a+bi Form Calculator

Understanding Rectangular and Polar Forms

A complex number can be visualized as a point on a two-dimensional plane. In rectangular form, we express it as a + bi, where a is the horizontal (real) coordinate and b is the vertical (imaginary) coordinate.

In polar form, we describe the same point using distance and direction: r·e^(iφ), where r is the distance from the origin and φ (phi) is the angle measured from the positive real axis.

Both representations describe identical numbers—just from different perspectives. Rectangular form is intuitive for addition and subtraction, while polar form simplifies multiplication and exponentiation.

Conversion Formulas

Converting from polar to rectangular form requires only two trigonometric relationships. Given a complex number with magnitude r and phase φ, apply these formulas:

• a — The real part of the rectangular form
• b — The imaginary part of the rectangular form
• r — The magnitude (modulus) of the complex number in polar form
• φ — The phase angle, typically in radians or degrees

Why These Formulas Work

Picture the complex number as a vector extending from the origin. The magnitude r represents the vector's length, and the angle φ represents its direction.

When you drop perpendicular lines to the real and imaginary axes, you create a right triangle. The horizontal leg has length r·cos(φ) and the vertical leg has length r·sin(φ). These are precisely the Cartesian coordinates a and b.

This geometric relationship—rooted in basic trigonometry—ensures the conversion is always accurate, whether working with small angles or large magnitudes.

Common Pitfalls and Considerations

Avoid these frequent mistakes when converting between forms.

1. Angle Units Matter — Ensure your phase angle matches your calculator's expected input. Most mathematical conventions use radians, but many engineering contexts employ degrees. A phase of 45° and π/4 radians represent the same direction, but inputting degrees into a radian-expecting calculator produces completely incorrect results.
2. Order of Operations — Always compute the trigonometric values first, then multiply by the magnitude. Computing cos(φ) yields a decimal between −1 and 1; multiplying by <code>r</code> scales this appropriately. Skipping steps or reversing the order introduces computational errors.
3. Negative Components — Both <code>a</code> and <code>b</code> can be negative, depending on which quadrant the complex number occupies. A phase angle between 90° and 180° produces a positive imaginary part but negative real part. This is correct—don't assume both components must be positive.
4. Special Cases — When the magnitude is zero, both components are zero regardless of phase. When the phase is exactly 0°, the imaginary part vanishes and you have a purely real number. These boundary cases are mathematically valid and often appear in practical applications.

Practical Example

Suppose you have a complex number in polar form: magnitude r = 5, phase φ = 60°.

First, convert the angle to radians if needed: 60° = π/3 radians.

Then calculate:

• a = 5 × cos(60°) = 5 × 0.5 = 2.5
• b = 5 × sin(60°) = 5 × 0.866... ≈ 4.33

The rectangular form is approximately 2.5 + 4.33i. You can verify this by computing the magnitude √(2.5² + 4.33²) ≈ 5 and the angle arctan(4.33/2.5) ≈ 60°.