Complex numbers are numbers that consist of a real number and an imaginary number. They come in the form of . (With and being real numbers and being the imaginary unit).
They add up very much like you would imagine. Take the two complex numbers (2+3i) and (3+4i). Adding the two together is as simple as adding the real parts (in this case: 2 + 3 = 5) and adding the imaginary parts (3i + 4i = 7i) to give us the final complex number of (5 + 7i).
The same applies for subtraction. Taking our example complex numbers of (2+3i) and (3+4i). We just subtract the real part (2 - 3 = -1) and subtract the imaginary part (3i - 4i = -i) to give us our new complex number (-1-i).
Multiplication is much like expanding quadratic and cubic equations. Simply, you just take the two complex numbers, sit them side by side and multiply out. Taking our example complex numbers we would do the following:
It really is as simple as thinking of like in our quadratics and treating it in exactly the same way, then replacing with , with and with .
The modulus of a complex number is the length of the line that joins the origin of an Argand diagram to the point that represents the complex number, and it is given by:
The argument of a complex number is the angle in radians of the complex number point from the real axis (the x-axis), measured counterclockwise. The argument for a complex number z, denoted arg(z), is given by:
if it is in the first quadrant.
When representing a complex number on an Argand diagram, we can see that
Since a complex number z can be represented by x+iy,
A complex number can be represented in either Cartesian or polar form. The polar form is:
where is and is the argument of z.
The polar form of a complex number is very similar to how we represent vectors. Accordingly, the modulus of a complex number is analogous to the resultant of the x and y components, and the argument is the direction of the resultant vector.
Euler's relationship is:
By substituting this into the polar form for a complex number, we can see that another way of writing the complex number z is:
Conjugates are just reflections of co-ordinates on an Argand diagram. They are reflected across the real axis and thus only the imaginary co-ordinate is changed (from, say, 2 + 2i to 2 - 2i). In other words, the sign of the imaginary part is changed (from negative to positive or vice versa).
Complex conjugates are useful when dividing complex numbers as the denominator can be made real by multiplying the top and bottom of the fraction by the complex conjugate of the denominator, for example:
This is called realising (the complex number version of rationalising) the denominator.
Conjugates are also useful when solving equations with real coefficients. If such an equation has an complex root, then the conjugate of the complex number will also be a root of the equation, allowing you to factorise the equation fully in most cases in exams.
If a polynomial has real coefficients and if any of its roots are complex, then these roots will occur in conjugate pairs.
Example: Given that is a solution, find the other roots of the equation .
We automatically know one of the other roots is (since the coefficients of the polynomial above are all real) so we can start to form a factorization of the equation:
Now that we know about complex numbers, we can start to solve quadratic equations whose determinant is negative (i.e. quadratics that do not cross or touch the x-axis on a Cartesian graph).
Example: Solve the equation .
Since we know that the equation does not have any real roots. Using the quadratic formula: