THE SQUARE ROOT OF MINUS ONE

David McNaughton

(Gulf Weekly, Al Nisr Publications, Dubai, UAE, 30 April 1992)

Note that the square root symbol is unfortunately rendered as Ö in Linux

What is the square root of minus one? In other words, is there a number which when multiplied by itself, produces an answer "- 1"?

In arithmetic, minus multiplied by minus gives plus. (Taking a debt away from someone is really like crediting a positive sum of money - and in a similar way, a sentence containing two negatives becomes positive). Thus, because (- 1) squared equals +1, the answer to Ö (-1) cannot be "minus one" (nor can the result be plus one).

As far as ordinary, everyday numbers go, the square root of minus one does not exist. However, mathematicians find it useful to invent a completely new kind of number to fulfil that role. They decided to assume that such a root really did exist, and wrote 'i' to represent it. ('i' was chosen because it stands for "imaginary").

We can now go on and find the square root of minus four: the answer is '2i', because

(2i)² = 4i² = - 4

In exactly the same manner, square roots can now be calculated for minus 25, minus 49 and minus 100. In this new field of mathematics, a totally new realm of numbers has been created which co-exists with our more familiar, everyday numbers (which do not of course contain any i's).

It is even possible for these two different types of number to appear side by side within the same entity. Consider (2+i) for example: it is just the quantity '2' added to 'i'. Such a mixture is called a "complex number". (Ordinary quantities containing no i's are named "real numbers").

As another illustration, (8-3i) is also a "complex number" - and there is no shorter way of writing it. It is never possible to multiply or add 'i' with an ordinary number to yield an ordinary (i.e. real) numeric answer.

It is intriguing however, that raising something to the power of 'i' does occasionally produce a real (or ordinary) answer. For example, ii is nearly 0.2079, although it would be too difficult here to try and show you why. In fact, this last calculation is rather meaningless, partly because ii can produce many other values.

However, there is a rather special number (approximately 23.1407) which raised to the power of 'i' gives, enigmatically, an answer "minus one" - and (please believe me!) this result is extremely important and useful in complex number theory. (23.1407 is ep , in case you are wondering).

It is illuminating to portray complex numbers on a graph. To plot the quantity (3+2i), measure three units to the right and two units upwards. For negative values, move downwards (if the i-component is negative), or to the left (when your real portion is negative). This graph, called an Argand Diagram, illustrates how cube roots and higher roots form a beautiful symmetry which is not at all obvious when they are written down as clumsy looking numbers.
 
 
 
 

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+2
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+1
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__ __ -2__ __ __ __ __-1 __ __ __ __ __ O __ __ __ __ +1__ __ __ __ __ *+2 __
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-1
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*. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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-2
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THE THREE CUBE ROOTS OF EIGHT

Marked by asterisks on this Argand Diagram, they form a perfect equilateral triangle.
They all lie at an equal distance from the zero-point, which is marked O.

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For example, what is the cube root of 8? Yes, one of the answers is '2', but there are actually a couple more solutions:

(1.732i -1), and (-1.732i -1).

Even if your knowledge of algebra is very elementary, try and verify those cube roots by multiplying three of them together, remembering that i² = -1 (so i³ = - i ). If you do that, it will be more accurate to work with Ö 3 (rather than with 1.732 which is only an approximation);

e.g. multiply out (Ö 3i - 1)(Ö 3i - 1)(Ö 3i - 1).

Now see what happens when you place those three cube roots on an Argand Diagram: they form an equilateral triangle. Furthermore, the distances are all equal from each of those root-points to the central zero-point. And something very similar happens with fifth-roots, or eighth-roots, or indeed any other roots.

The fifth-root of 32 is 2 (because 2 ´ 2 ´ 2 ´ 2 ´ 2 = 32), but if we allow complex solutions, then there are also four other 5th-roots. When all five are plotted on an Argand Diagram, they form a regular pentagon around the centre of the graph. It therefore seems that this drawing is a much clearer and better way of representing complex numbers, than writing down their numeric value.

Remember too that numbers have two square roots, not just one; e.g. for 9 they are +3 and -3, and for minus one they are +i and -i. The symmetry is still there when pairs like those are plotted on an Argand Diagram, but higher roots illustrate it in a more striking fashion.

You are probably asking "What use is all this?" It is of course impossible to collect 'i' objects, nor can you walk 'i' metres, nor can I lend you 'i' dollars or pounds.

However, it turns out that complex numbers are capable of providing analogies with certain situations, and helping to solve them. For example, when fluids are flowing or swirling round in a complicated pattern, the movement of each particle may be represented by a complex number on an Argand Diagram. Known rules for manipulating complex numbers (as compact, single entities) are then applied to examine what changes would be expected in the flow under specified conditions.

This is useful in meteorology. Forced to use ordinary algebra, two separate equations of motion are necessary, one for north-south airflow, and another for east-west flow. Coordinating two such equations is laborious, but with complex numbers just one equation will suffice, which turns out to be much simpler and easier.

Electricity is another branch of science which employs complex numbers. An alternating current always surges up with its maximum voltage, then decreases and swings the other way under negative voltage. That process may easily be represented on an Argand Diagram by a rotating arm. One end of the arm is fixed to the centre; the height of the other end shows the instantaneous voltage.

When an induction coil is inserted in a circuit, it pushes the alternating voltage out of phase by a quarter of a wavelength, which is equivalent to a 90-degree turn on our modified Argand Diagram. A capacitor produces a similar phase change, but in the opposite direction. The overall result then depends on the relative strengths of all coils and capacitors present.

On a drawing of a rotating arm, there is an amazingly simple rule for swinging it through 90 degrees - you just multiply the complex number by 'i'. Thus, when an electric circuit is considered in terms of its corresponding Argand Diagram, all associated formulae suddenly become quite neat and compact, even if they do contain a sprinkling of i's.

Of course it is not possible to have an i-number of volts or amps or ohms, but the size and phase of the resulting voltage may easily be deduced from those formulae.

Incidentally, electricians write 'j' for Ö (-1) instead of 'i', because they use 'i' for current. Often, they will be dealing with an extended network of several interacting circuits: that would be a really formidable problem without complex numbers.

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(2i-1)* . . . . . . . . . . . . +2 . . . . . . . . . . . . . . . . . .
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__ -2__ __ __ __ __-1 __ __ __ __ __ O __ __ __ __ +1 __ __ __ __ __+2 __
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THE EFFECT OF MULTIPLYING BY 'i'

The point in the upper right quadrant, (2+i), becomes (2i-1) if multiplied by i.
Join those two points (asterisks) to the zero-position - which is labelled "O".
Those two oblique lines will then be exactly 90 degrees apart - and are the same length.

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E-mail: DLMcN@yahoo.com

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