Mathematics

Complex Numbers

Study of imaginary and complex numbers: rectangular, polar, and trigonometric forms, arithmetic operations, De Moivre's theorem, and solving quadratic equations with complex roots.

Definition: The imaginary unit ii is defined as:
i=−1i = \sqrt{-1}
Powers of the imaginary unit ii

i1=−1=ii^1 = \sqrt{-1} = i
i2=−1⋅−1=−12=−1i^2 = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1}^2 = -1
i3=i2⋅i=−ii^3 = i^2 \cdot i = -i
i4=i2⋅i2=−1⋅(−1)=1i^4 = i^2 \cdot i^2 = -1 \cdot (-1) = 1
i5=i4⋅i=ii^5 = i^4 \cdot i = i
...
Note that starting from the 5th power, the values repeat. The powers of ii are cyclic with a period of 4. Powers with an exponent that is a multiple of 4 are equal to 1.
With this in mind, we can calculate any power of ii. Ex.:
i19=i16⋅i3=1⋅i3=−ii^{19} = i^{16} \cdot i^3 = 1 \cdot i^3 = -i
Definition: A complex number is composed of a real part and an imaginary part. They are denoted by the variable zz.
Complex number in standard (rectangular) form: z=a+biz = a + bi
Where aa is the real part, and bb is the imaginary part.
Complex number in Cartesian (vector) form: Denoted as an ordered pair (a,b)(a, b) on the complex plane, with the real axis R\mathbb{R} as the x-axis, and the imaginary axis ii as the y-axis.
Cartesian representation of a complex number
z=(a,b)z = (a,b)
Complex number in polar form: Written using the modulus mm (or rr) and the argument, or angle α\alpha, that the vector (a,b)(a,b) forms with the positive real axis; z=m∣αz = m_{|\alpha} (also written as r∠θr \angle \theta).
Polar representation of a complex number
Where:
m=a2+b2α=arctan⁡ba\begin{aligned} m &= \sqrt{a^2 + b^2} \\ \alpha &= \arctan\frac{b}{a} \end{aligned}
Ex: z=3+4i=(3,4)z = 3 + 4i = (3,4)
m=32+42=5α=arctan⁡43=53∘⇒z=5∣53∘m = \sqrt{3^2 + 4^2} = 5 \qquad \alpha = \arctan\frac{4}{3} = 53^\circ \quad \Rightarrow \quad z = 5_{|53^\circ}
Complex number in trigonometric form: This is the expanded form used to convert a complex number from polar to standard form:
z=m(cos⁡α+isin⁡α)z = m(\cos \alpha + i \sin \alpha)
Ex: z=5∣53∘=5(cos⁡53∘+isin⁡53∘)\text{Ex: } z = 5_{|53^\circ} = 5(\cos 53^\circ + i \sin 53^\circ)
Complex Conjugate zˉ\bar{z} of z=a+biz = a + bi: change the sign of the imaginary part, e.g. z=2−3i→zˉ=2+3iz = 2 - 3i \rightarrow \bar{z} = 2 + 3i
Opposite −z-z of z=a+biz = a + bi: change the sign of both the real and imaginary parts, e.g. z=2−3i→−z=−2+3iz = 2 - 3i \rightarrow -z = -2 + 3i
Addition and subtraction:
(2+3i)+(5−2i)=(2+5)+i(3−2)=(7+i)(2 + 3i) + (5 - 2i) = (2 + 5) + i(3 - 2) = (7 + i)
Multiplication: Same as multiplying binomials. Apply i2=−1i^2 = -1
(2+3i)⋅(5−2i)=10−4i+15i−6i2=16+11i(2 + 3i) \cdot (5 - 2i) = 10 - 4i + 15i - 6i^2 = 16 + 11i
Division: Multiply the numerator and denominator by the complex conjugate of the denominator, and simplify
2+3i5+2i=(2+3i)⋅(5−2i)(5+2i)⋅(5−2i)=16+11i52−(2i)2=16+11i29\frac{2 + 3i}{5 + 2i} = \frac{(2 + 3i) \cdot (5 - 2i)}{(5 + 2i) \cdot (5 - 2i)} = \frac{16 + 11i}{5^2 - (2i)^2} = \frac{16 + 11i}{29}
Inverse: z−1z^{-1} is calculated by doing the division 1/z1/z
(1−2i)−1=11−2i=1+2i(1−2i)(1+2i)=1+2i5(1 - 2i)^{-1} = \frac{1}{1 - 2i} = \frac{1 + 2i}{(1 - 2i)(1 + 2i)} = \frac{1 + 2i}{5}
Multiplication:
m1∣α1⋅m2∣α2=(m1⋅m2)∣α1+α2\displaystyle m_{1|\alpha_1} \cdot m_{2|\alpha_2} = (m_1 \cdot m_2)_{|\alpha_1+\alpha_2}

5∣53∘⋅3∣17∘=15∣70∘\displaystyle 5_{|53^\circ} \cdot 3_{|17^\circ} = 15_{|70^\circ}

Division:
m1∣α1m2∣α2=(m1m2)∣α1−α2\displaystyle \frac{m_{1|\alpha_1}}{m_{2|\alpha_2}} = \left(\frac{m_1}{m_2}\right)_{|\alpha_1-\alpha_2}

15∣53∘3∣17∘=5∣36∘\displaystyle \frac{15_{|53^\circ}}{3_{|17^\circ}} = 5_{|36^\circ}

Power:
(m∣α)n=m∣n⋅αn\displaystyle (m_{|\alpha})^n = m^n_{|n \cdot \alpha}

(5∣20∘)3=125∣60∘\displaystyle (5_{|20^\circ})^3 = 125_{|60^\circ}

De Moivre's Theorem:
(cos⁡α+isin⁡α)n=cos⁡(nα)+isin⁡(nα)(\cos \alpha + i \sin \alpha)^n = \cos(n\alpha) + i \sin(n\alpha)
Roots: An nn-th root of a complex number has nn solutions, given by:
m∣αn=mn∣α+360∘knfork=0,1,2...(n−1)\sqrt[n]{m_{|\alpha}} = \sqrt[n]{m}_{|\frac{\alpha+360^\circ k}{n}} \quad \text{for} \quad k = 0, 1, 2... (n-1)
27∣60∘3={3∣60∘+0∘3,3∣60∘+360∘3,3∣60∘+720∘3}={3∣20∘,3∣140∘,3∣260∘}\sqrt[3]{27_{|60^\circ}} = \left\{ 3_{|\frac{60^\circ+0^\circ}{3}}, 3_{|\frac{60^\circ+360^\circ}{3}}, 3_{|\frac{60^\circ+720^\circ}{3}} \right\} = \{ 3_{|20^\circ}, 3_{|140^\circ}, 3_{|260^\circ} \}
When the discriminant, b2−4acb^2 - 4ac, of a quadratic equation is negative, it has no real solutions; instead, it has two complex conjugate solutions:
x2−2x+5=0x=2±4−4⋅1⋅52=2±−162=2±4−12=1±2i\begin{aligned} x^2 - 2x + 5 &= 0 \\ x &= \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 5}}{2} = \frac{2 \pm \sqrt{-16}}{2} = \frac{2 \pm 4\sqrt{-1}}{2} = 1 \pm 2i \end{aligned}