A Bi Form Calculator - Polar to Rectangular

An a bi form calculator turns a complex number written in polar form r·exp(iφ) into the standard rectangular form a + bi, where a is the real part and b is the imaginary part. Use it whenever you have a magnitude and an angle and want to read off the (a, b) coordinates on the complex plane for graphing, arithmetic, or further calculation.

• • Convert polar to rectangular: Translate a magnitude r and a phase φ into the matching (a, b) point on the complex plane in a single step.
• • Read off the real and imaginary parts: See a = r·cos(φ) and b = r·sin(φ) reported as separate numbers so they can be used in further work.
• • Work in either degrees or radians: Switch the phase unit between degrees and radians without having to redo the math by hand.
• • Check a manual conversion: Use the magnitude and phase sanity checks to confirm that a hand-computed (a, b) pair round-trips back to the original polar inputs.

According to Wolfram MathWorld, the rectangular form of a complex number z = a + bi is related to the polar form z = r·exp(iφ) by a = r·cos(φ) and b = r·sin(φ), which is exactly the conversion the a bi form calculator performs.

The a + bi form is the one most algebra, engineering, and signal-processing courses use when they need to add, multiply, or graph a complex number, so converting out of polar form is a routine step before further work.

Once you have a and b in hand, the Complex Number Calculator takes care of adding, subtracting, multiplying, and dividing the same complex number with another one.

The calculator reads the magnitude r and the phase φ in the unit you choose, then multiplies the magnitude by the cosine and sine of the phase to get the real part a and the imaginary part b. It rounds each value to four decimal places and shows the a + bi form as a single string, with parentheses around b when it is negative.

• Magnitude r: The distance from the origin to the point on the complex plane. It is the radius of the polar representation and is always a non-negative real number.
• Phase φ: The angle from the positive real axis to the point, measured counterclockwise. Enter it in degrees or radians using the unit selector below.
• Real part a: The horizontal coordinate of the point, equal to r·cos(φ). It can be positive, negative, or zero.
• Imaginary part b: The vertical coordinate of the point, equal to r·sin(φ). It is positive above the real axis and negative below it.

When the input is in degrees, the calculator converts the phase to radians first using φ_rad = φ·π/180 before applying the cosine and sine, which matches the convention used in every standard engineering and precalculus textbook.

The output also shows a magnitude check √(a² + b²) and a phase check atan2(b, a) so you can confirm the round trip from polar to rectangular and back.

According to Wolfram MathWorld, the rectangular form of a complex number z = a + bi is related to the polar form z = r·exp(iφ) by a = r·cos(φ) and b = r·sin(φ).

If your textbook quotes the phase in radians but your calculator is set to degrees, the Radians to Degrees Calculator converts the angle without changing the math underneath.

These four ideas explain why the same (r, φ) pair can be written in either form, and how to read the result back out of the calculator.

The two forms describe the same point on the complex plane, so any polar pair (r, φ) has exactly one rectangular pair (a, b) and vice versa.

Wolfram MathWorld uses the term Cartesian form for the same rectangular a + bi expression, and many textbooks use the term standard form to mean the same thing.

For a quick unit switch on the phase value alone, the Angle Converter handles degree, radian, gradian, and turn conversions in one place.

Pick the magnitude and the phase, choose degrees or radians for the phase, and read the a + bi result. The expression, real part, and imaginary part all update at the same time.

1. 1 Enter the magnitude: Type the modulus r of the complex number. Leave it at 5 for the standard quick example.
2. 2 Enter the phase: Type the angle φ from the positive real axis. Positive values rotate counterclockwise; negative values rotate clockwise.
3. 3 Pick the phase unit: Choose degrees (0° to 360°) or radians (0 to 2π). The calculator uses the same unit for the phase check on the right.
4. 4 Read the a + bi expression: The rendered expression combines a, the sign, and b, with parentheses around a negative b so the imaginary unit is unambiguous.
5. 5 Use the magnitude and phase checks: The magnitude check should match the input r, and the phase check should match the input φ. If either drifts, double-check the input values.

If the magnitude is 4 and the phase is 60°, then a = 4·cos(60°) = 2, b = 4·sin(60°) ≈ 3.4641, so the a + bi form is 2 + 3.4641i. The magnitude check returns 4 and the phase check returns 60.

Because a and b are the legs of a right triangle whose hypotenuse is r, the Triangle Calculator is the natural follow-up when you need the side lengths or the other acute angle.

Putting the polar-to-rectangular conversion in one place saves a step every time you move between the two forms of a complex number.

• • Both forms side by side: The calculator shows the rendered a + bi string and the numeric (a, b) pair at the same time, so you can paste either form into further work.
• • Degrees and radians in one tool: Switch the phase unit without recomputing by hand, which is useful when a textbook uses degrees and a code library uses radians.
• • Round-trip checks included: The magnitude check √(a² + b²) and the phase check atan2(b, a) confirm the conversion did not lose precision.
• • Parentheses for negative b: The rendered expression uses parentheses around a negative imaginary part so the i cannot be misread as part of a coefficient.
• • Useful beyond algebra class: The same conversion shows up in AC circuit analysis, phasor diagrams, signal processing, and physics, so the result travels well beyond precalculus homework.

A textbook conversion usually only reports the numeric (a, b) pair, but the calculator also reports the rendered expression, which is the form most people paste into a follow-up computation.

The round-trip checks are especially useful in trigonometry and precalculus classes, where the same problem is often rephrased in both forms to test whether the student can move between them.

The same trig pair that produces a and b also solves the legs of a right triangle, so the Right Triangle Calculator reuses the cosine and sine inputs for the inverse problem.

A handful of input choices and a few edge values change the rendered expression in ways that are easy to miss. It helps to know which levers matter before trusting the result.

• • The calculator only handles one complex number at a time, so adding or multiplying two complex numbers is out of scope. Use a general complex arithmetic tool for that.
• • Floating-point math can return values like 1.0000000000000002 for inputs that should be exactly 1. The four-decimal rounding masks most of these cases but does not eliminate them.

According to Paul's Online Math Notes, the rectangular coordinates (a, b) of a point given in polar form are obtained by a = r·cos(θ) and b = r·sin(θ), which is the same conversion this calculator uses.

Wolfram MathWorld notes that the rectangular form is also called the Cartesian or algebraic form, so a tool that names a and b will line up with whichever term your textbook prefers.

According to Paul's Online Math Notes, the rectangular coordinates (a, b) of a complex number are obtained from its polar form by a = r·cos(θ) and b = r·sin(θ).

The magnitude check √(a² + b²) is the same operation the Vector Magnitude Calculator runs on 2D components.