FP1 – Complex Numbers

What are complex numbers? They’re numbers which are made of a real part, a, and imaginary part, bi, in the form a + bi, note: imaginary and real numbers can never be combined into one term. 

Imaginary number allow you to use the root of a negative number, for example √-4 = 2i.

Adding and subtracting complex numbers is simple; you just treat the real and imaginary parts separately.  

e.g. (2 + 5i) + (7 + 3i) = (2 + 7) + (5 + 3)i = 9 + 8i             

e.g. (6 + 3i) – (4 – 9i) = (6 – 4) + (3 – -9)i = 2 – 6i

Multiplying complex numbers is the same as multiplying in algebra, but beware: as i = √-1, i2 = -1.

e.g. (2 + 3i) x (4 + 5i); using FOIL/parrot’s beak method for multiplying brackets you get 8 + 10i + 12i + 14i2 which equals 8 + 22i – 14  so the final answer is – 6 + 22i.

In order to divide complex numbers, you must first know their complex conjugate. This is simply switching the sign between the real number and imaginary number from + to – or vice versa. Algebraically this means a + bi turns into a – bi. The pair of complex numbers is called the complex conjugate pair. The convention in mathematics is to call one of them z and the other one z*, it doesn’t matter which is which.

To divide two complex numbers, turn them into a fraction, then times both top and bottom by the complex conjugate of the denominator.

e.g. (10 +5i) divided by (1 + 2i) turns into (10 + 5i)/(1 + 2i) x (1 – 2i)/(1 – 2i) which turns into (10 + 5i)(1 – 2i)/(1 + 2i)(1 – 2i) and by using the method learnt for multiplication above it is easy to solve this fraction to get the answer of 4 – 3i. (Some of you may have met this technique before when rationalising denominators, and will see that the denominator of the combined fraction is the difference of 2 squares)

Sometimes you may be given the roots of a quadratic equation, which are always a conjugate pair. This means the equation will be (x – α)(x –β) or x2 – (α + β)x + αβ.

Complex numbers can be shown on a type of graph called an Argand diagram. This is the same as your normal type of graph, but now the x-axis is for the real part of your complex number, and the y-axis is for the imaginary part, and the vectors created by the complex numbers all begin from the origin.

The modulus of a complex number: │x + iy│= √(x2 + y2)

The argument of a complex number (arg z) is the angle θ (usually in radians) between the positive side of the real axis and the vector created by the complex number on an Argand diagram. If z = x + iy, then θ = arctan(y/x).  (Note: if when you draw the vector the complex number makes is to the left of the y (imaginary) axis, take you answer for θ away from 180 (if using degrees) or π (if using radians).). 

Complex numbers can also be written in the form  z = r(cos θ + isin θ) where r is the modulus, and where θ is between -180 and 180 degrees (or between – π and π).

Another property of complex numbers is that if you take any two complex numbers, z1 and z2, then │z1 z2│= │z1││z2│.

You can also find the square root of a complex number. To do this you create simultaneous equations.

e.g. z = 3 + 4i: (a + bi)2 = 3 + 4i then expand to get a2 – b2 + 2abi = 3 + 4i. Then equate the real numbers together in one equation, and the imaginary numbers together in another; a2 – b2 = 3 and 2ab = 4, then solve as you would normal simultaneous equations. In this case the answers are 2 + i and – 2- i.

The techniques you’ve learnt so far can also help you solve cubic or even quartic equations.

Cubic equations: either all 3 roots are real, or one is real and the other 2 roots are a conjugate pair, so to solve it divide the cubic by the real answer, then complete the square to find the conjugate pair.

Quartic equations: either all 4 roots are real, 2 are real and the other 2 are a conjugate pair, or the roots are 2 conjugate pairs.

This is all you need to know about complex numbers – I hope you’ve found it useful! And if there are techniques from GCSE and AS/A2 maths I’ve used or mentioned here that you are unsure about, such as diving cubics by (x + a), please comment and I’ll make another post explaining them – I’ve already completed my A-Level maths this year hence why I’ve started with a current I covered from Further Maths a few weeks ago.

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