Math History Notes

(Usually) Mostly a class for future teachers

• Name
• Major
• Why teach math? (if future teacher, if not, why this class?)
• What is goal in this class?
• Random fact.

• Name: John Hammond
• Major: Undergrad BSEd, lots of computer science (Missouri State 🧸), MA Math (Mizzou 🐯)
• Why teach math? Brief story.
• What is goal in this class? Help foster an appreciation of mathematics
• Random fact Wombats poop cubes.

• Name
• Major
• Why teach math? (if future teacher, if not, why this class?)
• What is goal in this class?
• Random fact.

I asked over email last week the following questions:

1. What do you think that math is? If someone asks, “what is math?” what would you say?
2. What is your favorite topic in mathematics? Such as favorite subject, favorite lesson, favorite fact, etc.
3. What is your goal for the class? Such as “at the end of the semester, I hope __”

You’ve already had access for a few days. We’ll talk highlights.

• Blackboard!

We’ll talk highlights.

• Syllabus here

… questions?

• Aztecs, Mayans* (base 20—vigesimal)
• Incans, Indians (base 10—decimal)
• Mesopotamians (e.g. Babylonians) (base 60—sexagesimal)

• Pick a base \(b\), then you can represent any number, \(N\), uniquely by: \[N = a_n b^n + a_{n-1}b^{n-1} + \dots + a_1 b + a_0\]

  For example, in base 10, \[6735 = 6\cdot 10^3 + 7\cdot 10^2 + 3 \cdot 10 + 5\]

• If we want base 2, the number \(13 = 1\cdot 2^3 + 1\cdot 2^2 + 1\), we write: \[13_{ten} = 1101_{two} = 1101_{2}\]

• Let’s convert into different bases to practice:
  • 782 to base 60
  • 782 to base 20

• What do you think these terms mean for number system?
• Can you give an example?

¹

• Where: Peru, Bolivia, Chile, Ecuador, Argentina
• Height of the empire in 15th century ~ 6 million people

²

• “Postal System” of runners (300 miles in 24 hours!)
• Decimal number system
• No (known) written language
• Number data recorded and transmitted with the quipu, a string of knotted colored cords
  • number of knots represents digits according to (Ifrah, 1985)
  • Here are some sample drawings from that author

• Quipu keepers in urban areas prepared, read, and stored quipu records for the government. ³

Drawing by Felipe Guaman Poma de Ayala of the Chief Accountant in Nueva corónica y buen gobierno (New Chronicle and Good Government) in 1615 (Guaman, n.d.) “Chief accountant and treasurer, Tawantin Suyu khipuq kuraka, authority in charge of the knotted strings, or khipu, of the kingdom”

… by the way, the whole book is available online here!

• Really long walk through
• Counting Beads (small objects)
• Sheet for a yupana

Try the following multiplication and division problems using the Egyptian system of “doubling-and-halving”:

• 57 * 63
• 364 * 28
• 181 ÷ 12
• 213 ÷ 13

• Ahmes was an Egyptian scribe, born about 1680 B.C.
• Wrote the papyrus
• … says he copied from earlier work circa 2000 B.C.
• The papyrus contains math:
  • division tables,
  • problems of area and volume,
  • methods for solving various types of problems and equations.
• Purchased in 1858 by Alexander Henry Rhind.
• Original likely 18 feet by 13 inches
• Rhind’s fragment partial;
  • rest found in New York Historical Society

Quote from Burton’s History of Mathematics (Burton, 2011)

The Rhind Papyrus starts with a bold premise. Its content has to do with “a thorough study of all things, insight into all that exists, knowledge of all obscure secrets.” It soon becomes apparent that we are dealing with a practical handbook of mathematical exercises, and the only “secrets” are how to multiply and divide.

⁴

⁵ View a full-size scrollable version on the British Museum Website

⁶

• Story of Darius I defeating a revolt in Persia in 516BC
• Carved on Bisotun Cliff in Iran
• four-hundred feet above the ground

• Written in Old Persian (Elamitic), Babylonian, and Persian

  # - (side note) Why are our numbers called Hindu-Arabic?⁷ :

• Translated in 1846 by Henry Rawlinson

• A common chart you’ll see in many texts such as (Eves, 1990; Katz, 2009):

• Let’s do better, with charts from Georges Ifrah (Ifrah, 1985, pp. 485–489)
• (and remember to share his story)

• Not everyone uses the Hindu-Arabic numerals
• Not even everyone in Eastern Arab or some Indian regions.
• Eastern Arabic numerals or the Arabic-Hindu numerals and the very similar Urdu numbers ⁸

How did the “Hindic” form of the numbers evolve? Looking at first five numbers (Ifrah, 2000, p. 467):

#

• Unit fractions, and special symbols fractions

⁹

• Example: Write 5/6 as an Egyptian fraction.
• Example: Double 1 1/15

• On the back is the “\(2/n\) table”
• Some examples:

• How do you divide 9 loaves of bread between 10 men?
  • Our way: 9 ÷ 10 = 9/10 of a loaf per man
• Mathematically, this is fair, but the guy with 10 tenths is probably not too happy!

• The Egyptian way:
  • 9 ÷ 10 = 2/3 + 1/5 +1/30
• The nine loaves of bread seem to be more “fairly” distributed this way!

An article search aside…

• Let’s say I’m looking for a particular article on reciprocal table.
• Got here: https://www.degruyter.com/document/doi/10.1515/za-2016-0016/html Darn! $42????
• To the library! (Copy the DOI or other information)
• No article. Darn again!
• Toggle switch. Hooray!
• Now it’s open, we can scroll through and see the tablet

• They avoided using fractions like 1/7 (why?)
  • Let’s do the division in base 60 and see!
• Estimating fractions with denominators that are not factors of 60

• Reciprocal table:

• Remember we have to determine number “place”

• 1548-1620 (modern Belgium), wrote in Dutch

• Decimal Fractions (in Europe)

  • Published De Thiende (’The Tenth’) in 1585
    • You can find it here (a 1606 English edition) (Stevin, 1608)

  • Operationswith decimals; circle-number notation for place value

    • (me: do an example specifically, and also look at the link above)
    • … Napier would use decimals in his work on logarithms soon after

A translation of a Babylonian tablet which is preserved in the British museum goes as follows:

4 is the length and 5 the diagonal. What is the breadth ? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9 ? 3 times 3 is 9. 3 is the breadth.

In the Zhoubi Suanjing (translated The Arithmetic Classic of the Gnomon and the Circular Paths of Heaven)

Here is a photo of a Ming dynasty 1603 copy:

And here is a more modern focused image

• This is functionally similar to Thabit Ibn Qurra’s proof without words mentioned in (Berlinghoff & Gouvêa, 2015)
  • (Chemla, 2005) notes that the two proofs are distinct in the manner of the interpretation of the diagrams.
    • Thabit physically moves the pieces to construct two ’unequal squares’
    • Liu Hui (commenting on the Zhou work) viewed it as two different ways to construct the same physical area (note that the external triangles are likely a drawing aid and not used in the proof, according to Chemla)
• As an aside, (Eves, 1972) attributes this to Bhaskara; I haven’t traced that down further.
• Let’s take a look at the proof Head to Geogebra!

• Pythagoreans loved numbers!
• Special number properties (composite numbers)
• Pythagorean triples (from Mesopotamia?)
• Triangular, square, pentagonal, etc. numbers
• “Perfect” numbers (along with “Deficient” and “Abundant” numbers)
• “Amicable”/“Friendly” numbers¹¹

• Our reading discusses level of accuracy
• According to (Edu, 2016), NASA uses 3.141592653589793

  • In the article, they note that where Voyager 1 was at the time, 12.5 billion miles from earth, if they used this number

    “It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches.”

With just a compass and unmarked straight-edge

• trisect an angle
• duplicate the cube
• square the circle

• Let’s find the areas of rectangles and triangles! (to Geogebra!)

Here’s an activity to try to see where \(A = (8/9 d)^2\) for the area of a circle might come from

• Show by experiment (using pennies, for example), that a circle of diameter 9 can essentially be filled by 64 circles of diameter 1.
• (Begin with one penny in the center, surround it with a circle of 6 pennies, and so on).

• Use the fact that 64 circles of diameter 1 also fill a square of side 8 to show how the Egyptians may have derived their formula for the area of a circle, \(A=(8/9d)^2\) where \(d\) is the diameter of the circle.

Note that \(\pi\) isn’t a constant Egyptians used, though, per (Imhausen, 2008):

Egyptians did not use \(\pi\). As can be seen from the symbolic representation of the text, above, the constant used to calculate the area of a circle was \(\frac19\), which was clearly not an extremely bad approximation of \(\pi\) , but rather the constant appropriate to the Egyptian method of mathematizing circles.

• What do we mean by “squaring”?
• 450 B.C. Hippocrates finds the area of a curved figure called a lune (crescent moon shape).
• First non-rectilinear area calculation
• The Geogebra activity!

• Inscribing and circumscribing polygons in/around a circle
• Compare the perimeter of the polygon with diameter of circle
• Since perimeter is close to circumference of the circle:
• \(\dfrac{\text{perimeter}}{\text{diameter}} \approx \dfrac{\text{circumference}}{\text{diameter}} = \pi\)
• inscribed polygon will be lesser
• circumscribed with be greater
• Gives as good an approximation as you could want
• Here’s a Geogebra demo

• Born in Syracuse 287 B.C. on the island of Sicily.
• Unlike any other ancients we’ve discussed so far, we know relatively a lot.
• Figures prominently in Plutarch’s Lives of the Noble Grecians and Romans (1st century AD)
• Archimedes also writes prefaces for each book
• On Floating Bodies
• On the Equilibrium of Planes
• Quadrature of the Parabola
• On the Measurement of the Circle
• On the Sphere and Cylinder
• On Spirals
• The Method

From (Gould, 1955) (NOTE TO SELF: Don’t advance the slide)

• William Jones in 1706 wrote Synpsis Palmoariorum Matheseos or A New Introduction to the Mathematics
  • “In a short and Easie Method” … “Design’d for the Benefit, and adapted to the Capacities of BEGINNERS”
• Here’s the book (Jones, 1706)

We saw various fractions to represent \(\pi\).

• What about infinite fractions
• Let’s talk about continued fractions.
• Term was first used by John Wallis in 1695
• idea is older

• Berlinghoff and Gouvea write: “Because solving linear equations is relatively easy, few standard history books have sections specifically on that subject.”
• We’ll take a look at linear equations, and
• things that are linear adjacent

The AO 8862 tablet

• old tablet - 1700’s BC.
• square prism with problems on four sides
• currently held at the Louvre

[fn: http://ressources.louvrelens.fr/EXPLOITATION/oeuvre-ao-8862.aspx]

Quoted in (Bunt et al., 1988)

Length, width. Length and width I have multiplied and thus formed the area. I have further added the excess of the length over the width to the area: 3,3. Further, I have added the length and width: 27. Required: length, width, and area.

• (pick some variables and talk, I give it away on the next line:)
• \(xy + y - x = 3,3\) and \(x + y = 27\).

The scribe’s solution:

• (Given) 27 and 3,3, the sums
• (Result) 15 length, 12 width, 3,0 area.
• You follow this method:
• 27 + 3,3 = 3,30
• 2 + 27 = 29
• Take half the 29 (14;30)
• 14;30 \(\times\) 14;30 = 3,30;15
• 3,30;15 - 3,30 = 0;15
• 0;15 has 0;30 as square root

(continued)

• 0;15 has 0;30 as square root
• 14;30 + 0;30 = 15 length.
• 14;30 - 0;30 = 14 width.
• Subtract the 2 you added to 27 from 14, the width:
• 12, the real width
• 15, length, 12, width; I have multiplied
• 15 \(\times\) 12 = 3,0 area
• 15 - 12 = 3
• 3,0 + 3 = 3,3

• Āryabhaṭa (born 476 AD) writes Āryabhaṭīya when he is 23
  • The Āryabhaṭīya is an astronomical book
  • The earliest existing work in Indian mathematics
  • he dates the manuscript and says how old he is.

  • As translated in (Clark, 1930), he thinks well of himself:

    By the grace of God the precious sunken jewel of true knowledge has been rescued by me, by means of the boat of my own knowledge, from the ocean which consists of true and false knowledge.

  • The math chapter is expanded and explained by Bhāskara I
  • Written in verse (sutras) with puns as memory devices

• The Rule of Three: Āryabhaṭa writes: (Keller, 2006)

  Now, when one has multiplied that fruit quantity in the Rule of Three by the desire quantity
  What has been obtained from that divided by the measure should be this fruit of the desire
  The denominators are respectively multiplied to the multipliers and the divisor.

• Huh???

• Bhaskara clarifies: (Keller, 2006)

  In order to bring about a Rule of Three the wise should know that in the dispositions
  The two similar quantities are at the beginning and the end. The dissimilar quantity is in the middle.

  • physically arrange, like one a table;
  • two right numbers are multiplied. The left divides that result.

• Here’s an example from Bhaskara (Keller, 2006)

  A lot of cattle is said to be made of eight tamed, three to be tamed.
  Out of one thousand and one, how many have been tames and how many are the others?

• This time, to get “similars” we need total collection of cows; 8+3=11 in the lot; 1001 in total:
• Write similars on outside, dissimilar inside:
• Tamed and untamed 11; tamed 8; total 1001
• Multiply the two right numbers and divide by the left number.
• 725; units? Tamed! (the units of the middle)
• So there are 1001-725 = 273 untamed.

• One more example? Merchants and earnings! (Keller, 2006)

  Merchants with respective investments of a half, a third and one eighth
  have a profit of seventy minus one. What are their respective profits?

Here’s an example (Let’s make it up):

• Pick a number
• Pick a dollar amount
• Pick a thing you’d buy
• I bought NUMBER of THING YOU’D BUY for $ DOLLARS. How much THING YOU’D BUY can be obtained for $1?

• Write similars on outside, dissimilar inside:
• $ DOLLAR QUANTITY $1
• Multiply the two right numbers and divide by the left number.
• Answer!

Jiuzhang suanshu or Nine Chapters on the Mathematical Art (around 200ish BC)

• Algorithms of finding square and cube roots
• Solutions to quadratic equations and systems
• Solutions of systems of up to five linear equations
  • The method given is basically Gaussian Elimination

Example problem (Problem 1 of Chapter 8):

There are three types of grain. Three bundles of the best, plus two bundles of the second best, plus one bundle of the worst give a total of 39 units of flour. But two bundles of the best, plus three bundles of the second best, plus one bundle of the worst give a total of 34 units of flour. Finally, one bundle of the best, plus two bundles of the second best, plus three bundles of the worst give a total of 26 units of flour. How many units of flour can be made from one bundle (for each type of grain)?

Represented as columns in a table form:

\(\left(\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 2 \\ 3 & 1 & 1 \\ 26 & 34 & 39 \end{array} \right)\)

… and do column-operations to reduce the matrix and solve the system.

Old Babylonian Story Problem ca. 1700 B.C, the solution to the question: “I have taken the area and two thirds of the side of my square and it is 0;35.”

Let’s work this one out.

You take 1 the “coefficient”. Two thirds of 1, the coefficient, is 0;40. Half of this, 0;20, you multiply by 0;20 (and the result) 0;6,40 you add to 0;35 and (the result) 0;41, 40 has 0;50 as its square root. 0;20, which you multiplied by itself, you subtract from 0;50 and 0;30 is the (side of) the square.”

• Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (c. A.D.780 - 850)
• Wrote on “algebra”, Hindu-Arabic numerals, spheres, and cones
• Worked as an astronomer and head librarian at the House of Wisdom under several caliphs
• Influential writings
  • kitab al-jam’ wa’l-tafriq al-ḥisāb al-hindī (Book of Addition and Subtraction with Indian Numbers)
    • original Arabic is lost, survived via Latin translations
  • al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (Compendious Book on Calculaion by Completion and Balancing)
    • Available here, 1831 translation (Rosen & Khuwarizmi, 1831)
  • Provided algebraic proofs followed by geometric proofs.

• al-jabra “restoration” – adding equal terms to both sides of an equation to remove negatives
• al-muqabala “reduction” – subtracting equal quantities from both sides

Using only positive numbers, he classified six types of quadratics “root” or “thing” refers to the variable, “number” is a constant.

1. roots equal squares \(bx = ax^2\)
2. roots equal numbers \(bx = c\)
3. squares equal numbers: \(ax^2 = c\)
4. squares and roots equal numbers: \(ax^2 + bx = c\)
5. roots and numbers equal squares: \(bx + c = ax^2\)
6. squares and numbers equal roots \(ax^2 + c = bx\)

(this was in our text as well) A problem and solution of al-Khwarizmi, as quoted in (Joseph, 2001, p. 477): “One square [mal] and 10 roots of the same equals 39 direhems.” Solution:

1. You halve the “number” of roots: result 5.
2. This you multiply by itself: result 25.
3. Add this to 39: result 64
4. Take the square root of this: result 8
5. Subtract from 8 the result given in step 1: result 3.
6. This is the root of the square you sought.

• We follow a similar method
• But we can have more than one solution
• Or no solution.
• (Me) read the translation from (Lévy, 2002)

:CUSTOM_ID: book-ii-of-elements

• Definition:

Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.

• Proposition II.1: (Bunt et al., 1988)

If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the divided line and the several parts of the divided line.

¹⁴

• Proposition II.4:

If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

¹⁴

• Proposition II.5:

If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

¹⁴

(Here’s the algebraic construction pre-done) https://www.geogebra.org/m/jutayp4b

• Proposition II.6:

If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.

¹⁴

(Here’s the pre-done construction…) Here’s the construction on Geogebra

• Proposition II.11:

To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.

¹⁴

Let’s do the construction on Geogebra from scratch.

• Khayyam discovered and extended a geometrical argument for solving cubics
• Leads to finding the real solution to cubic equations
• In Treatise on Demonstrations of Problems of al-jabra and al-muqabala:
  • “he emphasizes that the reader needs to be thoroughly familiar with the work of Euclid, Appolonius, and al-Khwarizmi in order to follow the solutions.” (Laubenbacher & Pengelley, 1999)
  • Showed strong mastery of geometry

• Classification of solutions of some cubics (from (Joseph, 2001))

We saw previously that \[ 4 = \sqrt[3]{2+\sqrt{-121}} - \sqrt[3]{-2+\sqrt{-121}} \] Here’s another example.

English translation (Cardano, 1993)

The second species of negative assumption involves the square root of a negative. I will give an example: If it should be said, Divide 10 into two parts the product of which is 30 or 40, it is clear that this case is impossible. Nevertheless, we will work thus…

• Head to Ars Magna chapter XXXVII, Rule II (in the Latin pdf, page 67)
• In the English translation by published by Dover, head to pg 219

Cardano concludes the discussion (Cardano, 1993, p. 220):

So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless.

• Wrote l’Algebra in 1572
  • Goal was to make algebra, inspired by Ars Magna, easier to understand
  • He wrote that Cardano wasn’t clear (ma nel dire f’ oscuro) (van der Waerden, 1985)
  • Include his newly discovered work by Diophantus

• He explained the solution to \(x^3 = 15x +4\), as quoted in (van der Waerden, 1985)
  • “At first, the thing seemed to me to be based more on sophism than on truth, but I searched until I found the proof”
• He was the first to observe that solutions with square roots of negatives appear in “conjugate pairs”
  • he didn’t call it that, though
• Notation for \(\sqrt{-1}\), and rules to work with it:
  • Our \(\sqrt{-1}\) is piú di meno and our \(-\sqrt{-1}\) is meno di meno
  • Symbolically: R.q.0m.1 and m.R.q.0m.1
• This was the subject of chapter 2 of l’Algebra

From (Barrow-Green et al., 2019):

This will seem to be more artificial than real, and I held the same opinion myself, until I found the geometrical demonstration“ … Plus times plus of minus, makes plus of minus

Minus times plus of minus, makes minus of minus

Plus times minus of minus, makes minus of minus

Minus times minus of minus, makes plus of minus

Plus of minus times plus of minus, makes minus

Plus of minus times minus of minus, makes plus

Minus of minus times minus of minus, makes minus

• Chapter 1 is about computing radicals
• Quote about the particular topic, but gives the feel:
• (this is really fun - it illustrates how math’s a bit different today)

…In short, I shall set forth the method which is the most pleasing to me today and it will rest in men’s judgment to appraise what they see: meanwhile I shall continue my discourse going now to the discussion itself

Translation from (Arcavi & Bruckheimer, 1991):

Let us first assume that if we wish to find the approximate root of 13 that this will be 3 with 4 left over. This remainder should be divided by 6 (double the 3 given above) which gives \(2/3\). This is the first fraction which is to be added to the 3, making 3 \(2/3\) which is the approximate root of 13. Since the square of this number is 13 \(4/9\) it is \(4/9\) too large, and if one wishes a closer approximation, the 6 which is the double of the 3 should be added to the fraction \(2/3\), giving 6 \(2/3\) and this number should be divided into the \(4\) which is the difference between 13 and 9, …

• This is the method which is most pleasing to him.
• Don’t despair! He explains what he’s doing in later paragraphs! (Quotes as in (Arcavi & Bruckheimer, 1991))

• Let us suppose we are required to find the root of 13. The nearest square is 9, which has root 3. I let the approximate root of 13 be 3 plus 1 tanto unknown.
• Its square is 9 plus 6 tanti p. 1 power. We set this equal to 13.
• Subtracting 9 from either side of the equation we are left with 4 equal to 6 tanti plus one power.
• Many people have neglected the power and merely set 6 tanti, equal to 4. The tanto then comes to \(2/3\).
• and the approximate value of the root is 3 \(2/3\) since it has been set equal to 3 p. 1 tanto.

• However, taking the power into account, if the tanto is equal to \(2/3\), the power will be \(2/3\) of a tanto, which, added to the 6 tanti, will give us 6 and \(2/3\) tanti, which are equal to 4.
• So the tanto will be equal to \(3/5\), and since the appropximate is 3 p. 1 tanto it comes to 3 \(3/5\).
• But if the tanto is equal to 3 \(3/5\), the power will be \(3/5\) of a tanto and we obtain 6 \(3/5\) tanti equal to 4 …
• e cosi procedendo si puo approssimare a una cosa insensible “and this process may be carried to within an imperceptible difference.”

Yale Babylonian Collection

:CUSTOM_ID: ybc-7289

¹⁵ presentation note: use two-finger pinch-out trackpad gesture to zoom on the laptop

• Leibniz (as quoted in (Laubenbacher & Pengelley, 1999, p. 216)): \[x^4 + a^4 = (x^2 + a^2\sqrt{-1})(x^2 - a^2\sqrt{-1})\]
• “Therefore, \(\int dx /(x^4 + a^4)\) cannot be reduced to the squaring of the circle or the hyperbola by our analysis above, but founds a new kind of its own”

• Nikolaus Bernoulli factors it.
• But this Bernoulli suggests in 1742 (Dunham, 1991a): \[x^4 - 4x^3 + 2x^2 + 4x + 4\] can’t be factored.
• Euler factors it.
• Euler attempts a proof in 1749
• Gauss finally resolves it in 1799.
  • Proves that every polynomial with real coefficients can be factored into linear and quadratic factors
  • his first of four proofs of the FTA (then 1816, 1816, and 1849)

Brief Bio

• 1802-1829 (Norway)
• Died of tuberculosis.
• Invented group theory
• No general algebraic formula for degree 5 polynomials (1824)

• 1811-1832 (France)
• Bad student
• Political revolutionary
• Mathematician

• Also invented group theory: (Quote in (Laubenbacher & Pengelley, 1999)):

Jump above calculations; group the operations, classify them according to their complexities rather than their appearances; this, I believe, is the mission of future mathematicians; this is the road on which I am embarking in this work

• Field called Galois Theory
• Solutions to polynomial equations (among his final 60 page work)

• Diophantus of Alexandria lives around 3rd century AD (probably)
• He quotes Hypsicles (150BC) and is quoted by Theon of Alexandria (350AD)
• A book by the Bishop of Laodicia dedicated a book to Diophantus around 278AD
• What we know is a puzzle:

Diophantus passed 1/6 of his life in childhood, 1/12 in youth, and 1/7 more as a bachelor. Five years after his marriage was born a son who died 4 years before his father, at 1/2 his father’s [final] age

quoted from (Eves, 1990)

What did he do?

• Wrote three works: Arithmetica, On Polygonal Numbers, and Porisms.
• The first we have a lot of (6 of 13 books)
• Second we have fragments
• Third is completely lost.

• A book on what we now call algebraic number theory
• Deals with 130 problems and solutions
• Problems involved 1st and 2nd degree equations (requiring positive rational solutions)
• No general methods, just solving individually (with clever tricks)
• The Greek contribution to algebra

• Book II, Problem 28

Find two numbers such that their product added to either gives a square number.

• His answer is \(\displaystyle \left(\frac34\right)^2, \left(\frac7{24}\right)^2\).

• Book III, Problem 6

Find three numbers such that their sum is a square and the sum of any pair is a square.

• His answer is \(80, 320, 41\).

• … do you think you’d try to find more?
• Are there more answers to a problem? A general formula?

• Book II, problem 8: To divide a given square number into two squares. As quoted in (Katz, 2003, p. 177):

Let it be required to divide 16 into two squares. And let the first square = \(x^2\); then the other will be \(16-x^2\); it shall be required therefore to make \(16-x^2=\) a square. I take a square of the form \((ax-4)^2\), a being any integer and 4 the root of 16; for example, let the side be \(2x-4\), and the square itself \(4x^2 + 16 - 16x\). Then \(4x^2 + 16 - 16x = 16 - x^2\). Add to both sides the negative terms and take like from like. Then \(5x^2 = 16x\), and \(x=16/5\). One number will therefore be 256/25, the other 144/25, and their sum is 400/25 or 16, and each is a square.

Solution to Book II, problem 8 suggests general method:

• To find two numbers \(x\) and \(y\) so that \(x^2 + y^2 = b^2\):
• Pick \(a\) to be any number.
• Set \(y=ax-b\)
• Then \(b^2 - x^2 = a^2 x^2 -2abx + b^2\)
• … \(2abx = (a^2+1)x^2\)
• … \(\displaystyle x = \frac{2ab}{a^2+1}\).
• What about other powers?

• Teacher at the University at Alexandria
• Her commentaries:
  • Conics of Appollonius
  • Edited Almagest of Ptolemy (astronomy / trig) (Almagest is from the Arabic for “Greatest”)
  • Edited Theon’s commentary of The Elements
  • Arithmetica by Diophantus
• Diophantus: “Father of algebra” and Hypatia: “Mother of algebra”

• 1463 - Regiomontanus: Called for a Latin translation of the remaining Greek texts
• 1570 Rafael Bombelli finds the manuscript in the Vatican
  • with Antonio Maria Pazzi translates it
  • … doesn’t publish it
  • uses the problems in his Algebra 1572
• 1575 - Xylander (formerly Wilhelm Holzmann) Translated the Greek and added his commentary
• 1621 - Bachet de Meziriac Published the Greek and Latin together with notes
• 1670 - Clement-Samuel Fermat “A second, carelessly printed, edition was brought out in 1670” (Eves, 1990, p. 180). This version included his father’s marginal notes.
  • Who was his father?

• Lived 1601-1665
• French lawyer
• Mathematician as a hobby
• Contributions include:
  • Calculus
  • Probability
  • Number Theory

Marginal notes in Diophantus’ Arithmetica

• Christian Goldbach and Leonhard Euler
• Fermat’s most famous note:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.[fn: http://www.textkit.com/greek-latin-forum/viewtopic.php?f=3&t=7108]

• Conjecture: The expression \(x^n + y^n = z^n\) has no integer solutions for \(n>2\).

• Conjecture: All elliptic curves are modular.
  • Conjectured by Yukata Taniyama and Goro Shimura in 1955.
• (I intended this slide to seem jarring and out of place)

• As a boy (of about 12) read about Fermat’s Conjecture
• … so he gets his PhD in Mathematics
• Becomes and still is a Professor of Mathematics at Oxford
• Worked in a specialized field involving elliptic curves
• Fermat’s Conjecture is a hard problem, don’t work on it

• 1987: “The Epsilon Conjecture” The Taniyama-Shimura conjecture implies Fermat’s Last Theorem.
• Wiles learns about this result…
• … and spends the next seven years of his professional life proving one of the most difficult results in modern mathematics.
• Proven. 1993.
• … an error.
• Patched up (100ish? pages later) 1994. Completely correct.
• BBC/Horizon’s Documentary on Fermat’s Last Theorem. You can find it on YouTube or Vimeo.
• Accolades for Wiles

The Mayan gods of the underworld were governed by the god of death - the God of Zero.

A human would be dressed in the regalia of the God of Zero, and then sacrified by having his lower jaw town off

Pliney the Elder.

• 60AD, Pliney is writing an astronomy work and dividing the sky in \(360^\circ\) as has been long done.
• However, he starts his positioning from \(1^\circ\).
• Romans counted Tuesday as three days after Sunday is Tuesday … because Sunday is counted as 1 day.
• Those musical among us use C to E as a “major third” because there are three notes: C, D, E even though there’s only two note increase from C to E.

Liber Abaci begins with:

(English quote from (Pisano, 1987))

The nine Indian figures are 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 which the Arabs call zephir any number whatsoever is written.

in writing this, we are getting the Hindu-Arabic numbers brought to the west in 1202, but he’s not including 0 as a first-class number.

• “The old figures alone are used because they cannot be easily falsified. ”(Venetian bookkeeping text)
• University of Paduo books: “non per cifras sed par literas claros” (no numbers, clear letters)
• 1494, Mayor of Frankfurt: “abstain from calculating with digits.”
• 1594, canon in Antwerp: don’t use numerals in contracts

• Basic idea: debits and credits and balance
• Luca Pacioli in 1494 formalized
• Pacioli’s bookkeeping showed readers that 0 is a value not just a sign. It has literal, tangible value like other numbers.

• Napier solving ab=0
• This is absolutely obvious to us, and our students do it rote without any thought.
• … but
• Multiplication is repeated addition. 5*7 = 7+7+7+7+7. Fine. a*b = a+a+a+a…+a b times or similarly b+b+b+…+b a times.
• Multiplication is just repeated addition.
• So all that’s really happening in \(ab\) is that we’re adding many things together.

• … but that addition sums to 0.
• This is a balanced ledger.
• This is a conservation of energy law.
• It requires us to hold that 0 is a value and that everything else should be in balance.
• This blew my mind!

Hundreds year long controversy to adopt Hindu-Arabic numerals

• To me, look in notes here
• (we’ll also talk about tally sticks later…)

• Woodcut by Gregor Reisch in Margarita Philosophica It’s called The Allegory of Arithmetic 1503
• Shows Boethius using Hindu Arabic numbers to do calculations while Pythagoras is using a counting board (and looks a bit concerned) (it is thought Pythagoras invented the Greek abacus)

[fn: Houghton Library / Public domain]

• Abacists favored Roman numerals and work with a physical counting device
• Algorists favored the Arabic numerals and algorithms

• Accountants against the change:

  • The numbers are highly susceptible to fraud (change a digit)
  • Too easy to use: (Stone, 1972) #+begin_quote [T]he flexibility of the new system would lead to inaccuracies and errors due to the less painstaking efforts on the part of the accountant.
  • To me, look here about tally sticks

Hundreds year long controversy to adopt Hindu-Arabic numerals

• To me, look in notes here
• (we’ll also talk about tally sticks later…)

• Woodcut by Gregor Reisch in Margarita Philosophica It’s called The Allegory of Arithmetic 1503
• Shows Boethius using Hindu Arabic numbers to do calculations while Pythagoras is using a counting board (and looks a bit concerned) (it is thought Pythagoras invented the Greek abacus)

[fn: Houghton Library / Public domain]

• Abacists favored Roman numerals and work with a physical counting device
• Algorists favored the Arabic numerals and algorithms

• Accountants against the change:

  • The numbers are highly susceptible to fraud (change a digit)
  • Too easy to use: (Stone, 1972) #+begin_quote [T]he flexibility of the new system would lead to inaccuracies and errors due to the less painstaking efforts on the part of the accountant.
  • To me, look here about tally sticks

• inside the orbrit of Saturn is a cube;
• the cube is circumscribed by orbit of Jupiter;
• tetrahedron between Jupiter and Mars,
• dodecahedron between Mars and Earth,
• icosahedron between Earth and Venus,
• octahedron between Venus and Mercury
• he worked out the size of orbits and size of the solids and was very close (except the Platonic solid part)

A neat application of an icosahedron.

• invented in 1943 by Buckminster Fuller
• a projection of the globe onto an icosahedron
• can be unfolded in different ways to illustrate different aspects of Earth
• Example images from Wikimedia:

¹⁶

¹⁷

¹⁸

Neat application for human migration

• standard Mercator-ish map
• Dymaxion map

• Check out On Plane Loci (Smith, 1929)
• Check out the Geogebra activity of Fermat’s geometry.

(Keller, 2006)

In a circle, the product of both arrows, that is the square of the half-chord, certainly for two bow fields.

“Just in this case, they relate to hawk and rat examples” (Keller, 2006),

Me: Go to (Keller, 2006, pp. 85–87)

• circa 180-125bc
• ’Father of Trigonometry’
• Created a table of chords
• … we don’t have them
  • Theon tells us there were 12 books
• Used \(\pi = 3;8,30\)
• Used a circle’s circumference divided into 360*60 parts, measured in minutes
  • This is (likely) the reason we use 360 parts of a circle
  • possibly due to the zodiac
  • also Babylonian fractions are better than others
    • Egyptian unit fractions were cumbersome
    • Greek fractions weren’t convenient otherwise
  • … gives diameter of about 6875’
  • and radius of about 3438’
    • or \(57^\circ 60'\)

• Ptolemy (circa 100-170AD)
• Actual Greek name of the book is is Mathematica Syntaxis
• Becomes Almagest in Arabic

• Astronomical model:
  • the heavens are spherical and rotate as a sphere
  • the Earth is a sphere
  • the Earth is the center of the cosmos
  • the distance from the Earth to the heavens is so big that Earth can be treated as a geometrical point
  • the Earth does not move

• Motion kind of makes sense
• Observed retrograde motion of Mars
  • Geogebra applet to see it
  • Ptolemy’s explanation (draw diagram)
  • Geogebra applet to observe his explanation
• … Copernicus explains it later with the Sun at the center.
  • Geogebra applet showing the Heliocentric model

• Kepler’s model:
  • Orbits of the planets lie on ellipses.
  • the universe is a bounded sphere
  • In his Mysterium cosmographicum (Secret of the Universe) 1596:
    • there are six planets
    • there are only five Platonic solids
    • Coincidence? Obviously not.
    • God separated the planets by Platonic solids
    • “God is always a geometer.” (Katz, 2003)

(Kozhamthadam, 2002) quoting a letter by Kepler

The sphere possesses a threefold quality: surface, central point, intervening space. The same is true of the motionless universe: the fixed stars, the sun, and the aura or intermediate ether; and it is also true of the Trinity: the Father, the Son, and the Holy Spirit.

• inside the orbrit of Saturn is a cube;
• the cube is circumscribed by orbit of Jupiter;
• tetrahedron between Jupiter and Mars,
• dodecahedron between Mars and Earth,
• icosahedron between Earth and Venus,
• octahedron between Venus and Mercury
• he worked out the size of orbits and size of the solids and was very close (except the Platonic solid part)

• Suggests his approach is similar to Hipparchus
• Also uses 360 subdivisions of a circle’s circumference
  • Further divides into (later Latin translation):
    • degree into sixty partes minutae primae
    • and these into sixty partes minutae secundae

For any quadrilateral inscribed in a circle: \[BD\cdot AC = AB \cdot CD + BC \cdot AD\] [[

./images/PtolemyTheorem.png]][By Dicklyon (Wikipedia en Inglés) [Public domain], via Wikimedia Commons]

• The side-lengths of the quadrilateral are chords
• …
• The side-lengths of the quadrilateral are sines of the angles!

[Kmhkmh (Own work) [CC BY 4.0 (http://creativecommons.org/licenses/by/4.0)], via Wikimedia Commons]

• … you’ll get into Ptolemy’s theorem and the “Star Trek” lemma next month

• Working in base-60, sexagesimal.
• Letting the diameter of the circle be 120
• This makes the radius 60, which is a unit in sexagesimal
• First, find the chord for \(36^\circ\)
  • Side of a regular decagon inscribed in the circle
  • Using Euclid II.6 (discussed in class in the algebra notes)
  • He finds the chord is \(37;4,55\)

• Second, he found the chord for \(72^\circ\), being \(70; 32, 3\)
• Next, the chord for \(60^\circ\) is \(60\).
• It’s worth remembering: \(\mbox{crd} \alpha = 2R \sin\dfrac\alpha2\)
• … and the values are pretty close
• We also need the Pythagorean Theorem! (To find supplement chords / complementary sines)

• To this point, we have sines for

• … he then repeatedly applies his half-angle formula and Pythagorean theorem and halves his way down to under \(1^\circ\).
• (I find this more delightful than you, perhaps…)
• (… but if you want, we can actually do the computations! I have slides for that)

• Scottish mathematician
• His father was knighted and was Master of the Mint (his father was 16 when Napier was born)
• Studied Theology
  • “violently anti-Catholic” according to (Eves, 1990)
  • 1593, A Plaine Discovery of the Whole Revelation of Saint John
    • attempts to mathematically prove the Pope was the Antichrist
    • end of the world occurs between 1688 and 1700.
    • 21 editions of the book were published
    • (Eves, 1990) says “Napier sincerely believed that his reputation with posterity would rest upon this book.”
  • numerology

• Wizarding stories from (Eves, 1990, p. 308)
  • “The black rooster”
  • Catching pigeons
• Rabdologia and “lattice multiplication”
• Published Mirifici logarithmorum canonis descripto in 1614
  • A description of the wonderful table of logarithms

Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers … I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.¹⁴

• \(2^3 \cdot 2^2 = \dots\)
• \(10^3 \cdot 10^2 = \dots\)
• \(10^{-7} \cdot 10^{7} = \dots\)
• \(10^7\cdot \left( 1 - 10^{-7}\right) = \dots\)

Let’s make a table! ¹⁴

Let’s compute a few things:

• \((\sin 90^\circ) \cdot 10,000,000\)
• \((\sin 89^\circ 59') \cdot 10,000,000\)
• \((\sin 89^\circ 58') \cdot 10,000,000\)
• \((\sin 89^\circ 57') \cdot 10,000,000\)

base 10 logarithms in 1624 in Arithmetica logarithmica

• Let’s explore logarithms and take a look at making a slide rule

• inverse shadow: \(\displaystyle \frac{\sin \theta}{\sin( 90^\circ - \theta)}\)
• shadow: \(\displaystyle \frac{\sin( 90^\circ - \theta)}{\sin \theta}\)

• Thoughts on Details Proper Construction
  • (Keller, 2006, p. 70) (You’ll notice that he amuses me)

• Working proportions of shadows of gnomons
  • think about ’shadow of tree’ type from trig class
• (Keller, 2006, p. 74)

• “Say the shadows of the two gnomons which stand respectively at eighty from the foot of a light on a pole whose height is seventy two; and at twenty from <a light whose height is> thirty”

“Setting Down:”

• Distance between the gnomon and base is 80;
• this multiplied by the gnomon is 960.
• The base is 72, the gnomon 12; their difference is 60
• The multiplied 960 is divided by this; “the shadow is obtained: 16.”

• Al-Biruni’s trigonometric lengths
• … Let’s construct all the lines first; then look at the relevant segments
• Imagine sun is in bottom left quadrant https://www.geogebra.org/classic/g8tzvkwa

Tangent isn’t ’tangent’ or a function, but I’m calling it that; it would be thought of as a shadow

• Formula \(\tan(\theta/2)\)
• \(\displaystyle \tan\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos \theta}\)

• Hippocrates : Quadrature of the Lune
• Eudoxus (408-355 B.C.)
  • Pioneered the Exhaustion Proof
• Archimedes (287-212 B.C.)
  • Took the Exhaustion Proof to the next level
  • Found the area of a circle (and other curved geometric figures) in organized steps of regular polygons
  • How was this done to find the area of the circle?
  • Found area of a parabolic sector by a geometric argument of \[\sum_{n=0}^\infty \dfrac{1}{4^n} = \dfrac{4}{3}\]
• Liu Hui (China 3rd century, A.D.) used a similar method of exhaustion re: circles

• Fermat is foundational in this:
• Found maximum and minimum values of curves
  • places where the slope of the tangent line is 0
  • we call this “Fermat’s theorem” in calculus
• Example: (using Viete’s notation of vowels are variables):
• Divide a quantity into two parts such their product is a maximum

• Let \(B\) be a given quantity; denote points on a curve by by \(A\) and \(B-A\). Assume \(E\) is a really small variable (like \(\epsilon\)):
• \[(A-E)[B-(A-E)] = A(B-A)\]
• because \(E\) is really, really small. Basically but not zero..
• Algebra…
• \[2AE - BE - E^2 =0\]

• \[2AE - BE - E^2 =0\]
• Divide through by E (which isn’t zero, remember?)
• \[2A - B - E = 0\]
• Now set \(E = 0\); (it wasn’t zero before! Promise!)
• \[2A = B\]
• The maximum of the curve occurs when \(A = B/2\).
• This is \(\displaystyle \lim_{\epsilon \to 0} \frac{f(x+\epsilon) - f(x)}{\epsilon} = 0\)

• Fermat’s Method of finding tangents to curves given a Cartesian equation by finding subtangents
  • the horizontal distance on the \(x\) axis from the \(x\) point to where the tangent would intersect. On the picture below, this tracing the value \(a\) to the left.

¹⁹

• assume we’re at point \((x,y)\) and want to find the subtangent \(a\); \(e\) is a small infinitessimal quantity
• Use The fact that we have similar triangles (whose hypotenuses) are the tangent: \[ \frac{a}{y} = \frac{e}{\text{vertical}}\]

• The Cartesian coordinates are given here; and now pretend it’s on the curve

• Set \(f(x + e, y + \frac{ey}{a}) = 0\)
• Algebra… divide everything by \(e\) (it’s not zero!)
• Then pretend that \(e\) was zero the whole time and solve for \(a\).

• Specific Fermat example is for the ’folium of Descartes’ (letter to Mersenne in June 1638) \[x^3 + y^3 = nxy\] (this picture has \(n=5\))

• Set \(f(x + e, y + \frac{ey}{a}) = 0\)
• \((x+e)^3 + y^3(1+\frac{e}{a})^3 - ny(x+e)(1+\frac{e}{a}) = 0\)
• … tedious algebra…
• \(e\left( 3x^2 + \frac{3y^3}{a} - \frac{nxy}{a} - ny\right) + e^2 \left( 3x + \frac{3y^3}{a} - \frac{ny}{a} \right) + e^3 \left( 1 + \frac{y^3}{a^3}\right) = 0\)
• divide everything by \(e\)
• … set \(e = 0\) and solve for \(a\)
• \(\displaystyle a = -\frac{3y^3 - nxy}{3x^2-ny}\)

• If \((x, y) = (1, 0.20164)\) and \(n=5\), we find the subtangent to be \(a= 0.493827\). Plotting the actual tangent to the curve, we find:

• Before we get to Newton… Edmond Halley in 1684 discussed Kepler’s third law of motion:

  The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

  • Halley suggested this implied the attraction of the sun was the inverse of the square of the distance to the planetary bodies.
  • Christopher Wren challenged Halley and Robert Hooke to show that the inverse square law implied the planet had an elliptic orbit (with a 40 shilling wager). (Cook, 1991)
  • Neither Halley and Hooke did it.
  • Halley poses the question to Newton.
  • Newton thinks a bit.
  • Within two years he “invents” physics and calculus.
  • He should publish a book!

• Now, let’s talk about Fishes
• Royal Society and Historia Piscium, published 1686
• Expensive for very detailed engravings
• Halley and Hooke being paid £50 for their work
  • 🐟How about 50 fish books!?!?🐠

Now let’s talk about Newton’s fluxional calculus

• A fluent is a point or a line in motion
• A fluxion is the velocity of that motion

The method “shortly explained rather than accurately demonstrated.” (Newton quoted in (Boyer, 1988))

We’ll work an example!

• “Arrange according to the dimensions of some fluent quantity, say x, and”
• Multiply its terms by any arithmetical progression [ending in 0]
• then by \(\dfrac{\dot{x}{x}}\)
• Carroy out this operation separately for each one of the fluent quantities, then put the sum of the products equal to nothing, and you have the desired equation.

We’ll “accurately demonstrate” it!

• Binomial theorem!

• Mesopotamian tablet BM 92687 (about 6th century BC)
• The oldest world map is flat Imago Mundi

[fn: Osama Shukir Muhammed Amin FRCP(Glasg) / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0)]

There were other ancient flat-Earthers

• Greece - Homer
• Ancient Norse/Germanic people.
  • Yggdrasil, the world tree, is actually the center of the disk

But the Earth is round.

Eratostothenes (276-194 BC)

• Learns about the sun at noon in Syrene, Egypt lighting up a well so the sun is straight up
• The next year, same day, in Alexandria at noon the sun cast a shadow of a vertical pole

• Ideas why this suggests the Earth is round?
• Here’s a picture:

• How can we compute the circumference of the Earth?
• (using kilometers, rather than stadia, which he used)

• He’s pretty close

Later, 1037AD, al-Biruni applied trig (from India?) to find the circumference within 1% of the correct value!

• using the astrolabe (thanks Hypatia) helps

Why did Columbus think he found Asia?

• using maps that were too small
• measured circumference 25% too small
• oops

• 1777 - 1855 in Germany
• School story (6 or 10 years old)
• at 14, Duke of Brunswick gives financial support to him
• Proves regular polygon of 17 sides can be constructed
• 1801, published Disquisitiones Arithmeticae foundation of modern number theory
• Proof of the fundamental theorem of algebra (age 22)
• Age 24, method of least squares and Ceres
• Married Johanna Ostaf in 1805
  • Legitimately in love. (quote in logseq notes)
• Becomes director of Gottingen observatory in 1807
• 1808, wife dies giving birth to second kid
  • Legitimately devastated. (quote in archive notes)

• marries Minna, best friend of first wife.
• 1809, work on celestial bodies
• Work in analysis in 1816
• 1818 geodesic (curved map) survey of Hanover;
• (non-Euclidean geometry; we’ll come back to this)
• 1830s physics, for example gravitation, magnetism
• as a hobby-ish, built a telegraph that could send over 5,000 feet
• differential geometry
• … lots of other things
• many famous students, including Riemann
• Minna’s death story

• Zeno and paradoxes
  • Example: Achilles and hare
  • Can’t accumulate to infinity; meaningless
• Aristotle says nah;

  • Actual infinite is bad

    \(\infty\) isn’t a number

  • Potential infinite is good

    Largest number ever? Add one.

• Euclid was with Aristotle
• “Infinitely many primes” is actually “Prime numbers are more than any assigned magnitude of prime numbers”
• Basically everyone had this view for 2000 years
• (probably people here, too)

• Fingers on hand
• chairs to humans
• etc
• bijections: one-to-one correspondence

• Line up every counting number with a square.
• Square numbers are ’sparse’
• So that line up is meaningless!
• PARADOX!!!
  • I mean, I disagree
• Can’t talk about \(<, >, =\) of infinite sets

• A set is said to be countable if there exists a one-to-one correspondence with the natural numbers.
• Examples
  • The set of even numbers, {2, 4, 6, 8, …}, is countable since we have the one-to-one correspondence n <-> 2n.
  • Galilleo’s paradox: The set of square numbers, is countable since we have the one-to-one correspondence \(n\) <-> \(n^2\)
  • The set of powers of 2, {2, 4, 8, 16, …}, is countable, with the one-to-one correspondence n <-> \(2^n\)
  • …

• Born March 3, 1845 in St. Petersburg, Russia, died January 6, 1918 in Halle, Germany.
• Family moved to Germany when he was 12
• Completed his doctorate at the University of Berlin in 1867 under Weierstrass
• In 1874 he published a paper which fundamentally changed the way we think of infinity.
• Even Gauss had said, “I protest above all against the use of an infinite quantity as a completed one, which is mathematics is never allowed. The Infinite is only a manner of speaking” [quoted from ref:Dunham1991]

• We can place the set of all integers (positive, negative, and zero) in one-to-one correspondence via the mapping: \[n \leftrightarrow \frac{1+(-1)^n(2n-1)}{4}\]
• (check examples)
• So they have the same cardinality (size of a set). He called this \(\aleph_0\)

• Draw the number array!

• A logical statement can only be true or false (it can’t be both).
• We assume the opposite of a statement (logical negation), and then through argumentation derive a logical contraction:
  • Ex: “Assume that the square root of two is rational…. We proved that it is not rational.”
  • It can’t be both rational and non-rational, and since our assumption (IS rational) was wrong, it must be that (NOT IS RATIONAL or irrational)

• The interval of real numbers between 0 and 1 is not denumerable. (Cantor in his 1874 paper)
• Proof by contradiction – the reductio ad absurdum:

• Our assumption: Let’s assume that the interval IS denumerable and find this gives a contradiction.
• If the interval (0, 1) IS countable, what does that mean? There is a one-to-one correspondence with the set of natural numbers.

• Every number between 0 and 1 is on the right side of the table. …. I’ll draw the table in a moment
• To arrive at the contradiction, we will construct number that is not on the list.
• Do the proof… Blow some minds.

• But the set of ALL real numbers is an infinite collection of all of these infinite, uncountable intervals. So surely it’s bigger…
• …Right?
• No. 
• The cardinality of all real numbers is the same as the cardinality of the interval (0, 1):
• Consider the one-to-one correspondence:
• \(f(x) = \dfrac{2x-1}{x-x^2}\) from \((0,1) \to \mathbb{R}\).

• To answer the question formally, we need only one more theorem:
• Theorem: If B and C are denumerable (countable) sets and A is the set of all elements belonging to B or C or both, then A is also denumerable.
• (We can prove it, but we’ll go without proof… unless you want one)

• We know that if B and C are countable, then the union A = B U C is countable.
• So if the set of rationals and irrationals were the same size (countably infinite), then their union would be countable:
• {Real numbers} = {rationals} union {irrationals}
• But the set of real numbers is NOT countable.
• So the set of irrational numbers is also not countable!

• An algebraic number is one which is a solution to some polynomial with rational coefficients.
• A transcendental number is one that is not algebraic. We know \(\pi\), \(e\), and some others.
• Cantor didn’t.
  • Lindemann wouldn’t prove \(\pi\) was transcendental for a decade!
• Cantor showed that the set of algebraic numbers is countable (a little beyond this class. If we had another class or two on the topic…)
• So all though the algebraic numbers (which include any conceivable roots of numbers) are nearly every number we can possibly imagine. And we can only imagine a handful of transcendentals
• There are vastly more transcendental numbers in the numeric universe!

• 1877 shows \(|(0,1) \times (0,1)| = |(0,1)|\)
• “I see it, but I don’t believe it.”

• Is told to quit by respected colleague Mittag-Leffler: (Laubenbacher & Pengelley, 1999)

  I am convinced that the publication of your new work… will greatly damage your reputation among mathematicians… but in this way, you will exercise no significant influence, which you naturally desire as does everyone who carries out scientific research.

• Quits mathematics
• Back to theology!
• Theologians love his work on infinite sets!
• 1895 publishes Beiträge zur Begründung der transfiniten Mengenlehre
• Discussing cardinalities of sets, showed that:
• \(|\mathcal{P}(A)| > |A|\) for any set \(A\). (Called “Cantor’s Theorem”)
• “In place of a given set L, another set M can be placed which is of greater power than L”
• … our term ’power sets’
• \(\aleph_0 < |(0,1)| < |\mathcal{P}((0,1))| < |\mathcal{P}(\mathcal{P}((0,1)))| < \dots\)
• an infinite chain of transfinite numbers
• … changed math forever

• Aristotle Quote in (Ifrah, 2001, p. 105)

  Suppose every instrument could by command or by anticipation of the need execute its function on its own; suppose that spindles could weave of their own accord, and plectra strike the strings of zithers by themselves; then craftsmen would have no need of hand-work, and masters have no need of slaves.

• Abacus
• Napier’s Bones (interactive html5)
• These don’t mechanize computing
  • They help us count
  • We still have to do manual work

• Greeks had mechanics (Archimedes, for example)
• Heron built steam-powered machines
• One theory from (Ifrah, 2001):
  • lack of a positional number system with zero
  • lack of a need to do tons of computations

• Science progressed (exploded) in the 16th and 17th centuries
  • see also Newton, the Bernoullis, Lagrange, Laplace, …
• doing science was tied to mathematical computation

• 1525 - a pedometer

  • Invented by Jean Errard de Bel-le-Duc (Ifrah, 2001, p. 124):

    A new geographical instrument which, attached to a horse’s saddle, uses the horse’s steps to display the length of the journey one has made … by which one can exactly measure the circuit of a place or the length of a journey“

  • He made the old-timey FitBit!

• First automatic calculator: the “calculating clock” 1623 by German astronomer Wilhelm Schickard (1592-1635)

  • Here’s a working model built from Schickhard’s notes in 1960

• Strickard described it to Kepler in 1623 (letter dated September 20, 1623)
• Only copy of the machine was burned in a fire on February 22, 1624.

• Pascaline 1642 - how it works (linked on “Additional Resources”): Recommend to watch it on at least 1.5 speed because he talks really slowly https://www.youtube.com/watch?v=3h71HAJWnVU
• Leibniz stepped-reckoner 1694

• lots of time passes, lots of machines
• Thomas de Colmar’s Arithmometer in 1822
  • reliable, functional, successful commercially
  • marketed in many countries with different names:
    • “Saxonia, Archimedes, Unitas, TIM”
    • guess what the last one means
    • “Time is Money”

• … side track
• Keyboards were invented in the mid-1800’s
• but were fiddly; error prone
• (side note: why QWERTY?)
• First printer invented in 1872
• First calculator with a keyboard and printer in 1882
  • Marketed as a cash register
  • help solve disputes with customers

• Babbage’s difference engine with relaxing music https://www.youtube.com/watch?v=be1EM3gQkAY
• This video demos the math behind the machine https://www.youtube.com/watch?v=PFMBU17eo_4

Really great photo collection from the Computer museum

History of Computing in ComputerWorld 1981 https://archive.org/details/TheHistoryOfComputing/mode/2up

• Pascaline - how it works:

https://www.youtube.com/watch?v=3h71HAJWnVU Recommend to watch it on at least 1.5 speed because he talks really slowly

• Babbage’s difference engine with relaxing music https://www.youtube.com/watch?v=be1EM3gQkAY
  • This video demos the math behind the machine https://www.youtube.com/watch?v=PFMBU17eo_4
• ENIAC Really great photo collection from the Computer museum
  • setting it up https://www.computerhistory.org/revolution/birth-of-the-computer/4/78/316
  • operating it https://www.computerhistory.org/revolution/birth-of-the-computer/4/78/321
  • implemented in a circuit chip in 1995: https://www.computerhistory.org/revolution/birth-of-the-computer/4/78/327

• Leibniz invents binary number system - using 0’s and 1’s published in 1703 Explication de l’Arithmétique Binaire
• Has addition, subtraction, multiplication, and division
• Acknowledges the earlier work of Fuxi and the “mystery of lines” (Note: Fuxi is said to be the inventor of Chinese writing)

• Leibniz’ binary system was philosophical - Laplace wrote, as in (Ifrah, 2001, p. 89):

  In the binary system which he invented, Leibniz sought to perceive the Creation. He imagined that the Unity stood for God and Zero for the Void, that the Supreme Being had drawn every existing thing out of the void just as unity and zero suffice to represent every possible number in the system…

  … I mention this merely to show how childish fancies can cloud the visions of even the most distinguished men.

  Old-timey burn!

• Yijing and yin and yang
  • yin is “female” and represented by a broken line: - -
  • yang is “male” and represented by a contiuous line: —
• Use trigrams (three lines) or hexagrams (six lines)

• Summer, the South, the sky, the Sun (solid lines…)

    ---
    ---
    ---
  

• Winter, the North, the Earth, the Moon:

    - -
    - -
    - -
  

• (boolean negations)

• Thunder, the Northeast, coming of Spring:

  - -
  ---
  - -
  

• Wind, the Southwest, end of Summer:

  ---
  - -
  ---
  

• trigrams represent \(2^3 = 8\) possibilties
• hexagrams represent \(2^6 = 64\) possibilities
• the ability to negate terms, as well as combine them
  • we’d call them join and meet
  • … or call them and and or
• Later this is a Boolean Algebra we study in Math 322
  • equivalent to \(\cup, \cap\) in set theory in 501.

Laws of Thought https://www.gutenberg.org/files/15114/15114-pdf.pdf

Formal Logic by de Morgan https://archive.org/details/formallogicorthe00demouoft

Trig and double algebra https://archive.org/details/ost-math-trigonometry_and_double_algebra/page/n7/mode/2up

Memoirs (by his wife) https://archive.org/details/memoirofaugustus00demouoft/page/n11/mode/2up

A Budget of Paradoxes https://archive.org/details/budgetofparadoxe00demouoft/page/22/mode/2up

• Mesopotamian tablet BM 92687 (about 6th century BC)
• The oldest world map is flat Imago Mundi

[fn: Osama Shukir Muhammed Amin FRCP(Glasg) / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0)]

There were other ancient flat-Earthers

• Greece - Homer
• Ancient Norse/Germanic people.
  • Yggdrasil, the world tree, is actually the center of the disk

But the Earth is round.

Eratostothenes (276-194 BC)

• Learns about the sun at noon in Syrene, Egypt lighting up a well so the sun is straight up
• The next year, same day, in Alexandria at noon the sun cast a shadow of a vertical pole

• Ideas why this suggests the Earth is round?
• Here’s a picture:

• How can we compute the circumference of the Earth?
• (using kilometers, rather than stadia, which he used)

• He’s pretty close

Later, 1037AD, al-Biruni applied trig (from India?) to find the circumference within 1% of the correct value!

• using the astrolabe (thanks Hypatia) helps

Why did Columbus think he found Asia?

• using maps that were too small
• measured circumference 25% too small
• oops

• 1777 - 1855 in Germany
• School story (6 or 10 years old)
• at 14, Duke of Brunswick gives financial support to him
• Proves regular polygon of 17 sides can be constructed
• 1801, published Disquisitiones Arithmeticae foundation of modern number theory
• Proof of the fundamental theorem of algebra (age 22)
• Age 24, method of least squares and Ceres
• Married Johanna Ostaf in 1805
  • Legitimately in love. (quote in logseq notes)
• Becomes director of Gottingen observatory in 1807
• 1808, wife dies giving birth to second kid
  • Legitimately devastated. (quote in archive notes)

• marries Minna, best friend of first wife.
• 1809, work on celestial bodies
• Work in analysis in 1816
• 1818 geodesic (curved map) survey of Hanover;
• (non-Euclidean geometry; we’ll come back to this)
• 1830s physics, for example gravitation, magnetism
• as a hobby-ish, built a telegraph that could send over 5,000 feet
• differential geometry
• … lots of other things
• many famous students, including Riemann
• Minna’s death story

• Zeno and paradoxes
  • Example: Achilles and hare
  • Can’t accumulate to infinity; meaningless
• Aristotle says nah;

  • Actual infinite is bad

    \(\infty\) isn’t a number

  • Potential infinite is good

    Largest number ever? Add one.

• Euclid was with Aristotle
• “Infinitely many primes” is actually “Prime numbers are more than any assigned magnitude of prime numbers”
• Basically everyone had this view for 2000 years
• (probably people here, too)

• Fingers on hand
• People in chairs
• etc
• bijections: one-to-one correspondence

• Line up every counting number with a square.
• Square numbers are ’sparse’
• So that line up is meaningless!
• PARADOX!!!
  • I mean, I disgree
• Can’t talk about \(<, >, =\) of infinite sets

• A set is said to be countable if there exists a one-to-one correspondence with the natural numbers.
• Examples
  • The set of even numbers, {2, 4, 6, 8, …}, is countable since we have the one-to-one correspondence n <-> 2n.
  • Galilleo’s paradox: The set of square numbers, is countable since we have the one-to-one correspondence \(n\) <-> \(n^2\)
  • The set of powers of 2, {2, 4, 8, 16, …}, is countable, with the one-to-one correspondence n <-> \(2^n\)
  • …

• Born March 3, 1845 in St. Petersburg, Russia, died January 6, 1918 in Halle, Germany.
• Family moved to Germany when he was 12
• Completed his doctorate at the University of Berlin in 1867 under Weierstrass
• In 1874 he published a paper which fundamentally changed the way we think of infinity.
• Even Gauss had said, “I protest above all against the use of an infinite quantity as a completed one, which is mathematics is never allowed. The Infinite is only a manner of speaking” [quoted from ref:Dunham1991]

• We can place the set of all integers (positive, negative, and zero) in one-to-one correspondence via the mapping: \[n \leftrightarrow \frac{1+(-1)^n(2n-1)}{4}\]
• (check examples)
• So they have the same cardinality (size of a set). He called this \(\aleph_0\)

• Draw the number array!

• A logical statement can only be true or false (it can’t be both).
• We assume the opposite of a statement (logical negation), and then through argumentation derive a logical contraction:
  • Ex: “Assume that the square root of two is rational…. We proved that it is not rational.”
  • It can’t be both rational and non-rational, and since our assumption (IS rational) was wrong, it must be that (NOT IS RATIONAL or irrational)

• The interval of real numbers between 0 and 1 is not denumerable. (Cantor in his 1874 paper)
• Proof by contradiction – the reductio ad absurdum:

• Our assumption: Let’s assume that the interval IS denumerable and find this gives a contradiction.
• If the interval (0, 1) IS countable, what does that mean? There is a one-to-one correspondence with the set of natural numbers.

• Every number between 0 and 1 is on the right side of the table. …. I’ll draw the table in a moment
• To arrive at the contradiction, we will construct number that is not on the list.
• Do the proof… Blow some minds.

• But the set of ALL real numbers is an infinite collection of all of these infinite, uncountable intervals. So surely it’s bigger…
• …Right?
• No. 
• The cardinality of all real numbers is the same as the cardinality of the interval (0, 1):
• Consider the one-to-one correspondence:
• \(f(x) = \dfrac{2x-1}{x-x^2}\) from \((0,1) \to \mathbb{R}\).

• To answer the question formally, we need only one more theorem:
• Theorem: If B and C are denumerable (countable) sets and A is the set of all elements belonging to B or C or both, then A is also denumerable.
• (We can prove it, but we’ll go without proof… unless you want one)

• We know that if B and C are countable, then the union A = B U C is countable.
• So if the set of rationals and irrationals were the same size (countably infinite), then their union would be countable:
• {Real numbers} = {rationals} union {irrationals}
• But the set of real numbers is NOT countable.
• So the set of irrational numbers is also not countable!

• An algebraic number is one which is a solution to some polynomial with rational coefficients.
• A transcendental number is one that is not algebraic. We know \(\pi\), \(e\), and some others.
• Cantor didn’t.
  • Lindemann wouldn’t prove \(\pi\) was transcendental for a decade!
• Cantor showed that the set of algebraic numbers is countable (a little beyond this class. If we had another class or two on the topic…)
• So all though the algebraic numbers (which include any conceivable roots of numbers) are nearly every number we can possibly imagine. And we can only imagine a handful of transcendentals
• There are vastly more transcendental numbers in the numeric universe!

• 1877 shows \(|(0,1) \times (0,1)| = |(0,1)|\)
• “I see it, but I don’t believe it.”

• Is told to quit by respected colleague Mittag-Leffler: ref:laubenbacher1999

  I am convinced that the publication of your new work… will greatly damage your reputation among mathematicians… but in this way, you will exercise no significant influence, which you naturally desire as does everyone who carries out scientific research.

• Quits mathematics
• Back to theology!
• Theologians love his work on infinite sets!
• 1895 publishes Beiträge zur Begründung der transfiniten Mengenlehre
• Discussing cardinalities of sets, showed that:
• \(|\mathcal{P}(A)| > |A|\) for any set \(A\). (Called “Cantor’s Theorem”)
• “In place of a given set L, another set M can be placed which is of greater power than L”
• … our term ’power sets’
• \(\aleph_0 < |(0,1)| < |\mathcal{P}((0,1))| < |\mathcal{P}(\mathcal{P}((0,1)))| < \dots\)
• an infinite chain of transfinite numbers
• … changed math forever

• What’s an interesting thing you learned?
• How might you use it in a classroom (or not)?
• How has your appreciation of math changed?
  • Autumn
  • Noha
  • Abe
  • Jacob
  • Lizzie
  • Amelia

What’s an interesting thing you learned? How might you use it in a classroom (or not)? How has your appreciation of math changed?

• John
• Isaac
• Sam
• Liz
• Quincy
• Tony

• You might not be a super math nerd.
• I’m not

Future teachers of America!

• famous (and not famous) mathematicians were once kids
• share your enthusiasm
• Let’s inspire people
• And let’s have fun.

Future engineers of America!

• You can share your love of math, too!
• And please make bridges and buildings and electrical systems safe.

• Thanks for your insight and your passion.
• Keep doing great work, and good luck in your future!
• Thanks for a fun class.