118. Imaginary Numbers


There is one kind on numbers which defy even our imagination! They are aptly named as Imaginary Numbers. What are they?

All perfect squares have whole numbers as their square roots. Square roots of 1 are (+ 1) and (-1). Square roots of 4 are (+ 2) and (-2). Square roots of 9 are (+3) and (-3). Square roots of 16 are (+4) and (-4) etc.

Also two negative numbers when multiplied, always yield a positive number. So (-1) x (-1) = +1 always and never equal to (-1).

In that case, what will be the strange number, which when multiplied by itself, will give us (-1)? In other words, what is the square root of the number (-1)?

Girolamo Cardano, a 16th century Italian, was the first one to talk about Imaginary Numbers. He thought that they were no doubt ingenious, but were quite useless.

Imaginary numbers are not absolutely useless though! There are certain calculations which are impossible to make, unless we make use of these Imaginary Numbers.

Useful or not, we find the Imaginary numbers in every High School curriculum, stretching the imagination of the baffled children, to their very limits!

Visalakshi Ramani

16 Responses to 118. Imaginary Numbers

  1. Arvind says:

    Maami, after many years of being dissatisfied with complex numbers, I finally came across a system of thinking about numbers that make sense to me. I thought I will share that with you.

    Think first of real number x as a piece of flexible material that extends from a fixed point, called 0 to x. Then k*x extends the material k times over. (This way of thinking allows k to be irrational and avoids the pitfalls that arise when thinking of multiplication as repeat addition.) x/k simply shrinks. Now come two interesting things.
    The first is more common what is -x. This is a 180 degree rotation of the flexible material from 0 to x while holding the end at 0 constant.
    The second is i*x. Its a 90 degree rotation. What about i*(ix). Thats just the x vector first rotated by 90 and then by 90 more. That is the same as a 180 degree rotation and therefore gives us -x.

    After much pain, I finally understand, at least to a greater extent, why i*i is -1. Of course, the pedagogical way of defining i as sqrt(-1) therefore i*i = -1 is not very satisfying or illuminating.

    Arvind

  2. Dear Arvind,
    very interesting way to express square root of (-1)!
    Still we need a lot of imagination to express (k*x) when k happens to be irrational and the concept of repeated addition will not be possible!
    Multiplication need not be always imagined as repeated additions-especially when it involves huge numbers and also irrational or imaginary numbers!
    Regarding the 90 degrees rotations, can it be in XY plane or should it be XZ plane?
    Well, the imaginary numbers demand a lot of imagination from the mathematicians, expressed either way!
    with best wishes,
    Visalakshi Ramani.

  3. Arvind says:

    The choice of the plane does not matter for rotations. Basically any axis perpendicular to the line from 0 to x.

    Arvind

    • Dear Mr. Arvind,
      I too thought that XY and XZ planes should be equally good for the rotation through a right angle.
      Looking forward to more thought provoking comments from you on other topics as well!
      with best wishes,
      Visalakshi Ramani.

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