Complex algebra

If I tell you there are imaginary numbers would you believe me? like imaginary creatures such as Dracula, or Leprechaun? Well isn't it all about our own mind that created numbers? Isn't it like all the numbers are imaginary after all and we believe in them only because they fit into our little convention of real mathematics? 

Imagine the square root of a negative number just by another convention. That's the power of imagination, though it is not very sensible only because we have not let it have a place in our mind as usual as the square root of four. When we deal with such imaginary numbers we come across to a whole new field of mathematics which is not normal and called complex algebra. As a result, a complex number is a sum of a real number and an imaginary number and is shown as

z = x + yj

where, x is the real component and y is the imaginary component of the complex number z. So what makes it imaginary? The j factor which is the square root of -1. Of course, this is one way to show a complex number which is named the rectangular form or trigonometric form. This name is given because the complex number can be shown in a rectangular coordinate system named complex plane. 

The other ways to represent a complex number is in the phasor form or polar form and exponential form. In this form z is presented using a magnitude r and an angel θ (in radians) from the axis of real component:

z = r∠θ     or     z = rejθ     or     z = r[cos(θ) + j Sin (θ)]

According to the Euler's equation:

ejθ = cos(θ) + j Sin (θ)

e-jθ = cos(θ) - j Sin (θ)

Cos(θ) = (ejθ + e-jθ) / 2

Sin(θ) = (ejθ - e-jθ) / 2

Using the rectangular form the x and y can be found by using the cosine and sine of the θ, respectively, as shown in the picture below. Also, if we have the x and y, we can derive the complex number's magnitude using the Pythagorean triangle. So we can write:

z = √(x2+y2) ∠ tan-1(y/x)

In summation of two complext number, similar to normal calculations, when adding two complex numbers, the real components add together and the imaginary components add together. That is, for example, 

(z1= 2+3j) + (z2 = 1-2j ) = (z3= 3 + j). 

In multiplication of two complex number, according to the algebraic distributive law each component of the first number will be multiplied with each of the components of the second number. In this process j2 can be replaced by -1. 

In division of two complex number, complex conjugate of the number has to be used. The complex conjugate of a complex number (x + yj) is (x - yj). In this process a technique is used called rationalizing denominator in which, both the numerator and denominator of the division is multiplied by the complex conjugate of the denominator which results in a real number in the denominator and makes the calculation straight forward. It is much easier to do the multiplication and division of complex numbers if they are presented in polar format. If z1 = r1∠θ1 and z2 = r2∠θ2 then:

z1 × z2 = r1 × r2 ∠ θ1 + θ2 

z1 / z2 = r1 / r2 ∠ θ1 - θ2

Next topic in Electricity and Magnetism for the FE exam is Electrostatics.