Fields and complex numbers

The real part of the complex number $z=a+jb$ , written as $\Re (z)$ , equals $a$ . We consider the real part as a function that works by selecting that componentof a complex number not multiplied by $j$ . The imaginary part of $z$ , $\Im (z)$ , equals $b$ : that part of a complex number that is multiplied by $j$ . Again, both the real and imaginary parts of a complex number are real-valued.

The complex conjugate of $z$ , written as $\overline{z}$ , has the same real part as $z$ but an imaginary part of the opposite sign.

Using Cartesian notation, the following properties easily follow.

• If we add two complex numbers, the real part of the result equals the sum of the real parts and the imaginary part equals the sum of the imaginary parts. This property follows from the laws of vector addition. $$a_1+j{b}_1+a_2+j{b}_2=a_1+a_2+j(b_1+b_2)$$ In this way, the real and imaginary parts remain separate.
• The product of $j$ and a real number is an imaginary number: $ja$ . The product of $j$ and an imaginary number is a real number: $jjb=-b$ because $j^2=-1$ . Consequently, multiplying a complex number by $j$ rotates the number's position by $90$ degrees.

Complex numbers can also be expressed in an alternate form, polar form , which we will find quite useful. Polar form arises arises from the geometric interpretation of complex numbers.The Cartesian form of a complex number can be re-written as $$a+jb=\sqrt{a^2+b^2}(\frac{a}{\sqrt{a^2+b^2}}+j\frac{b}{\sqrt{a^2+b^2}})$$ By forming a right triangle having sides $a$ and $b$ , we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle. We thus obtain the polar form for complex numbers. $$\begin{array}{l}z=a+jb=r\times \angle \theta \\ r=\left|z\right|=\sqrt{a^2+b^2}\\ a=r\cos \theta \\ b=r\sin \theta \\ \theta =\arctan \left(\frac{b}{a}\right)\end{array}$$ The quantity $r$ is known as the magnitude of the complex number $z$ , and is frequently written as $\left|z\right|$ . The quantity $\theta$ is the complex number's angle . In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.

Euler's formula

Surprisingly, the polar form of a complex number $z$ can be expressed mathematically as

Calculating with complex numbers

Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts andimaginary parts separately.

Division requires mathematical manipulation. We convert the division problem into a multiplication problem by multiplyingboth the numerator and denominator by the conjugate of the denominator.

The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form.