Converting Normal to Polar Form

Transform complex numbers from rectangular (normal) to polar coordinates

Introduction to Normal-to-Polar Conversion

In many practical applications, complex numbers are given in normal (rectangular) form \(z = a + bi\), but we need them in polar form to better understand their magnitude and direction.

Converting from normal to polar form requires calculating two key parameters:

  • The magnitude (distance from origin)
  • The argument (angle from positive real axis)
Key Insight:

The conversion process combines the Pythagorean theorem (for magnitude) with inverse trigonometric functions (for argument), but requires careful attention to quadrants.

Conversion Formulas

Step 1: Calculate the Magnitude

Magnitude Formula:

The magnitude is the distance from the origin to the point \((a, b)\):

Magnitude:
\(\displaystyle r = |z| = \sqrt{a^2 + b^2}\)

This formula comes directly from the Pythagorean theorem.

Step 2: Calculate the Argument

Argument Formula:

The argument is the angle from the positive real axis. Several equivalent formulas exist:

Using Cosine:
\(\displaystyle \varphi = \arccos\left(\frac{a}{r}\right)\)
Using Tangent:
\(\displaystyle \varphi = \arctan\left(\frac{b}{a}\right)\)
Important: Quadrant Consideration

The simple formulas above don't always give the correct angle. You must consider which quadrant the complex number is in and adjust the angle accordingly.

Quadrant-Based Angle Adjustment

The complex plane is divided into four quadrants. The argument depends on the signs of the real and imaginary parts.

Complex Number Quadrants:

Quadrant diagram for complex numbers

Different quadrants require different angle calculations

Quadrant-Specific Formulas

Quadrant I (+ real, + imag)
\(\displaystyle \varphi = \arctan\left(\frac{b}{a}\right)\)

\(0° < \varphi < 90°\)
Quadrant II (- real, + imag)
\(\displaystyle \varphi = 180° - \arctan\left(\frac{b}{|a|}\right)\)

\(90° < \varphi < 180°\)
Quadrant III (- real, - imag)
\(\displaystyle \varphi = 180° + \arctan\left(\frac{|b|}{|a|}\right)\)

\(180° < \varphi < 270°\)
Quadrant IV (+ real, - imag)
\(\displaystyle \varphi = 360° - \arctan\left(\frac{|b|}{a}\right)\)

\(270° < \varphi < 360°\)
Alternative Notation:

The argument can also be expressed as a value between \(-180°\) and \(+180°\) instead of \(0°\) to \(360°\). In this case, Quadrants III and IV use negative angles.

Step-by-Step Conversion Process

1Identify the real and imaginary parts
For \(z = a + bi\), determine \(a\) and \(b\)
2Calculate the magnitude
\(\displaystyle r = \sqrt{a^2 + b^2}\)
3Determine the quadrant
Check the signs of \(a\) and \(b\) to identify which quadrant
4Calculate preliminary angle
Use the appropriate arctangent or arccosine
5Adjust angle based on quadrant
Add or subtract 180° or 360° as needed
6Write polar form
Express as \(z = r(\cos\varphi + i\sin\varphi)\)

Conversion Examples

Example 1: Quadrant I - \(z = 3 + 4i\)

Convert \(3 + 4i\) to polar form

Step 1: \(a = 3\), \(b = 4\)

Step 2: Calculate magnitude

\(\displaystyle r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Step 3: Quadrant I (both positive)

Step 4-5: Calculate argument

\(\displaystyle \varphi = \arctan\left(\frac{4}{3}\right) \approx 53.13°\)

Result: \(z = 5(\cos 53.13° + i\sin 53.13°)\)

Example 2: Quadrant II - \(z = -3 + 4i\)

Convert \(-3 + 4i\) to polar form

Step 1: \(a = -3\), \(b = 4\)

Step 2: Calculate magnitude

\(\displaystyle r = \sqrt{(-3)^2 + 4^2} = 5\)

Step 3: Quadrant II (negative real, positive imag)

Step 4-5: Calculate argument

\(\displaystyle \varphi = 180° - \arctan\left(\frac{4}{3}\right) = 180° - 53.13° = 126.87°\)

Result: \(z = 5(\cos 126.87° + i\sin 126.87°)\)

Example 3: Quadrant III - \(z = -3 - 4i\)

Convert \(-3 - 4i\) to polar form

Step 2: Magnitude remains \(r = 5\)

Step 3: Quadrant III (both negative)

Step 4-5: Calculate argument

\(\displaystyle \varphi = 180° + \arctan\left(\frac{4}{3}\right) = 180° + 53.13° = 233.13°\)

Result: \(z = 5(\cos 233.13° + i\sin 233.13°)\)

Example 4: Quadrant IV - \(z = 3 - 4i\)

Convert \(3 - 4i\) to polar form

Step 2: Magnitude remains \(r = 5\)

Step 3: Quadrant IV (positive real, negative imag)

Step 4-5: Calculate argument

\(\displaystyle \varphi = 360° - \arctan\left(\frac{4}{3}\right) = 360° - 53.13° = 306.87°\)

Alternate notation (±180°): \(\varphi = -53.13°\)

Result: \(z = 5(\cos 306.87° + i\sin 306.87°)\)

Conversion Summary Table

Normal Form Quadrant Magnitude Argument (0-360°) Argument (±180°)
\(3 + 4i\) I \(5\) \(53.13°\) \(53.13°\)
\(-3 + 4i\) II \(5\) \(126.87°\) \(126.87°\)
\(-3 - 4i\) III \(5\) \(233.13°\) \(-126.87°\)
\(3 - 4i\) IV \(5\) \(306.87°\) \(-53.13°\)

Key Points to Remember

  • Magnitude formula: \(r = \sqrt{a^2 + b^2}\)
  • Always identify which quadrant the complex number is in
  • Quadrant I: Use \(\arctan\left(\frac{b}{a}\right)\) directly
  • Quadrant II: Add 180° to the calculated angle
  • Quadrant III: Add 180° to the calculated angle (or subtract from 360°)
  • Quadrant IV: Subtract from 360° (or use negative angle)
  • Arguments can be expressed as 0-360° or ±180°
  • Always verify using: \(a = r\cos\varphi\) and \(b = r\sin\varphi\)


























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