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Complex Numbers

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Wize University Linear Algebra Textbook > Complex Numbers

Basics of Complex Numbers

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A complex number z∈Cz \in \mathbb{C} is a number of the form z=a+biz = a+bi where a and b are real numbers and i=p−1i = p−1. The number a is called the real part of zz and the number b is called the imaginary part of z.z.

Operations on Complex Numbers Consider two complex numbers z1=a1+ib1z1 = a_1 + ib_1 and z2=a2+ib2z_2 = a_2 + ib_2. Then

1. [Addition and Substraction] =z1±z2=(a1±a2)+i(b1±b2)=z_1 ± z_2 = (a_1 ± a_2) + i(b_1 ± b_2)

2. [Multiplication] z1z2=(a1+ib1)(a2+ib2)=(a1a2−b1b2)+i(a1b2+a2b1)z_1z_2 = (a_1 + ib_1)(a_2 + ib_2) = (a_1a_2 − b_1b_2) + i(a_1b_2 + a_2b_1)

3. [Division] =z1z2=a1+ib1a2+ib2=(a1+ib1)(a1−ib2)(a2+ib2)(a2−ib2)=(a1a2+bab2)+i(a2b1−a1b2)a22+b22=\frac{z_1}{z_2}=\frac{a_1+ib_1}{a_2+ib_2}=\frac{(a_1+ib_1)(a_1-ib_2)}{(a_2+ib_2)(a_2-ib_2)}=\frac{(a_1a_2+b_ab_2)+i(a_2b_1-a_1b_2)}{a_2^2+b_2^2}

4. [Complex Conjugate] z1ˉ=a1−ib1\bar{z_1}=a_1-ib_1

5. [Absolute Value] ∣z∣=∣a1+ib1∣=a12+b12|z|=|a_1+ib_1|=\sqrt{a_1^2+{b_1^2}}

Polar Form Consider a complex number z=x+iyz = x+iy. Let x=r cosθx = r \ cos \theta and y=r sinθy = r \ sin \theta. Then the radius r is equal to the absolute value of z:r=∣z∣=x2+y2z: r = |z| =\sqrt{ x^2 + y^2}. Also, the angle θ\theta is called the argument of zz and is denoted by θ=arg(z)\theta = arg(z). Operations in Polar Form The formula

eiθ=cosθ+i sinθe^{i\theta}=cos\theta+i\ sin\theta
applied to the polar form of a complex number z=r cosθ+ir sinθz = r \ cos \theta + ir\ sin \theta gives the convenient form z=reiθz = re^{i\theta}. Consider two complex numbers in this form z1=r1eiθ1z_1 = r_1e^{i\theta1} and z2=r2eiθ2 z_2 = r_2e^{i\theta2}. Then

1. [Multiplication - 2nd Form] z1z2=r1r2ei(θ1+θ2)z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}

2. [Division - 2nd Form] z1z2=r1r2ei(θ1−θ2)\frac{z_1}{z_2}=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)} provided r2≠0.r^2\neq0.

3. [Power - 2nd Form: De Moivre’s Theorem] zn=(reiθ)n=rn einθz^n=(re^{i\theta})^n=r^n\ e^{in\theta}

4. [Root - 2nd Form] zn=rnexp=[i(θn+k2πn]{\sqrt[n]{z}}={\sqrt[n]{r}}exp=[i(\frac{\theta}{n}+k\frac{2\pi}{n}] where k=0,1,2,...,n−1.k = 0, 1, 2, . . . ,n − 1.

More Basics of Complex Numbers Questions:
Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers
Practice: Operations with Complex Numbers
Perform the following operations.
Given the following complex number
z=3+4i1−iz = \frac{3 + 4i}{1 - i}
a) Write zz in Cartesian coordinates z=a+biz = a + bi .
Basics of Complex Numbers
Practice: Operations with Complex Numbers
Perform the following operations.
If z=3+2iz = 3+2i and w=1−4iw = 1- 4i, compute z−1z^{-1}and zwzw.
If z=3+2iz = 3+2i and w=1−4iw = 1- 4i, compute z−1z^{-1}and z⋅wz\cdot w
Simplify the expression 21+i+11+3i3−i\frac{2}{1+i}+\frac{11+3i}{3-i}
Practice: Operations With Complex Numbers
Perform the following operations:
(a) (−1)(−i)+(−i)(-1)(-i) + (-i)
(b) i(a+bi)‾\overline{i(a+bi)}