Complex Numbers: Real and Imaginary! Quiz

Answer: a

The best way to look at a complex number ('a + bi') is in two parts, the first part is 'a' and you'll notice it is not multiplied by the imaginary unit, 'i'. It, 'a', is therefore not imaginary and is thus real, however, 'b' is multiplied by the imaginary unit and so 'b' is the imaginary part of the complex number.

a = Real part

b = Imaginary part

i = Imaginary unit

Answer: (-1)^(1/2)

The imaginary unit is easier to define in words by first playing around with the algebra:

i = (-1)^(1/2)
i^2 = -1

As can be seen above, the imaginary unit squared is equal to negative one. When students first start off studying mathematics and come across the quadratic equation (which features a square root), they are told that you can't take the square root of a negative number. Later on in one's studies however, it is found out that you can indeed square root a negative number, and 'i' is the result!

Answer: (a + c) + (b + d)i

The addition of complex numbers is nice in the fact that it is very much similar to the addition of real numbers. The only extra process is to add the imaginary terms of complex numbers up as well. In 'Z1' the real part of the complex number is 'a' and in 'Z2' the real part is 'c'. The addition of these real parts is 'a + c', but you then have to add the imaginary terms which is basically the same. The imaginary term in 'Z1' is 'b' and in 'Z2' it is 'd' and the addition of these two terms is 'b + d'. The only thing left to do is bolt on the imaginary unit, 'i':

Z1 + Z2 = (a + c) + (b + d)i

Answer: Complex conjugate of Z2 (Z2*)

In order to divide one complex number by another, the complex conjugate of the denominator needs to be discovered. If 'Z1' is to be divided by 'Z2' then 'Z2' is the complex denominator ('Z1' will be the numerator). We have defined 'Z2' as being 'c + di' and the complex conjugate of this is found by simply changing the sign associated with the imaginary part of the complex number.

Z2 = c + di

Z2* = c - di = complex conjugate

Answer: Abraham de Moivre

Below is de Moivre's Formula:

[cos(x) + sin(x)i]^n = cos(nx) + sin(nx)i; where 'cos' and 'sin' are trigonometric functions, 'x' is an angle in radians, 'n' is a constant number and 'i' is the imaginary unit equal to the square root of negative one.

The enigmatic 'i' appears in this formula which means that complex numbers aren't far away. In fact if you arbitrarily pick an angle 'x' in radians and a random number 'n', the right hand side of the equation can be nicely reduced to the simplest form of a complex number, 'a + bi'.

Further inspection of the right hand side of de Moivre's formula leads one to notice that both 'cos(nx)' and 'sin(nx)' are examples of multiple angles. That is, the argument of both cosine and sine (the terms in the brackets) is a multiple, 'n', of the angle 'x'. If 'n = 2' then we get 'cos(2x)' and 'sin(2x)' and these are examples of double angles. The formula proposed by de Moivre can help quantify such multiple angles in terms of single angles, such as 'cos(x)'.

Answer: Leonhard Euler

Euler's Formula, 'e^(ix) = cos(x) + sin(x)i' is perhaps the most celebrated work of genius in the field of mathematics. Physicist and science icon, Richard Feynman, said of the formula, "[it is] one of the most remarkable, almost astounding, formulas in all of mathematics".

Euler's Formula is intricately linked with geometry and trigonometry; the former in the respect of the complex plane and Argand diagrams and the latter with regards to the derivation of hyperbolic functions.

Answer: Leonhard Euler

Euler's identity, '[e^(i*pi)] + 1 = 0'*, really is extraordinary. Even if the aesthetics of the identity don't tickle your fancy, the truly beautiful and inspiring aspect of Euler's identity is hiding beneath the surface.

The irrational number 'e' is named Euler's number and is the inverse of the natural logarithm, 'ln'. The value of 'e' cannot be accurately expressed in numbers zero to nine because there is no end; it struts off to an infinite number of decimal places. To illustrate the point I wish to make however, 'e = 2.71828182845904523536' to twenty decimal places (but there are infinitely many more).

Pi is a well known irrational number, which also has infinite decimal place values, and to twenty decimal places it is given the value, 'pi = 3.14159265358979323846'. The imaginary unit, 'i', is equal to the square root of negative one. The beauty is in the fact that the numbers 'e' and pi which both go on for an infinite number of decimal places can be placed into an expression as simple as '[e^(i*pi)] + 1' and equate to zero! The mathematics behind how these two phenomenal numbers end up interacting to equal zero is available but I'll leave it as an exciting mystery here!

*I'd encourage anybody, if they have not seen the expression before, to have a look around the internet or in mathematical books for this identity as it is more aesthetically pleasing in those formats.

Answer: No

The value of the real part of a complex number dictates how far along the x-axis (abscissa) the plot point is. The value of the imaginary part of a complex number corresponds to where the plot point is positioned with regards the y-axis (ordinate). The complex plane forms the basis of an Argand diagram and is thus sometimes referred to as the Argand plane.

Answer: Modulus

The modulus of a complex number 'z', '|z|', is another way of saying the absolute value of 'z'. If 'z = a + bi', then '|z| = ([a^2] + [b^2])^1/2'. That is, the absolute value of a complex number is found by squaring the value of the real part, squaring the value of the imaginary part, adding the squares together and finally taking the square root of that value. You may notice that this is just applying Pythagorean theorem to complex numbers, 'z^2 = a^2 + b^2'.

Answer: Fourier analysis

One of the most basic uses of complex numbers in the physics of waves is best demonstrated with the use of a sine wave with a given frequency. By finding the modulus, or absolute value, of a complex number 'z = a + bi', '|z|', you have simultaneously found the amplitude of the sinusoidal wave. By finding the argument of 'z' (inverse tangent of the quotient of 'b/a') you find the phase of the wave.