Multiplying one positive number by another gives a positive result. Multiplying by 0 gives 0. It follows that multiplying a positive number by a negative one gives a negative result. Given the last point, and the fact that multiplying a negative number by 0 gives 0, one can deduce that multiplying together two negative numbers gives a positive result.

From the above, it can be deduced that the square of a number is always positive (except that of 0, which is 0). It may therefore be said that negative numbers do not have square roots. On the other hand, it can be decided that such square roots do exist but do not have numerical values. This brings us to the subject of imaginary numbers.

Imaginary numbers are multiples of *i*, which denotes √-1. Their use enables any equation to be solved. If an equation simplifies to x²=-(a²), then x=±ai.

Complex numbers consist of both a real and an imaginary part, for example 4+3i. They can be manipulated in the same way as purely real numbers. Treat i as a variable, then simplify powers of i using such rules as i²=-1 and i³=-i.

If z denotes the complex number a+bi, ¯z denotes its complex conjugate a-bi (The bar should be over the z, but I do not know the code for that symbol). Conversely, z is the conjugate of ¯z.

Complex numbers can be represented graphically in what is termed the complex plane. If the real axis corresponds to the x-axis and the imaginary axis to the y-axis, z=a+bi is represented by a point at (a,b). The modulus *r* of the number is equal to the length of the straight line connecting that point to the origin (0,0), and its argument (Arg z) is the angular distance of that line from the positive real axis. z can therefore be written in terms of polar co-ordinates: z=r cos θ + ir sin θ, where r=|z| and θ=Arg z.

It will be noticed that when a number is multiplied by i, the point denoting that number in the complex plane is rotated anticlockwise through 90° about the origin. This makes complex numbers a useful tool when dealing with quantities that vary sinusoidally, for example alternating electric current. There, the i acts as a coordinate and is not intended to imply that such quantities literally have an imaginary component.

z can also be expressed in terms of e. cos x=1-(x²/2!)+(x^{4}/4!)-(x^{6}/6!)+... and sin x=x-(x³/3!)+(x^{5}/5!)-(x^{7}/7!)+... In addition, it will be remembered that e^{x}=1+x+(x²/2!)+(x³/3!)+... Substituting ix for x and simplifying gives e^{ix}=1+ix-(x²/2!)-(ix³/3!)+(x^{4}/4!)+(ix^{5}/5!)... It therefore seems that e^{ix}=cos x + i sin x; this is known as Euler's formula. Finally, z=r cos θ + ir sin θ=r(cos θ + i sin θ). Substituting gives z=|z|e^{iθ}.

If nθ is substituted for θ, e^{inθ}=cos nθ + i sin nθ. However, e^{inθ}=(e^{iθ})^{n}=(cos θ + i sin θ)^{n}. Therefore, (cos θ + i sin θ)^{n}=cos nθ + i sin nθ. Given that z=re^{iθ}, z^{n}=r^{n}e^{inθ}=r^{n}(cos nθ + i sin nθ). This is De Moivre's Theorem.

Given the above, e^{-iθ}=cos (-θ) + i sin (-θ)=cos θ - i sin θ. Therefore, e^{iθ}+e^{-iθ}=cos θ + i sin θ + cos θ - i sin θ=2 cos θ. Rearranging gives cos θ=0.5(e^{iθ}+e^{-iθ}). Similarly, working out e^{iθ}-e^{-iθ} gives sin θ=(e^{iθ}-e^{-iθ})/2i. It will be remembered that cosh θ=0.5(e^{θ}+e^{-θ}) and sinh θ=0.5(e^{θ}-e^{-θ}). By comparison, therefore:

cosh iθ=cos θ. sinh iθ=i sin θ.

Also:

cos iθ=cosh θ. sin iθ=i sinh θ.