Complex Numbers and Quadratic Equations

Introduction

Need for Complex Numbers

In real numbers, some equations like $x^2+1=0$ have no real solution. To solve such equations, we extend the number system to complex numbers.

Imaginary Unit $i$

Define $i$ such that: $$i^2=-1$$ So:

• $i=\sqrt{-1}$ (symbolically)
• $i^3=i^2\cdot i=-i$
• $i^4=(i^2)^2=1$
• Powers repeat every 4: $$i^{4k}=1,\ i^{4k+1}=i,\ i^{4k+2}=-1,\ i^{4k+3}=-i$$

Definition

A number of the form $a + ib$, where $a$ and $b$ are real numbers, is defined to be a complex number.

• $a$ is the real part, denoted by Re($z$).
• $b$ is the imaginary part, denoted by Im($z$).
• $i$ (iota) = $\sqrt{-1}$.

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Algebra of Complex Numbers

Equality of Complex Numbers

If $z_1=a+ib$ and $z_2=c+id$, then: $$z_1=z_2 \iff a=c \text{ and } b=d$$

Basic Operations

Let $z_1=a+ib$ and $z_2=c+id$.

Addition: $z_1 + z_2$

$$(a+ib)+(c+id)=(a+c)+i(b+d)$$

Subtraction: $z_1 - z_2$

$$(a+ib)-(c+id)=(a-c)+i(b-d)$$

Multiplication:$z_1 z_2$ $$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$$

Division:$\frac{z_1}{z_2}$ $$\frac{a+ib}{c+id} = \frac{(a+ib)(c-id)}{(c+id)(c-id)} = \frac{(ac+bd)+i(bc-ad)}{c^2+d^2}$$ Key idea: multiply numerator and denominator by the conjugate.

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Conjugate of a Complex Number

For $z=a+ib$, its conjugate is: $$\overline{z}=a-ib$$

Important properties

1. $z\overline{z}=a^2+b^2$ (always real and non-negative)
2. $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$
3. $\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}$
4. $\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z_1}}{\overline{z_2}}$ (if $z_2\ne0$)

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Modulus (Magnitude)

For $z=a+ib$: $$|z|=\sqrt{a^2+b^2}$$

Properties

• $|z|\ge 0$
• $|z|=0 \iff z=0$
• $|z_1z_2|=|z_1||z_2|$
• $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ (if $z_2\ne 0$)

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Argand Plane and Polar Representation

Complex numbers can be represented geometrically in the Argand plane (Complex plane).

• $x$-axis: Real axis
• $y$-axis: Imaginary axis

Polar Form of a Complex Number

If $z=a+ib$ and $r=|z|$, then define angle $\theta$ such that: $$\cos\theta = \frac{a}{r},\quad \sin\theta = \frac{b}{r}$$ Then: $$z=r(\cos\theta+i\sin\theta)$$ where $r>0$.

• $r$ is modulus.
• $\theta$ is argument (principal value generally taken).

Finding Argument $\theta$: Let $\alpha = \tan^{-1}|\frac{y}{x}|$.

• Quadrant I: $\theta = \alpha$
• Quadrant II: $\theta = \pi - \alpha$
• Quadrant III: $\theta = -(\pi - \alpha)$
• Quadrant IV: $\theta = -\alpha$

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5. Quadratic Equations

For the quadratic equation $ax^2 + bx + c = 0$ with real coefficients: If Discriminant $D = b^2 - 4ac < 0$, the roots are complex conjugates.

Formula: $$x = \frac{-b \pm i\sqrt{|D|}}{2a}$$

Complex Numbers

Example 1

Express $(5i)(-\frac{3}{5}i)$ in the form $a+ib$.

Solution: $5 \times (-\frac{3}{5}) \times i^2 = -3(-1) = 3$. Form: $3 + 0i$.

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Example 2

Express $i^9 + i^{19}$ in standard form.

Solution: $i^9 = (i^4)^2 \cdot i = 1 \cdot i = i$. $i^{19} = (i^4)^4 \cdot i^3 = 1 \cdot (-i) = -i$. Sum $= i - i = 0$.

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Example 3

Find the multiplicative inverse of $2 - 3i$.

Solution: $z^{-1} = \frac{\bar{z}}{|z|^2} = \frac{2+3i}{2^2+(-3)^2} = \frac{2+3i}{13}$. Ans: $\frac{2}{13} + \frac{3}{13}i$.

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Example 4

Express in form $a+ib$: $\frac{5 + \sqrt{2}i}{1 - \sqrt{2}i}$.

Solution: Rationalize: $\frac{(5+\sqrt{2}i)(1+\sqrt{2}i)}{(1)^2+(\sqrt{2})^2} = \frac{5 + 5\sqrt{2}i + \sqrt{2}i - 2}{3} = \frac{3 + 6\sqrt{2}i}{3} = 1 + 2\sqrt{2}i$.

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Example 5

Find the modulus of $z = \frac{1+i}{1-i}$.

Solution: $|z| = \frac{|1+i|}{|1-i|} = \frac{\sqrt{2}}{\sqrt{2}} = 1$.

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Example 6

Find the conjugate of $z = (3-2i)(2+3i)$.

Solution: $z = 6 + 9i - 4i - 6i^2 = 6 + 5i + 6 = 12 + 5i$. Conjugate $\bar{z} = 12 - 5i$.

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Example 7

Find the argument of $z = 1 + i\sqrt{3}$.

Solution: $x=1, y=\sqrt{3}$ (Quadrant I). $\tan \alpha = |\sqrt{3}/1| = \sqrt{3} \implies \alpha = \frac{\pi}{3}$. $\theta = \frac{\pi}{3}$.

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Example 8

Convert $z = -1 - i$ to polar form.

Solution: $r = \sqrt{(-1)^2+(-1)^2} = \sqrt{2}$. $\tan \alpha = 1 \implies \alpha = \frac{\pi}{4}$. Quadrant III: $\theta = -(\pi - \frac{\pi}{4}) = -\frac{3\pi}{4}$. $z = \sqrt{2}(\cos \frac{-3\pi}{4} + i\sin \frac{-3\pi}{4})$.

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Example 9

Solve $x^2 + 3 = 0$.

Solution: $x^2 = -3 \implies x = \pm \sqrt{-3} = \pm i\sqrt{3}$.

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Example 10

Solve $x^2 + x + 1 = 0$.

Solution: $D = 1^2 - 4(1)(1) = -3$. $x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}$.

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Example 11

Find the modulus of $(1+i)^{10}$.

Solution: $|(1+i)^{10}| = |1+i|^{10} = (\sqrt{2})^{10} = 2^5 = 32$.

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Example 12

If $x + iy = \frac{a+ib}{a-ib}$, prove $x^2 + y^2 = 1$.

Solution: Taking modulus on both sides: $|x+iy| = |\frac{a+ib}{a-ib}| = \frac{\sqrt{a^2+b^2}}{\sqrt{a^2+b^2}} = 1$. Squaring: $x^2+y^2=1$.

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Example 13

Find real $\theta$ such that $\frac{3+2i\sin\theta}{1-2i\sin\theta}$ is purely real.

Solution: Multiply by conjugate $1+2i\sin\theta$. Imaginary part of numerator must be 0. Im part: $6\sin\theta + 2\sin\theta = 8\sin\theta = 0$. $\sin\theta = 0 \implies \theta = n\pi$.

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Example 14

Evaluate $\sqrt{-25} \times \sqrt{-49}$.

Solution: $5i \times 7i = 35i^2 = -35$. (Note: $\sqrt{a}\sqrt{b} \neq \sqrt{ab}$ if both $a, b < 0$).

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Example 15

Find the least positive integer $n$ for which $(\frac{1+i}{1-i})^n = 1$.

Solution: $\frac{1+i}{1-i} = i$. $i^n = 1 \implies n=4$.

Questions and Answers (Board Exam)

Q1. Define modulus of a complex number.

Answer: The modulus of $z = x+iy$ is the non-negative real number $\sqrt{x^2+y^2}$, denoted by $|z|$. It represents the distance from the origin.

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Q2. What is the conjugate of $z = x+iy$?

Answer: The conjugate is $\bar{z} = x-iy$. It represents the reflection of $z$ across the real axis.

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Q3. State the value of $i^{4k}$.

Answer: $i^{4k} = 1$ for any integer $k$.

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Q4. What is the fundamental theorem of algebra?

Answer: A polynomial equation of degree $n$ has exactly $n$ roots in the complex number system.

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Q5. Write the polar form of a complex number.

Answer: $z = r(\cos \theta + i \sin \theta)$, where $r=|z|$ and $\theta = \arg(z)$.