Let us state complex number definition or complex numbers definition. Any number which can be expressed in the form of a + i b is called a complex number. Here i is called iota.
So we can easily say that any number can be expressed in complex number form. Here a is the real part and b is the complex part.
Take an example of an integer 2. 2 can be expressed as 2 + i 0. So it is very clear that any number can be expressed in this form. Also we must remember that i 2 is equals to 1. Here is how we simplify complex numbers. As we know i 2 is equals to 1 so i 4 = 1.
Hence whenever we encounter the complex number we must divide it by the 4 and the remainder should be written as power of iota. Let us take an example. i 6. Divide it by 4. The remainder is 2. Hence i 6 = i 2 = -1.
I hope it is clear how to simplify complex numbers. Let us now move to operations on complex number like division of complex Numbers and Adding and Subtracting Complex Numbers. In case of addition the real parts are added individually and the complex parts are added individually. For example (2 + 6i) + (4 + 7i) = (2+4) + (6+7)i = 6 + 13i.
Similarly let us take an example of how subtraction is done. (2 + 6i) - (4 + 7i) = (2-4) + (6-7)i = -2 - 1i.
Similar is the multiplication. We need to multiply the real part individually and the complex parts individually. Let us take an example. (2 + 6i) * (4 + 7i) = (2 * 4) + (6 * 7)i = 8 + 42i.
What about division? It is a little complex but easily understandable. It is method similar to that applied in solving square roots. Yes you are right, Multiplication of both numerator and denominator by the conjugate of denominator. And then the problem is simplified. Let us take an example to show the processing.
3 + 4i / 2 + 3i = [(3 + 4i) ( 2 – 3i )] / [( 2 + 3i ) ( 2 – 3i )] = (6 – 12 i) / (4 – 9i2 ) = (6 – 12i) / 13. This is how we solve division of complex numbers. It is quite easy if we practice it once. Also take other examples to clear them in your mind.