Math Chat: Algebra, Voting Systems, Logarithms, and why .999... = 1

Math Chat is a potpourri of diverse mathematical topics to discuss. This month we are highlighting three videos that focus on the nature and history of algebra, voting systems, logarithms, and why .999... = 1. If you have a favorite math topic or a math question, bring it and we can chat about it too!

To start the discussion, here are three videos that explore algebra, voting, logarithms, and the fact that .999... = 1. 

• This 1¾ hour video of Keith Devlin discusses the nature and history of algebra and ends with a survey of the mathematical paradoxes of voting systems. In addition, the last 18 minutes is a general Q&A on the Nobel Prize in economics, math education, and game theory which we can also discuss. Read my notes on Devlin's lecture.

http://www.youtube.com/watch?v=FME9avU3u2Y

Note: Keith Devlin's MOOC "Introduction to Mathematical Thinking" begins on Feb 3rd (10 weeks long deadline-driven course): https://www.coursera.org/course/maththink

Vi Hart's video of 9.999... reasons that .999... = 1

http://www.youtube.com/watch?v=TINfzxSnnIE

Vi Hart's video on logarithms

http://www.youtube.com/watch?v=N-7tcTIrers

Discussion topics and some questions:

• The Nature and history of algebra

• How does algebra differ from arithmetic and algorithm?

• How did algebra develop historically?

• What is the difference between algebraic expression, algorithm, and a model?

• What are the differences between plurality voting, single transferable vote (STV), and approval voting? Which method do you think is most fair?

• Devlin asserts "you tell me who you want to win, and I can give you a fair way to count the votes so that person will win": do you believe it?

• In single transferable vote (STV), is it possible that by adding support to a candidate they could shift from winning to losing? Is it fair that the order in which candidates are eliminated can make that much difference (in STV)?

• Is the expression "the will of the people" meaningful?

• What are the implications of Arrow's Impossibility Theorem to democracy?

• Does .999... = 1?

• Which of Vi Hart's reason's most appeals to you?

• Are there any of her reasons that you do not understand?

• Does Vi Hart's video on logarithm's give a good picture of how they work?

• Do you have any questions about logarithms?

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  • Lynn

    I had a great time and met some interesting people

    1 · January 26, 2014

  • Greg G.

    The people were very nice very intelligent and very motivated to do more I'm looking forward to the next meeting

    1 · January 25, 2014

  • CJ F.

    Did anyone have any questions or comments about the videos, so I can think about them before the meetup?

    Watching the videos is optional. If you have time and missed 1 or 3 of them, here are the links one last time:

    1) 1¾ hour video of Keith Devlin on the history and meaning of algebra and voting systems (with asides on models, algorithms, game theory, & nobel prize in economics): http://www.youtube.com/watch?v=FME9avU3u2Y

    2) 10 minute Vi Hart video on logarithms: http://www.youtube.com/watch?v=N-7tcTIrers

    3) 10 minute Vi Hart video on reasons that .999... = 1: http://www.youtube.com/watch?v=TINfzxSnnIE

    If you watched them all, let us know if you had any questions or comments, so I can muse on them overnight.

    See you at 11 AM tomorrow morning!

    Take SEPTA to avoid worries about driving with 1-2 inches of snow predicted in the afternoon. The meeting should last 2 hours until 1 PM which is when the snow is expected to begin (warning: predictions can be wrong!).

    January 24, 2014

  • Lynn

    I really hope to make it out to this meetup. I have certain issues with transportation and such so forgive me if I don't make it. Hope to see ya there!

    January 23, 2014

    • CJ F.

      Septa might make it possible. I mean how can we talk about algebra without al-Khwarizmi's participation?

      1 · January 24, 2014

  • A former member
    A former member

    Gotta work this Sat morning...

    1 · January 24, 2014

  • Martin C.

    Judging from the number of responses I have gotten (0), I am probably just talking to myself, but here goes anyway.

    I was wondering if Devlin was right in saying that game theory is only used in economics and did a Web search. I found some mention of computer science applications to things like controlling network traffic and cryptography. I also found this article: http://sciblogs.co.nz/misc-ience/2010/05/12/smart-agents-for-smart-grids-an-application-of-game-theory/ , explaining how you may eventually have a small computer attached to your electric meter using game theory to save on electric bills and help the electric grid make most efficient use of electric power generation.

    January 20, 2014

    • Martin C.

      I think that Devlin ended his history of algebra prematurely. See for example, http://www.ucs.louisi...­ Scroll down to the section on European math after 1500. If Omar Khayyam is to be credited for his geometric solution to the cubic equation then certainly credit should be given to the Italian Renaissance mathematicians who found closed form solutions to both the cubic and quartic equations. Someone not listed who should also be included was John Napier, who came up with logarithms in the beginning of the 17th century.

      January 21, 2014

    • Lynn

      It really depends on how you define economics. If you use the very broad definition proposed by Lionel Robbins—broadly speaking, decision making under scarcity—then virtually anything can fall under the category of "economics".

      January 22, 2014

  • CJ F.

    As I mentioned that the Keith Devlin video discusses the meaning of algebra. Devlin calls algebra "logically reasoning about numbers qualitatively". In this 10m video about logarithms (http://www.youtube.com/watch?v=N-7tcTIrers), Vi Hart says "elementary algebra is just fancy counting". On Saturday, let us know what you think!

    I think Vi's video is exquisite: so elementary, so entertaining, so conceptually clear! Her concept and execution are remarkable. Though I get Martin Cohen's point that it is perhaps simpler to see multiplication as repeated addition and exponentiation as repeated multiplication. Her poetic approach keeps it all on the number line which has its own elegance. Martin's point about exponentiation not being commutative is interesting, but might not attract the viewer with the allure of controversy!

    Read my deconstruction of Vi's video at https://plus.google.com/104222466367230914966/posts/UWVypeeZHKk

    Watch Vi's video at http://www.youtube.com/watch?v=N-7tcTIrers

    January 22, 2014

  • CJ F.

    On Saturday Math Chat will engage your math topics and questions. Since we might not have much to say about some things, this 1¾ hour video with Keith Devlin will give a topic we can discuss in more depth: http://www.youtube.com/watch?v=FME9avU3u2Y

    The 2nd part of Devlin's video compares three voting systems: plurality voting, single transferable vote (STV), & approval voting.

    Devlin outlines Kenneth Arrow's Impossibility Theorem: No voting system can satisfy these three fairness criteria: 1) Unanimity, 2) Individual Sovereignty, and 3) Nondictatorship.

    As Devlin puts it "you tell me who you want to win, and I can give you a fair way to count the votes so that person will win."

    Does voting represent "the will of the people" or is it a coin toss? Can we invent a fair voting system or does Arrow's theorem preclude it?

    Read my full notes on the video at https://plus.google.com/104222466367230914966/posts/Xteeo2w4d39

    Watch Devlin's video at http://www.youtube.com/watch?v=FME9avU3u2Y

    January 21, 2014

  • Martin C.

    Do we need to believe in complex numbers?

    The matter of dealing with complex numbers arose out of the study of algebra. They did not arise from the study of quadratic equations, as many believe. Those early mathematicians who studied quadratic equations just said that a quadratic equation had no solution if the number under the radical in the equation for the solution was negative.

    Cubic equations were a different matter. There were cases where all the roots are real, but in order to extract them it is necessary to take square roots of negative numbers. It was the work on cubic equations of the mathematicians of the Italian Renaissance that forced people to take imaginary numbers seriously. See for example http://www.mathisbeauty.org/cubicequationsandcomplexnumbers.pdf.

    January 21, 2014

  • CJ F.

    During Saturday's Math Chat, there will be an opportunity to talk about any math topic you want. To give us some common ground, we suggest watching this 1¾ hour video with Keith Devlin of Stanford: http://www.youtube.com/watch?v=FME9avU3u2Y

    In the first half of the video Devlin discusses the meaning and history of algebra. How does algebra differ from arithmetic? Is there a difference between logical thinking and numerical thinking? When did algebra enter math practice? Before or with Diophantus? Brahmagupta? Al-Khwarizmi? Omar Khayyám?

    What is the difference between algebraic expression, algorithm, and a model?

    Did you have any questions about the first half of the Devlin video? What did you learn about the history of algebra from Devlin?

    How do you define algebra?

    Wikipedia credits François Viète: https://en.wikipedia.org/wiki/History_of_algebra

    Did Devlin merely discuss the precursors to algebra?

    Watch the Devlin video here: http://www.youtube.com/watch?v=FME9avU3u2Y

    January 20, 2014

  • Martin C.

    Vi Hart's counting approach is in sharp contrast to the approach that Devlin recommends for teaching multiplication and exponentiation, for which he, justifiably as far as I am concerned, got a lot of flak. For more about this see Devlin's page at http://devlinsangle.blogspot.com/. Scroll down to the What is Multiplication section.

    January 15, 2014

  • Martin C.

    How to compute a logarithm using only basic algebra.

    Suppose you want to find log 2.
    10^x = 2
    Since x < 1, x can be written as
    x = /(n1 + r1) where n1 is an integer and 0 < r1 <1
    10^(1/(n1+r1)) = 2
    2^(n1+r1) = 10
    Since 2^3 = 8 < 10 < 2^4 =16
    n1 = 3
    We therefore have so far:
    x=1/3+r1

    2^(3+r1) = 10
    8 * 2^r1 = 10
    2^r1 = 10/8 = 5/4

    This has the same form as the original problem and we proceed in the same way.
    r1 = 1/(n2+r2)
    (5/4)^(n2+r2) = 2
    With a little work, you will find that n2=3.
    I am going to stop here, but we could go on indefinitely to greater precision.
    Dropping r2, we have r1 = 1/3
    log 2 = x = 1(3 + 1/3) = .3
    log 2 = .301 to three places.

    January 12, 2014

    • Martin C.

      The first line should be x = 1/(n1+r1)

      January 12, 2014

  • Martin C.

    a * 3 = a + a + a
    a^3 = a * a * a
    Why stop there? We can create a new operator, a^^3 that will be a^(a^a).
    For more see http://www.greatplay.net/essays/large-numbers-part-i-magnitude-and-simple-functions

    1 · January 12, 2014

  • Martin C.

    Regarding Vi Hart's logarithm video. I like the way that she shows that arithmetic derives from counting. That is what is done formally by the Peano axioms. I think her descriptions of multiplication and exponentiation would have been simplified by simply saying that multiplication is repeated additions (2+2+2 = 2 x3) and exponentiation is just repeated multiplication (2^3 = 2x2x2). Her dislike of roots notation overlooks an important point. Unlike addition, exponentiation is not commutative. There is an important distinction to be made. The solution of x^2 = 3 is different from 2^x=3. We use radicals for the first solution and logarithms for the second.

    1 · January 11, 2014

  • Martin C.

    Here is a site on approval voting: http://rangevoting.org/. Although obviously biased, it has some interesting information, particularly the results of simulations

    On whether .9999... =1, if you think not, consider the following. Should .11111... in binary also be different from 1? .1111... in binary is the same as the series 1/2 + (1/2)^2 + (1/2)^3 + ... Does every convergent series have a value different from the value it converges to?

    1 · January 11, 2014

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