Complex Number to Rectangular Form Calculator

Convert complex numbers from polar form (r×e^iφ) to rectangular form (a + bi)

Convert Polar to Rectangular Form

Distance from origin (always non-negative)

Angle from positive real axis

Rectangular Form Result

0
Rectangular Form (a + bi)
Real part (a):0.0000
Imaginary part (b):0.0000

Conversion formulas:

• Real part: a = r × cos(φ) = 0 × cos(0.0000) = 0.0000

• Imaginary part: b = r × sin(φ) = 0 × sin(0.0000) = 0.0000

Common Example Conversions

1∠0°
1 × e^(i0°) = 1
1∠90°
1 × e^(i90°) = 0 + i
1∠180°
1 × e^(i180°) = -1 + 0i
2∠45°
2 × e^(i45°) = 1.414 + 1.414i
3∠60°
3 × e^(i60°) = 1.5 + 2.598i
1∠30°
1 × e^(i30°) = 0.866 + 0.5i
5∠135°
5 × e^(i135°) = -3.536 + 3.536i
2∠-45°
2 × e^(i-45°) = 1.414 - 1.414i

Complex Number Forms

Rectangular Form

z = a + bi

Where a = real part, b = imaginary part

Polar (Exponential) Form

z = r × e^(iφ)

Where r = magnitude, φ = phase

Trigonometric Form

z = r(cosφ + isinφ)

Expanded polar form

Conversion Formulas

Polar → Rectangular

a = r × cos(φ)

b = r × sin(φ)

Rectangular → Polar

r = (a² + b²)

φ = atan2(b, a)

Quick Tips

Magnitude r is always non-negative

Phase φ can be in radians or degrees

cos(0) = 1, sin(0) = 0 for real numbers

cos(π/2) = 0, sin(π/2) = 1 for pure imaginary

Understanding Complex Number Conversion

What is Rectangular Form?

The rectangular form (also called Cartesian form) represents a complex number as z = a + bi, where 'a' is the real part and 'b' is the imaginary part. This corresponds to a point (a, b) on the complex plane.

Why Convert from Polar Form?

  • Addition and subtraction are easier in rectangular form
  • Direct identification of real and imaginary components
  • Standard form for many mathematical operations
  • Easy plotting on the complex plane

Conversion Process

From: z = r × e^(iφ)

To: z = a + bi

Step 1: Calculate real part using a = r × cos(φ)

Step 2: Calculate imaginary part using b = r × sin(φ)

Step 3: Combine as z = a + bi

Remember: Ensure your angle is in radians when using trigonometric functions!

Complex Plane Visualization

In the complex plane, the rectangular form directly gives you the coordinates: the real part 'a' is the x-coordinate (horizontal axis), and the imaginary part 'b' is the y-coordinate (vertical axis). This makes it easy to plot and visualize complex numbers.

Quadrant I

a > 0, b > 0

0 < φ < π/2

Quadrant II

a < 0, b > 0

π/2 < φ < π

Quadrant III

a < 0, b < 0

π < φ < 3π/2

Quadrant IV

a > 0, b < 0

3π/2 < φ < 2π