a+bi Form Calculator: Polar to Rectangular Conversion
Enter a complex number in polar form (magnitude r and angle phi) to get its rectangular a+bi representation, or switch to rectangular mode and convert the other way. The calculator also shows the modulus, argument, complex conjugate, and Euler exponential form, with a step-by-step breakdown of every computation.
Formula
Worked example
Convert r=5, phi=45 deg to rectangular form: a = 5*cos(45 deg) = 5*(sqrt(2)/2) = 3.535534; b = 5*sin(45 deg) = 5*(sqrt(2)/2) = 3.535534. So z = 3.535534 + 3.535534i. Modulus check: sqrt(3.535534^2 + 3.535534^2) = sqrt(25) = 5. Conjugate: 3.535534 - 3.535534i.
What is a+bi form?
Every complex number can be written as a+bi, where a is the real part and b is the imaginary part. The letter i represents the imaginary unit, defined so that i^2 = -1. This rectangular form is the most common way to express complex numbers in algebra because addition and subtraction are straightforward: you simply combine real parts and combine imaginary parts separately. The complex plane (Argand diagram) plots a on the horizontal axis and b on the vertical axis, so each complex number corresponds to a unique point in the plane.
Polar form and the conversion formulas
The polar form expresses the same complex number using its distance from the origin (the modulus r) and its angle from the positive real axis (the argument phi). The two representations are linked by trigonometry: a = r * cos(phi) and b = r * sin(phi). Going the other way, r = sqrt(a^2 + b^2) and phi = atan2(b, a), the two-argument arctangent that automatically places the angle in the correct quadrant. This is also written as r * cis(phi) or, using Euler's formula, r * e^(i*phi). Euler's formula is e^(i*phi) = cos(phi) + i*sin(phi), the link that makes exponential, polar, and rectangular forms interchangeable.
The complex conjugate and its uses
The complex conjugate of z = a+bi is z* = a-bi. It is the mirror image of z reflected across the real axis. The product z * z* = a^2 + b^2 = r^2, which is always a non-negative real number. This property makes conjugates essential for rationalising complex fractions: to divide by a+bi, multiply numerator and denominator by its conjugate a-bi, turning the denominator into a pure real number. Conjugates also appear in polynomial roots: if a polynomial with real coefficients has a complex root a+bi, then a-bi is also a root.
Modulus, argument and De Moivre's theorem
The modulus |z| = r = sqrt(a^2 + b^2) measures the size of the complex number. The argument arg(z) = phi = atan2(b, a) is the signed angle in the range (-pi, pi]. Polar form makes multiplication and division easy: to multiply two complex numbers, multiply their moduli and add their arguments; to divide, divide moduli and subtract arguments. This leads directly to De Moivre's theorem: (r * cis(phi))^n = r^n * cis(n*phi), which gives a simple formula for powers and, by extension, the nth roots of any complex number.
Common complex numbers in a+bi and polar form
| Degrees | Radians | cos(phi) | sin(phi) | z = 1*cis(phi) | Quadrant |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 + 0i | Real axis (+) |
| 30 | pi/6 | sqrt(3)/2 | 1/2 | 0.866025 + 0.5i | Q1 |
| 45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 0.707107 + 0.707107i | Q1 |
| 60 | pi/3 | 1/2 | sqrt(3)/2 | 0.5 + 0.866025i | Q1 |
| 90 | pi/2 | 0 | 1 | 0 + 1i | Imag axis (+) |
| 120 | 2pi/3 | -1/2 | sqrt(3)/2 | -0.5 + 0.866025i | Q2 |
| 135 | 3pi/4 | -sqrt(2)/2 | sqrt(2)/2 | -0.707107 + 0.707107i | Q2 |
| 150 | 5pi/6 | -sqrt(3)/2 | 1/2 | -0.866025 + 0.5i | Q2 |
| 180 | pi | -1 | 0 | -1 + 0i | Real axis (-) |
| 210 | 7pi/6 | -sqrt(3)/2 | -1/2 | -0.866025 - 0.5i | Q3 |
| 225 | 5pi/4 | -sqrt(2)/2 | -sqrt(2)/2 | -0.707107 - 0.707107i | Q3 |
| 240 | 4pi/3 | -1/2 | -sqrt(3)/2 | -0.5 - 0.866025i | Q3 |
| 270 | 3pi/2 | 0 | -1 | 0 - 1i | Imag axis (-) |
| 315 | 7pi/4 | sqrt(2)/2 | -sqrt(2)/2 | 0.707107 - 0.707107i | Q4 |
| 360 | 2pi | 1 | 0 | 1 + 0i | Real axis (+) |
Reference values for angles that appear frequently in mathematics and engineering. Angles in degrees; exact forms where possible.
Frequently asked questions
What does a+bi mean in mathematics?
In a+bi, a is the real part and b is the imaginary part of a complex number. The symbol i denotes the imaginary unit, where i^2 = -1. Together, a and b specify a unique point on the complex plane, with a measured along the horizontal axis and b along the vertical.
How do I convert polar form to a+bi?
Given a magnitude r and an angle phi, the real part is a = r * cos(phi) and the imaginary part is b = r * sin(phi). Make sure phi is in radians if your trig functions expect radians, or convert degrees to radians by multiplying by pi/180 first. The result is the complex number a + bi.
How do I convert a+bi to polar form?
The modulus (magnitude) is r = sqrt(a^2 + b^2). The argument (angle) is phi = atan2(b, a), which gives the angle in the range (-180, 180] degrees or (-pi, pi] radians, placing it in the correct quadrant automatically.
What is the complex conjugate of a+bi?
The conjugate is a-bi. It has the same real part but the opposite sign on the imaginary part. The product of a complex number and its conjugate is always a real number: (a+bi)(a-bi) = a^2 + b^2. This makes conjugates indispensable for simplifying division of complex numbers.
What is Euler's formula and how does it relate to a+bi?
Euler's formula states e^(i*phi) = cos(phi) + i*sin(phi). Multiplying both sides by r gives r*e^(i*phi) = r*cos(phi) + i*r*sin(phi) = a + bi. So the exponential form r*e^(i*phi), the polar form r*cis(phi), and the rectangular form a+bi are all equivalent representations of the same complex number, connected by Euler's identity.
What quadrant does my complex number fall in?
If a > 0 and b > 0, the number is in quadrant 1. If a < 0 and b > 0, it is in quadrant 2. If a < 0 and b < 0, it is in quadrant 3. If a > 0 and b < 0, it is in quadrant 4. Special cases: b = 0 means the number lies on the real axis; a = 0 means it lies on the imaginary axis.
Why does atan2 give a different answer than atan(b/a)?
The single-argument atan(b/a) cannot distinguish between (a, b) and (-a, -b), because dividing negates both, so it always returns an angle in (-90, 90) degrees. The two-argument atan2(b, a) uses the signs of a and b separately to place the angle in the full range (-180, 180] degrees, giving the correct quadrant every time.