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A clever use of a parabola to perform multiplication

I read lots of Math books and I've run into many many interesting Math "things" in my travels but here's something very clever I've never encountered before. Let's say you want to multiply 5 by 8. Do the following:

  1. Plot the graph of y=x^2.
  2. Draw a line that crosses the parabola where x = -5 Parabola used for multiplicationand where x = 8 on the parabola. (Ignore the fact that x = -5 and not +5 at the left intersection point; this calculator does not do signed arithmetic!)
  3. Note the value of y where the line crosses the y-axis.
  4. The value of y is 40 and indeed 5 x 8 = 40.

Can you figure out why this trick works? Never mind that it's much more work to plot the graphs and determine where the line crosses the y-axis than it is to do the arithmetic in the first place!

This clever exploration, plus a number of other nice explorations for high school students come from the book Mathematics: A Human Endeavor by Harold Jacobs.

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  1. I looked at this once before, but it’s only just occurred to me that this is actually a special case of a multiplication nomogram (there are many different nomograms that do multiplication).

    If you mark the right arm of the parabola with its abscissas and the left arm with the negatives of the abscissas, then the three scales of the nomogram are the y-axis and the left and right sides of the parabola.

    This is a nomogram I (re-) discovered a few years ago. It turns out to be a transformed version of the circle nomogram (which I’d give a link to if I could still find the page – it seems to have disappeared).

    By the classification scheme on Winchell Chung’s pages (http://www.projectrho.com/nomogram/standardForms.html) it’s a Genus II nomogram (with w and u interchanged). I forget which book the classification scheme is from; Winchell will know.

    This particular parabolic-multiplication nomogram has basic determinantal equation

    | -u u^2 1 |
    | v v^2 1 | = 0
    | 0 w 1 |

    (Top row is left arm of parabola, middle row is
    right arm of parabola and bottom row is y-axis up the middle).

    expanding the bottom row
    0 – w.(-u-v) -u.v^2 – v.u^2 = 0

    => w(u+v) = uv(u+v)

    divide by (u+v) (so need u+v non-zero)

    => w = uv

    That is, such a nomogram encodes multiplication of positive real u and v.

    [By appropriate linear transformations of the 3×3 matrix in the basic determinant above, we can convert the graph into any of the conic sections – circle(+diameter), ellipse(+axis), hyperbola(+axis).]

    This is different from the most common (parallel-scale) nomogram, which encodes multiplication as addition of logarithms.

  2. It’s not the link I was looking for, but the first page of article at http://myreckonings.com/wordpress/2008/01/09/the-art-of-nomography-i-geometric-design/
    has a circular nomogram. It’s not hard to convert between your parabolic multiplication nomogram and a (slightly more common) circular multiplication nomogram.

  3. Yes It is great problem. We always can see one thing from various angles.
    Try this:
    consider y = x^2 the equation of the parabola
    y = mx + c the equation of a straight line that will cut the the y-axic on (0, c) and the parabola on M1(x1, y1) and M2(x2, y2)

    to find the intersections between the two graphs make y = y that means:
    x^2 = mx + c therfor x^2 – mx -c = 0
    m will be the sum of x1 and x2 and c = x1*x2.

    (Give me 5)!!!!!!!!!!

  4. I tried this problem with my students ( Year 8 to year 11)in Australian school.
    Students in years 8 & 9 used graph papers,Drew the graph of y = x^2 and then they drew a straight line that cuts the parabola. They could figure out the situation but could not figure out way this works.
    Students in years 10 & 11 used algebra and wrote equation like x^2 = m*x + c and the changed it to:
    X^2 – mx – c= 0
    Horray they said: c = x1*x2 where c is the y-intercept of the straight line with the y-axis.
    I hope you are happy with that!!!!!!

  5. Do you have more problem like that?

  6. Here is the answer.

    The equation of a parabola symmetric about the y axis is We can set
    Let us pick two values of y. These two values are the numbers we want to multiply.

    To draw a line, we need to define 2 points, as given one point we can draw many lines.
    Let these points be b and c.
    The parabola and line are then equal at these 2 points:

    I have to submit a pdf file with the math.

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