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
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
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