1. Complex numbers
There is no number which squares to make -1, so there is no ‘real’ answer!
What is ?
Mathematicians have realised that by defining the imaginary number ,
many previously unsolvable problems could be understood and explored.
If , what is:
A number with both a real part and an imaginary one is called a complex number Eg
The imaginary part of z, called Im z is 3
A complex number in the form is said to be in Cartesian form
Complex numbers are often referred to as z, whereas real numbers are often referred to as x
The real part of z, called Re z is 2

2. Expand & simplify as usual, remembering that i2 = 1
Manipulation with complex numbers
Techniques used with real numbers can still be applied with complex numbers:
WB1 z = 5 – 3i, w = 2 + 2i
Express in the form a + bi, where a and b are real constants,
(a) z2 (b) a)
Expand & simplify as usual, remembering that i2 = 1 b)
An equivalent complex number with a real denominator can be found by multiplying by the complex conjugate of the denominator
If then its complex conjugate is

3. Modulus and argument Im Re The complex number
can be represented on an Argand diagram by the coordinates Eg Eg Eg Re
The modulus of z, Eg
The principal argument Eg
is the angle from the positive real axis to in the range Eg
Remember the definition of arg z

4. Im Re WB2 The complex numbers z1 and z2 are given by
Find, showing your working,
(a)
in the form a + bi, where a and b are real,
(b) the value of
The modulus of is
(c) the value of
, giving your answer in radians to 2 decimal places. Im
The principal argument
is the angle from the positive real axis to in the range Re

5. Im Re Im Re WB3 z = 2 – 3i (a) Show that z2 = −5 −12i.
Find, showing your working,
(b) the value of z2,
(c) the value of arg (z2), giving your answer in radians to 2 decimal places. Im Re
(d) Show z and z2 on a single Argand diagram. Im Re

6. Complex roots In C1, you saw quadratic equations that had no roots.
Quadratic formula Eg
If then
We can obtain complex roots though
We get no real answers because the discriminant is less than zero
We could also obtain these roots by completing the square:
This tells us the curve
will have no intersections with the x-axis

7. Im Re Im Re WB4 z1 = − 2 + i (a) Find the modulus of z1
(b) Find, in radians, the argument of z1 , giving your answer to 2 decimal places. Im Re
The solutions to the quadratic equation z2 − 10z + 28 = 0 are z2 and z3
(c) Find z2 and z3 , giving your answers in the form p  iq, where p and q are integers.
(d) Show, on an Argand diagram, the points representing your complex numbers Im Re

8. WB6
Given that , where a and b are real constants,
(a) find the value of a and the value of b.
Comparing coefficients of x2
Comparing coefficients of x0
(b) Find the three roots of f(x) = 0.
either or
(c) Find the sum of the three roots of f (x) = 0.
So sum of the three roots is -1

9. Problem solving with roots
If a is a root of f(x)
then is a factor
In C2 you met the Factor Theorem:
Eg Given that x = 3 is a root of the equation ,
(a) write down a factor of the equation,
(b) Given that x = -2 is the other root, find the values of a and b
is the other factor
is the equation factorised
expanding
In FP1 you apply this method to complex roots…

10. Problem solving with complex roots
We have seen that complex roots come in pairs: Eg
This leads to the logical conclusion that if a complex number
is a root of an equation, then so is its conjugate
We can use this fact to find real quadratic factors of equations:
WB5 Given that 2 – 4i is a root of the equation z2 + pz + q = 0,
where p and q are real constants,
(a) write down the other root of the equation,
(b) find the value of p and the value of q.
Factor theorem:
If a is a root of f(x)
then is a factor

11. Im Re WB7 Given that 2 and 5 + 2i are roots of the equation
(a) write down the other complex root of the equation.
(b) Find the value of c and the value of d.
(c) Show the three roots of this equation on a single Argand diagram. Im Re

12. Problem solving by equating real & imaginary parts
Eg Given that
where a and b are real, find their values
Equating real parts:
Equating imaginary parts:

13. WB8 Given that z = x + iy, find the value of x and the value of y such that
where z* is the complex conjugate of z.
z + 3iz* = −1 + 13i
then
Equating real parts:
Equating imaginary parts:

14. Eg Find the square roots of 3 – 4i in the form a + ib, where a and b are real
Equating real parts:
Equating imaginary parts:
as b real
Square roots are -2 + i and 2 - i

15. Eg Find the roots of x4 + 9 = 0
Equating real parts:
Equating imaginary parts:
Roots are

16. known as the modulus-argument form of a complex numberIm If
and
then
known as the modulus-argument form of a complex number Re
Eg express in the form
Eg express
in the form
From previously,
and so

17. It can also be shown that:
The modulus & argument of a product
It can be shown that:
It can also be shown that:
Eg if and
Eg if and Im Re
This is easier than evaluating and then finding the modulus…

18. It can also be shown that:
The modulus & argument of a quotient
It can be shown that:
It can also be shown that:
Eg if and
Eg if and
From previously,
This is much easier than evaluating and then finding the modulus…

19. Im Re WB9 z = – 24 – 7i (a) Show z on an Argand diagram.
(b) Calculate arg z, giving your answer in radians to 2 decimal places. Re
It is given that w = a + bi, a  ℝ, b  ℝ.
Modulus-argument form
Given also that and
where
(c) find the values of a and b
and
(d) find the value of
given

20. Complex numbers Using: Manipulation with complex numbers
Modulus and argument Im
Also Re
Complex roots
w is a root of Find the values of a and b
Equating real & imaginary parts
Find the values of p and q
Equating real parts:
Equating imaginary parts: