as K)
Find 3 similar triangles.
Use them to set up 2 proportions that can
be used to express a2 and b2 in terms of c
and x.
Explain how this shows that a2 + b2 = c2.
How did Euclid prove the Pythagorean
Theorem (Elements I.47)?
Show that the area of triangle AGB is half
the area of the square on side b.
Show that the area of triangle AHC is half
the area of AHJK.
Show that the shaded triangles AGB and
AHC are congruent.
What can we conclude about the areas of
the square on side b and AHJK?
Use a similar approach to show that the area
of BIJK is a2.
Explain how this all shows that a2 + b2 = c2.
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Liu Hui, a Chinese mathematician of the
3rd century AD, wrote a commentary on
the Jiuzhang Suanshu (Nine Chapters on
the
Mathematical
Arts,
originally
published in the first century BC) that
came to be considered part of that book.
He
provided
explanations
and
brief
justifications for the rules in the Nine
Chapters, taking it beyond a mere book of
“recipes.”
This diagram resembles the one Liu Hui
used to justify the “Gougu rule” (what the
Chinese called the theorem).
a) Show that all the triangles in the diagram are similar to one another.
b) Find 3 triangles that are inside the squares on the legs of triangle ABC but outside the square on
the hypotenuse of ABC.
c) Find 3 triangles that are outside the squares on the legs of triangle ABC but inside the square on
the hypotenuse of ABC.
d) Show that each triangle from part b) has a “congruent partner” from part c).
e) Explain how this shows that a2 + b2 = c2.
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III.
Archimedes’ Estimate of π
Archimedes used the “method of exhaustion” to find his approximation for π: he “trapped” the
circumference of a circle of radius r between the perimeters of inscribed and circumscribed regular
polygons with increasingly larger numbers of sides: Pinscribed < 2"r < Pcircumscribed (Archimedes is often
credited with “foreshadowing” integral calculus, since the method he used is very similar to the
approach used in developing the Riemann integral, where the area under the graph of a function is
!
“trapped” between sums of rectangles under the graph and rectangles over the graph with
increasingly more (and skinnier) rectangles.)
a) Archimedes started by using regular hexagons inscribed and circumscribed on a circle of radius r.
Without loss of generality, and to simplify our calculations, let’s use a unit circle with r = 1 in what
follows. Show that for this unit circle the perimeter of the inscribed hexagon is 6 and the perimeter
of the circumscribed hexagon is 4√3, and that this leads to an “interval estimate” of
3 < " < 2 3 # 3.46 . (See Figure a) on next page.)
b) We have an advantage over Archimedes, because we know trigonometry and he didn’t. Use
!
trigonometric methods to show that the length of one side of a regular n-gon inscribed in a unit circle
"180° %
is 2sin(180/n), and thus n sin$
' < ( . (See Figure b) on next page.)
# n &
c) Similarly, use trigonometric methods to show that the length of one side of a regular n-gon
# 180° &
circumscribed!around a unit circle is 2tan(180/n), and thus " < n tan%
( . (See Figure c) on next
$ n '
page.)
d) Using amazing ingenuity and effort, given the number system and computational methods
!
available to him, Archimedes eventually found the perimeters of inscribed and circumscribed regular
96-gons, which resulted in his famous approximation for π: 3
10
1
< " < 3 . Use your results from
71
7
parts b) and c) to get an interval estimate for π based on n = 96. How does it compare with
Archimedes’ approximation?
!
e) Use your results from parts b) and c) to generate interval estimates for π, taking n to be multiples
of 100. How large must n be to get π accurate to 4 decimal places (3.1416)?
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Figure a)
Regular hexagons inscribing and circumscribing unit
circle.
How large are the central angles for each hexagon?
If GC = 1, how long is DC, and what’s the perimeter of
the inscribed hexagon?
What are the measures of the angles of triangle GDJ?
If GD = 1, how long is DJ? How long is KJ? What’s the
perimeter of the circumscribed hexagon?
If GC = 1, what’s the circumference of the circle?
Use the three perimeters you found to estimate π.
Figure b)
BC is one side of a regular n-gon inscribed in a unit
circle with center A; D is midpoint of BC.
What’s the measure of angle BAD?
If AB = 1, how long is BD? How long is BC?
What is the perimeter of the regular n-gon?
Use this perimeter to get an “underestimate” for π.
Figure c)
EF is one side of a regular n-gon circumscribing
a unit circle with center A; D is midpoint of EF.
What’s the measure of angle EAD?
If AD = 1, how long is ED? How long is EF?
What is the perimeter of the regular n-gon?
Use this to get an “overestimate” for π.
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