Concept of Imaginary Numbers
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’
[pic]
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.
[pic]
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
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’
[pic]
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.
[pic]
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