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Revision:Complex Numbers

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics Revision Notes > Complex Numbers

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Complex Numbers

Up until this point, you will probably have been told that you cannot square a number and come out with -1. However, often it is useful to give it an answer.

Complex, Imaginary and Real Numbers

Mathematicians have decided that if we see -1 then we can replace this with the symbol i^2. From this, we can give every answer to every mathematical function a value.

Any number in the form bi is called an imaginary number. Any number without i is called a real number. Any number with a combination of both is called a complex number, which is normally written a + bi.

If two complex numbers are equal, then their real parts are equal, and the imaginary parts are equal. That is a + bi = c + di if and only if a = c and b = d

Complex Real Imaginary
3+2i 7 4i
4-0.5i \pi i\sqrt{3}

Argand Diagram

As well as numerically, you can portray complex numbers on an Argand diagram. These diagrams consist of two axes - the real axis (horizontal) and the imaginary axis (vertical), labelled as such. Each complex number is defined as a point on this diagram. The complex number 'a' + 'b'i is 'a' along the real axis; and 'b' up the imaginary axis.

Modulus and Arguments

Using this diagram, complex numbers can be defined in another way, by the angle they make with the Real axis, and the length of vector to that point.

The modulus of a + bi is defined as its length.

If z = a + bi, then the modulus of z, written |z| = \sqrt{a^2 + b^2}.

The argument of a complex number is defined as the angle it makes with the Real (x) axis. This could be an infinite number of things (\theta,\, \theta + 2\pi, \theta + 4\pi etc.) You are normally asked for the principal argument, between \pi and -\pi.

This is normally worked out by finding the triangle on the Argand diagram with the angle in and using trigonometry.

Complex Number Functions

Re and Im

Given a complex number a + bi,

Re(a + bi) = a Im(a + bi) = b

Example: z = 2 - 3i \rightarrow Im(z) = -3, Re(z) = 2

Complex Conjugate

Given a complex number z = a + bi, its complex conjugate, shown by z^* is equal to a - bi.

It is important to note that the only number you can multiply a complex number z by to get a non-zero real number is its conjugate, z^* (or any non-zero real multiple of it).

Example: z = -3 + 2i \rightarrow z^* = -3 - 2i

Adding or Subtracting Complex Numbers

This is simply done by adding the component parts:

(a + bi) + (c + di) = (a + c) + (b + d)i

Example: (3 + 2i) - (2 - 5i) = 1 + 7i

Multiplying Complex Numbers

Multiplying two complex numbers is done as though expanding two brackets. Remember that i^2 = -1

(a + bi)(c + di) = ac + adi + cbi + bdi^2 = (ac - bd) + (ad + bc)i

Example: (3 + i)(5 - 2i) = 15 - 6i + 5i - 2i^2 = 17 -i

Dividing Complex Numbers

It is much trickier to divide complex numbers than multiply them. As such, division is often turned into multiplication by multiplying top and bottom of a fraction by its conjugate:

\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{(c^2 + d^2)}

Since c^2 + d^2 is a real number, you can simply divide each term by it.

Example: \frac{2 + 3i}{4 - i} = \frac{(2 + 3i)(4 + i)}{(4 - i)(4 + i)} = \frac{(8 - 3) + (12 + 2)i}{(17)} = \frac{5}{17} + \frac{14}{17}i

Equations using Complex Numbers

Often you are given an equation and are required to solve it to find complex roots. This is done in two ways.

  • Factor Theorem: Once you have discovered roots you can simplify the equation by factorising.
  • If a complex number z is the root of an equation with real coefficients, then so is z*. This result must be memorised, though you don't have to know how to prove it.
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