Complex Numbers and Quadratic Equations Ex 5.2

Recap: What are Complex Numbers?

A complex number is a number that can be expressed in the form $z = x + iy$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit, satisfying $i^2 = -1$. Here, $x$ is called the real part of $z$, denoted as Re($z$), and $y$ is called the imaginary part of $z$, denoted as Im($z$).

Geometrically, a complex number $z = x + iy$ can be represented as a point $(x, y)$ in the complex plane (also known as the Argand plane). The x-axis is called the real axis, and the y-axis is called the imaginary axis. This representation is crucial for understanding the concepts of modulus and argument, which are the building blocks of the polar form. Think of it like coordinates for numbers that extend beyond the simple number line, allowing us to represent roots of equations like $x^2 + 1 = 0$ which have no real solutions.

Modulus and Argument of a Complex Number

Modulus (r or |z|)

The modulus of a complex number $z = x + iy$, denoted as $|z|$ or $r$, is the distance of the point $(x, y)$ from the origin $(0, 0)$ in the complex plane. It is always a non-negative real number. Mathematically, $|z| = \sqrt{x^2 + y^2}$.

Argument (θ or arg(z))

The argument of a non-zero complex number $z = x + iy$, denoted as arg($z$) or $\theta$, is the angle made by the line segment connecting the origin to the point $(x, y)$ with the positive x-axis, measured in the counter-clockwise direction. The principal argument, which is what we usually refer to, lies in the interval $(-\pi, \pi]$ (or sometimes $[0, 2\pi)$). It is found using $\tan\alpha = |y/x|$, where $\alpha$ is the reference angle, and then adjusting based on the quadrant of $(x, y)$.

The Polar Form of a Complex Number: $z = r(\cos\theta + i\sin\theta)$

The polar form provides an alternative way to represent a complex number using its distance from the origin (modulus) and its angle with the positive real axis (argument). If we have a complex number $z = x + iy$, we can relate $x$ and $y$ to $r$ and $\theta$ using basic trigonometry in a right-angled triangle formed by the origin, the point $(x, y)$, and its projection on the x-axis.

From the Argand plane, we can observe that:

• $x = r\cos\theta$
• $y = r\sin\theta$

Substituting these into the Cartesian form $z = x + iy$, we get:
$z = r\cos\theta + i(r\sin\theta)$
$z = r(\cos\theta + i\sin\theta)$

This is the polar form of the complex number $z$. Here, $r$ is the modulus $|z|$ and $\theta$ is the argument arg($z$). This form is particularly useful for visualizing complex numbers and simplifying operations like multiplication, division, and finding powers or roots of complex numbers, as it turns algebraic operations into geometric transformations. Understanding how to accurately find $\theta$ based on the quadrant of $z$ is paramount.

Steps to Convert a Complex Number $z = x + iy$ to Polar Form

1. Step 1: Calculate the Modulus (r) — Find the modulus $r$ using the formula: $r = |z| = \sqrt{x^2 + y^2}$. This represents the distance of the complex number from the origin.
2. Step 2: Determine the Reference Angle (α) — Calculate the reference angle $\alpha$ using $\tan\alpha = |y/x|$. Always use the absolute values of $x$ and $y$ to find $\alpha$, which will be an acute angle ($0 < \alpha < \pi/2$). This reference angle helps locate the principal argument.
3. Step 3: Identify the Quadrant of $z$ — Based on the signs of $x$ (real part) and $y$ (imaginary part), determine which quadrant the complex number $z = x + iy$ lies in: Quadrant I: $x > 0, y > 0$ Quadrant II: $x < 0, y > 0$ Quadrant III: $x < 0, y < 0$ Quadrant IV: $x > 0, y < 0$
4. Step 4: Find the Principal Argument (θ) — Using the reference angle $\alpha$ and the quadrant, determine the principal argument $\theta$ (usually in $(-\pi, \pi]$): Quadrant I: $\theta = \alpha$ Quadrant II: $\theta = \pi - \alpha$ Quadrant III: $\theta = -(\pi - \alpha)$ or $\theta = \alpha - \pi$ (whichever falls in the range $(-\pi, \pi]$) Quadrant IV: $\theta = -\alpha$ If $x=0$ or $y=0$, special cases apply: $z = iy$ ($x=0$): If $y > 0$, $\theta = \pi/2$. If $y < 0$, $\theta = -\pi/2$. $z = x$ ($y=0$): If $x > 0$, $\theta = 0$. If $x < 0$, $\theta = \pi$.
5. Step 5: Write the Polar Form — Substitute the calculated values of $r$ and $\theta$ into the polar form: $z = r(\cos\theta + i\sin\theta)$.

Worked Examples: Converting to Polar Form

• Example 1: Convert $z = 1 + i\sqrt{3}$ to polar form. Step 1: Identify $x$ and $y$. Here, $x = 1$ and $y = \sqrt{3}$. Step 2: Calculate the modulus $r$. $r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$. Step 3: Determine the reference angle $\alpha$. $\tan\alpha = |y/x| = |\sqrt{3}/1| = \sqrt{3}$. Since $\tan(\pi/3) = \sqrt{3}$, $\alpha = \pi/3$. Step 4: Identify the quadrant and find $\theta$. Since $x > 0$ and $y > 0$, $z$ lies in Quadrant I. Therefore, $\theta = \alpha = \pi/3$. Step 5: Write the polar form. $z = r(\cos\theta + i\sin\theta) = 2(\cos(\pi/3) + i\sin(\pi/3))$.
• Example 2: Convert $z = -1 - i$ to polar form. Step 1: Identify $x$ and $y$. Here, $x = -1$ and $y = -1$. Step 2: Calculate the modulus $r$. $r = \sqrt{x^2 + y^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$. Step 3: Determine the reference angle $\alpha$. $\tan\alpha = |y/x| = |-1/-1| = 1$. Since $\tan(\pi/4) = 1$, $\alpha = \pi/4$. Step 4: Identify the quadrant and find $\theta$. Since $x < 0$ and $y < 0$, $z$ lies in Quadrant III. Therefore, $\theta = -(\pi - \alpha) = -(\pi - \pi/4) = -(3\pi/4)$. Step 5: Write the polar form. $z = r(\cos\theta + i\sin\theta) = \sqrt{2}(\cos(-3\pi/4) + i\sin(-3\pi/4))$.
• Example 3: Convert $z = -\sqrt{3} + i$ to polar form. Step 1: Identify $x$ and $y$. Here, $x = -\sqrt{3}$ and $y = 1$. Step 2: Calculate the modulus $r$. $r = \sqrt{x^2 + y^2} = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2$. Step 3: Determine the reference angle $\alpha$. $\tan\alpha = |y/x| = |1/-\sqrt{3}| = 1/\sqrt{3}$. Since $\tan(\pi/6) = 1/\sqrt{3}$, $\alpha = \pi/6$. Step 4: Identify the quadrant and find $\theta$. Since $x < 0$ and $y > 0$, $z$ lies in Quadrant II. Therefore, $\theta = \pi - \alpha = \pi - \pi/6 = 5\pi/6$. Step 5: Write the polar form. $z = r(\cos\theta + i\sin\theta) = 2(\cos(5\pi/6) + i\sin(5\pi/6))$.
• Example 4: Convert $z = -4i$ to polar form. Step 1: Identify $x$ and $y$. Here, $x = 0$ and $y = -4$. Step 2: Calculate the modulus $r$. $r = \sqrt{0^2 + (-4)^2} = \sqrt{0 + 16} = \sqrt{16} = 4$. Step 3: This is a special case. Since $x=0$, the number lies on the imaginary axis. Step 4: Find $\theta$. Since $y < 0$, the point is on the negative imaginary axis. The angle is $- \pi/2$. Therefore, $\theta = -\pi/2$. Step 5: Write the polar form. $z = r(\cos\theta + i\sin\theta) = 4(\cos(-\pi/2) + i\sin(-\pi/2))$. Note that $\cos(-\pi/2) = 0$ and $\sin(-\pi/2) = -1$, so $z = 4(0 + i(-1)) = -4i$, confirming the conversion.

Exam Tips & Common Mistakes in Exercise 5.2

When solving problems from Exercise 5.2, pay close attention to these critical points:

1. Quadrant Identification: This is the most common pitfall! Always correctly identify the quadrant of the complex number $z = x + iy$ based on the signs of $x$ and $y$. A wrong quadrant means a wrong argument, even if your modulus is correct.
2. Principal Argument Range: Remember that the principal argument $\theta$ is usually defined in the interval $(-\pi, \pi]$. Make sure your calculated $\theta$ falls within this range. If you get a value outside, adjust it by adding or subtracting $2\pi$.
3. Reference Angle vs. Argument: Differentiate between the reference angle $\alpha = \tan^{-1}(|y/x|)$ (always acute) and the actual argument $\theta$ (which depends on the quadrant). Do not use $\tan^{-1}(y/x)$ directly unless you are in the first quadrant, as it won't give the correct angle for other quadrants.
4. Special Cases ($x=0$ or $y=0$): If the complex number lies on an axis (e.g., $z=3i$, $z=-5$, etc.), the argument can be directly determined as $0, \pi/2, \pi,$ or $-\pi/2$ without using the $\tan\alpha$ formula. For instance, for $z = 3i$, $x=0, y=3$, so $\theta = \pi/2$. For $z=-5$, $x=-5, y=0$, so $\theta = \pi$.
5. Exact Values: Use exact values for trigonometric ratios (like $\pi/6, \pi/4, \pi/3$) whenever possible, rather than decimal approximations, unless explicitly asked.

Practice Questions with Solutions

• Q: Convert $z = 1 - i$ to polar form. A: Step 1: Identify $x=1, y=-1$. Step 2: Modulus $r = \sqrt{1^2 + (-1)^2} = \sqrt{1+1} = \sqrt{2}$. Step 3: Reference angle $\alpha = \tan^{-1}(|-1/1|) = \tan^{-1}(1) = \pi/4$. Step 4: Quadrant IV ($x>0, y<0$). So, $\theta = -\alpha = -\pi/4$. Step 5: Polar form is $\sqrt{2}(\cos(-\pi/4) + i\sin(-\pi/4))$. Final answer: $\sqrt{2}(\cos(-\pi/4) + i\sin(-\pi/4))$
• Q: Convert $z = -1 + i\sqrt{3}$ to polar form. A: Step 1: Identify $x=-1, y=\sqrt{3}$. Step 2: Modulus $r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1+3} = \sqrt{4} = 2$. Step 3: Reference angle $\alpha = \tan^{-1}(|\sqrt{3}/-1|) = \tan^{-1}(\sqrt{3}) = \pi/3$. Step 4: Quadrant II ($x<0, y>0$). So, $\theta = \pi - \alpha = \pi - \pi/3 = 2\pi/3$. Step 5: Polar form is $2(\cos(2\pi/3) + i\sin(2\pi/3))$. Final answer: $2(\cos(2\pi/3) + i\sin(2\pi/3))$
• Q: Convert $z = -7$ to polar form. A: Step 1: Identify $x=-7, y=0$. Step 2: Modulus $r = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7$. Step 3: This is a special case ($y=0$, on the negative real axis). Step 4: For $x < 0, y=0$, the argument $\theta = \pi$. Step 5: Polar form is $7(\cos(\pi) + i\sin(\pi))$. Final answer: $7(\cos(\pi) + i\sin(\pi))$
• Q: Convert $z = -\sqrt{3} - i$ to polar form. A: Step 1: Identify $x=-\sqrt{3}, y=-1$. Step 2: Modulus $r = \sqrt{(-\sqrt{3})^2 + (-1)^2} = \sqrt{3+1} = \sqrt{4} = 2$. Step 3: Reference angle $\alpha = \tan^{-1}(|-1/-\sqrt{3}|) = \tan^{-1}(1/\sqrt{3}) = \pi/6$. Step 4: Quadrant III ($x<0, y<0$). So, $\theta = -(\pi - \alpha) = -(\pi - \pi/6) = -5\pi/6$. Step 5: Polar form is $2(\cos(-5\pi/6) + i\sin(-5\pi/6))$. Final answer: $2(\cos(-5\pi/6) + i\sin(-5\pi/6))$

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