Concept of Imaginary Numbers

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Complex Numbers

All complex numbers consist of a real and imaginary part.
The imaginary part is a multiple of i (where i =[pic] ).

We often use the letter ‘z’ to represent a complex number eg. z = 3 +5i

The conjugate of z is written as z* or [pic]

If z1 = a + bi then the conjugate of z (z* ) = a – bi

Similarly if z2 = x – yi then the conjugate z2* = x + yi

z z* will always be real (as i2 = -1)

For two expressions containing complex numbers to be equal, both the real parts must be equal and the imaginary parts must also be equal.

If z1 = a + bi , z2 = x + yi and 2z1 = z2 + 3 then

2( a + bi) = x + yi + 3 hence 2a + 2bi = x + 3 + yi

so 2a = x + 3 (real parts are equal) and 2b = y (imaginary parts are equal)

When adding/subtracting complex numbers deal with the real parts and the imaginary parts separately

eg. z1 + z2 = a + bi + x + yi
= a + x + (b + y)i

When multiplying just treat as an algebraic expression in brackets

eg. z1 z2 = (a + bi)(x + yi)
= ax + ayi + bxi + byi2
= ax - by + (ay + bx)i (as i2 = -1)

Division by a complex number is a very similar process to ‘rationalising’ surds – we call it ‘realising’


Argand Diagrams

We can represent complex numbers on an Argand diagram. This similar to a normal set of x and y axes except that the x axis represents the real part of the number and the y axis represents the imaginary part of the number.


The argand diagrams allow complex numbers to be expressed in terms of an angle (the argument) and the length of the line joining the point z to the origin (the modulus of z). Hence the complex number can be expressed in a polar form. The argument is measured from the real axis and ranges...
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