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What is the Square Root of i?

Asked by a student at Eastwood High on May 9, 1997:
What is the square root of i?
There are two square roots of i: (1/sqrt(2)) (1 + i) and (-1/sqrt(2)) (1 + i). You can check that these are indeed square roots of i: just square each of them, and you get i.

The important question is, how are these answers obtained?

An elementary, but not the best, way to obtain this answer is to solve the equation (a+bi)^2 = i for a and b. If you expand this equation using the rules for complex multiplication, you get (a^2 - b^2) + (2ab)i = 0 + 1i. Equating real and imaginary parts gives a^2-b^2=0 and 2ab = 1.

The equation a^2-b^2=0 means a = +/- b. However, if you plug a=-b into the second equation you get -2b^2 = 1 which can not be satisfied by any real number b. Therefore, the case a = -b is not possible, meaning a must equal b. Then the second equation becomes 2a^2 = 1. This means either a = b = 1/sqrt(2) or a = b = -1/sqrt(2). These are the answers that were given above.

However, the easiest and most insightful way to take the square root of a complex number (as well as any higher order roots) is to use the geometric representation of the complex numbers. Just as we can plot real numbers as points on a line, we can think of complex numbers as lying on a plane. The horizontal (real) axis corresponds to the real part of the complex number and the vertical (imaginary) axis corresponds to the imaginary part. For example, the number 3 + 2i is located at the point (3,2) which is two units above and three units to the right of the origin.

What is most useful about this visualisation of the complex numbers is how addition and multiplication behave. To add two numbers a + bi and c + di, we can think of shifting the first one c units to the right and d units up.

Multiplication is a bit more difficult to see. Before we go on, it is useful to introduce another means of specifying points in the plane. Suppose that X is any point in a plane and suppose that O is the origin. If we know the distance r from X to the origin O and we also know the angle t between the positive real axis and the line segment OX, we can locate the point X. This angle and distance are known as the polar coordinates of X. Multiplication corresponds to adding the angles of the two points and multiplying their lengths.

                   * |
                  *  |
                 *   |
              r *    | r sin t
               *     |
              *angle |
             * t     |
            O=========--------------------> x-axis
             r cos t

The reason for this has to do with trigonometry: the coordinates of the point whose distance is r and whose angle is t are given by x = r cos t, y = r sin t. That means it is the complex number r(cos t+ i sin t) which equals re^(it) because of de Moivre's formula (which is explained in the answer to another question). If you multiply two numbers like that, you get

re^(it) se^(iw) = (rs) e^(i(t+w))
(the laws of exponents still hold for complex exponential). In other words, the product is a complex number whose distance from the origin is rs (the products of the distances of the factors), and whose angle is t+w (the sum of the angles of the factors).

Understanding multiplication helps us understand other operations such as taking the square root. For the sake of simplicity, first consider the case in which the number we are interested in has a distance of 1 from the origin. The square root of this number also has a distance of 1 from the origin and forms an angle with the real axis which is 1/2 of the angle corresponding to the original number. In the event that the number is not 1 unit away from the origin, we obtain new distance from the origin by taking the square root of the old distance from the origin (here the lengths are positive real numbers and the notion of "square root" is already defined).

Now we can understand why we got the answer we did for the square root of i. It is easily seen that i forms an angle of 90 degrees with the real axis and has distance of 1 from the origin. Its square root is the number with a distance of 1 from the origin and an angle of 45 degrees from the real axis (which is 1(cos 45 degrees + i sin 45 degrees) = (1/sqrt(2)) (1 + i)).

A cautious reader will note that there is some ambiguity in choice of the angle in our definition of polar coordinates. A point of distance 1 from the origin creating an angle of 45 degrees with the real axis is the same point which is 1 unit from the origin and forms an angle of 405 degrees with the real axis. Generally we always insist that the angle be between 0 and 360 degrees. Note however that when taking the square root of a complex number it is also important to consider these other representations. For instance, i can also be viewed as being 450 degrees from the origin. Using this angle we find that the number 1 unit away from the origin and 225 degrees from the real axis ((-1/sqrt(2)) (1 + i)) is also a square root of i.

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