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Thread: Can you identify this equation?

  1. #1
    Oct 2007

    Can you identify this equation?


    While I was back in school, during a maths class I had finished some coursework early and had to sit out the remaining time. For some reason related to the work, there was a list of square numbers in front of me. After staring at them for a while I picked up on pattern when doing some mental arithmetic and formed a rule which I've used ever since to help with long multiplication. I'm pretty sure it is either a famous rule/sequence or just an obvious statement to a mathematician. Anyway, after mentioning it to someone recently, I decided to transcribe it to a formula which goes as follows:

    $\displaystyle xy = ((x+y)/2)^2 - ((x-y)/2)^2)$

    I'm pretty sure I've written that right. Does anyone know what this is or if it is obvious. Like I said, I find it useful for doing multiplication in my head for numbers in the tens, aslong as you know a midpoint (between x and y) square number.

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  2. #2
    MHF Contributor
    Apr 2005
    It is the intersection of two hyperbolas.

    xy = N is a hyperpola.

    [(x -a)/2]^2 -[(y -b)/2]^2 = N is a hyperbola too.
    But [(x +y)/2]^2 -[(x-y)/2]^2 = N, I am not very sure. Maybe someone using a graphing calculator could investigate that.
    I don't know how to use any graphing calculator.
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  3. #3
    Super Member

    May 2006
    Lexington, MA (USA)
    Hello, Morello!

    $\displaystyle xy \:= \:\left(\frac{x+y}{2}\right)^2 - \left(\frac{x-y}{2}\right)^2$
    You discovered this yourself . . . Good for you!

    I've made many "discoveries" myself,
    . . but they always turn out to have been known for centuries.

    And I wouldn't call it "obvious".

    I think I've seen it before, but not in any textbook.
    More likely a "recreational" book with interesting formulas, shortcuts, etc.
    [So I don't think it could be called "Morello's Law",
    . . where your name will be cursed by students for several millenia.]

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  4. #4
    Oct 2007

    Smile Follow Up


    I was just sorting through some old emails and followed a link to my original post - never saw the second reply! Just thought I'd let anyone know who might be interested, I did get a further answer:

    Someone had recognised the formula as a variation of a Babylonian equation. Here was the base of the reply:

    An overview of Babylonian mathematics

    Here is a quote:

    "Perhaps the most amazing aspect of the Babylonian's calculating
    skills was their construction of tables to aid calculation. Two
    tablets found at Senkerah on the Euphrates in 1854 date from 2000
    BC. They give squares of the numbers up to 59 and cubes of the
    numbers up to 32. The table gives 82 = 1,4 which stands for

    8^2 = 1, 4 = 1 60 + 4 = 64

    and so on up to 59^2 = 58, 1 (= 58 60 +1 = 3481).

    The Babylonians used the formula

    ab = [(a + b)^2 - a^2 - b^2]/2

    to make multiplication easier. Even better is their formula

    ab = [(a + b)^2 - (a - b)^2]/4

    which shows that a table of squares is all that is necessary to
    multiply numbers, simply taking the difference of the two squares
    that were looked up in the table then taking a quarter of the
    Just thought I'd share the result to my query! Just goes to show what you can think of in your spare classroom time!
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