Complex numbers were first developed in the mid 16th Century, as a means for solving certain cubic equations. They consist of an imaginary part (in terms of i, or ) and a real part ('traditional' numbers, so to speak - numbers such as 1, -324 or ). Since then, they have become a frequently used type of number in solving polynomial equations and in, unusually, calculations by engineers. In FP1, we consider fairly basic uses of Complex Numbers in solving quartic, cubic and quadratic equations. Similarly, we consider basic arithmetic of complex numbers - adding, subtracting, multiplying and dividing complex numbers.
Each Complex Number has a 'Complex Conjugate'. Despite it's rather complicated name, it is a very simple conecpt:
The Complex Number has the Complex Conjugate .
The Complex Conjugate has multiple uses. For instance, if an equation has the complex root (1+3i), the complex conjugate (1-3i) will also be a root. Similarly, in order to divide complex numbers, one must use the complex conjugate, as we will later see, to 'rationalise' the denominator. One does this by multiplying the fraction by (which has the same effect, value wise, of multiplying by one). This then allows for the quotient to be re-written in the form a+ib.
For any complex number, the complex conjugate is denoted with an *. For instance, the complex number 'a' has the complex conjugate 'a*'. The complex number 'y' has the complex number 'y*'. Mostly, the examination will use the complex numbers z and w.
This is a slightly more complex process, similar to rationalising the denominator in situations involving surds.
For instance: . In order to 'divide' the top complex number (on the numerator) by the bottom complex number (on the denominator), one must first make the denominator into a real number. In order to do this, one must use the 'complex conjugate' of the denominator. This is, as already seen in section 2, similar to the denominator, only the imaginary part of the number has the opposite polarity. So, in our example, the complex conjugate of would be . Since, however, one cannot simply introduce terms to the fraction. If, however, we use the fraction , we know that this has value one. Yet, multiplying our original fraction by this will change it, allowing for a simplification to take place.
This can by mathematically solved:
Whilst this process was much expanded for clarity, it does perhaps stress the need for care and the need for accuracy when dividing complex numbers. Similarly, whilst the numbers may be a little complex, the process is fairly straightforward. It is also very similar to 'rationalising the denominator' in that one eliminates the 'i' term from the denominator, allowing the demoninator to become a factor of the overall expression.
There are only two scenarios for solutions of Quadratic Equations (that is, those where the highest power of is ). If the co-efficients in the quadratic are all real, one can either have two real roots (which also includes a repeated root, as in the case of ) or two complex/imaginary roots. Only equations of this type (with real coefficients) are needed in FP1.
Consider now the quadratic equation . Previously, one would have been unable to solve this as it would require one to root . Since we now represent this value as , this quadratic can now be solved with relative ease:
Using the quadratic formula gives:
We can then use the factor theorem to prove that either or are factors of .
Let is a factor of
The same result is achieved for , since .
The same can be done with Complex Roots:
Consider the quadratic equation:
Using the quadratic formula, one can see that:
If one was to now factorise this expression:
It should be noted, once again, that is a conjugate pair (since is the complex conjugate of and vice versa). This property is extremly useful, and is often examined in FP1.
Graphical Representations of Complex Numbers
This is the general Argand Diagram, displaying the complex number z = a + bi
This is an interesting concept, when one considers that a part of the complex number is totally imaginary. How can one represent something, in a mathematic manner, that is totally imaginary? An 'Argand Diagram' is the tool used to graphically represent a complex number. It does so by treating the real part of the number as an x-co-orderinate and the imaginary part of the number as a y-co-ordinate.
Argand Diagrams are quite helpful when it comes to determining the 'Argument' of the number. It will also allow for a more visual representation of what is meant by the 'Modulus' of the complex number.
This is very similar to finding the length of any given straight-line on a graph. Imagine that you had two points - the origin (0,0) and point A (3,4). The line OA is drawn, connecting the two points. To find the length of this line, one would use an application of Pythagoras, as learned in C1 and C2: