a+bi Form Calculator
Convert complex numbers between rectangular (a+bi) and polar (r×e^iφ) forms
Complex Number Converter
Polar Form: r × e^(iφ)
Distance from origin (≥ 0)
Angle from positive real axis
Rectangular Form (a+bi)
Step-by-step Solution
- 1.Given: r = 0, φ = 0°
- 2.Formula: a = r × cos(φ), b = r × sin(φ)
- 3.Convert φ to radians: φ = 0.000000 rad
- 4.Calculate cos(φ): cos(0.000000) = 1.000000
- 5.Calculate sin(φ): sin(0.000000) = 0.000000
- 6.Real part: a = 0 × 1.000000 = 0.000000
- 7.Imaginary part: b = 0 × 0.000000 = 0.000000
- 8.Result: z = 0
Example Conversions
Unit Complex Number
Polar: 1 × e^(i·0°) = 1
Rectangular: 1 + 0i = 1
Explanation: The number 1 on the real axis
Imaginary Unit
Polar: 1 × e^(i·90°) = e^(i·π/2)
Rectangular: 0 + 1i = i
Explanation: The imaginary unit on the positive imaginary axis
Euler's Identity Example
Polar: 1 × e^(i·π) = e^(iπ)
Rectangular: -1 + 0i = -1
Explanation: Famous case: e^(iπ) + 1 = 0
45° Angle Example
Polar: √2 × e^(i·45°) = √2 × e^(i·π/4)
Rectangular: 1 + 1i
Calculation: a = √2·cos(45°) = 1, b = √2·sin(45°) = 1
Complex Number Forms
Rectangular Form
z = a + bi
Point (a,b) on complex plane
Polar Form
z = r × e^(iφ)
Distance r and angle φ from origin
Trigonometric Form
z = r(cos φ + i sin φ)
Equivalent to polar form
Conversion Formulas
Polar → Rectangular
a = r × cos(φ)
b = r × sin(φ)
Rectangular → Polar
r = √(a² + b²)
φ = arctan(b/a)
Understanding Complex Numbers
What are Complex Numbers?
Complex numbers extend the real number system by including the imaginary unit i, where i² = -1. They can represent points on a two-dimensional plane, with real and imaginary components.
Rectangular Form (a + bi)
The rectangular form expresses a complex number as a + bi, where 'a' is the real part and 'b' is the imaginary part. This form directly shows the coordinates on the complex plane.
Polar Form (r × e^iφ)
The polar form expresses the same number using magnitude (r) and phase angle (φ). This form is particularly useful for multiplication and division of complex numbers.
Key Relationships
Euler's Formula: e^(iφ) = cos(φ) + i·sin(φ)
Magnitude: |z| = √(a² + b²)
Argument: arg(z) = arctan(b/a)
Conjugate: z* = a - bi
Applications
- •Electrical engineering (AC circuits)
- •Signal processing and Fourier transforms
- •Quantum mechanics and physics
- •Computer graphics and rotations