1.414214
0.785398
rad
—
deg
0.707107
0.707107
0.707107
0.707107
1.414214
0.785398
rad
—
deg
0.707107
0.707107
0.707107
0.707107
The Complex Number Polar Form Calculator converts a complex number from rectangular (Cartesian) form $$a + bi$$ to polar (trigonometric) form $$r(\cos\theta + i\sin\theta)$$. The polar representation expresses a complex number in terms of its distance from the origin (modulus) and direction (argument), providing a geometric interpretation that simplifies many complex number operations.
Every non-zero complex number $$z = a + bi$$ can be uniquely represented in polar form as $$z = r(\cos\theta + i\sin\theta)$$, where $$r = |z| = \sqrt{a^2+b^2}$$ is the modulus and $$\theta = \arg(z) = \text{atan2}(b, a)$$ is the principal argument. The modulus $$r$$ is the radial distance from the origin, and $$\theta$$ is the angle measured counterclockwise from the positive real axis.
The polar form is particularly powerful for multiplication, division, and exponentiation of complex numbers. De Moivre's theorem states that $$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$$, making power calculations trivial in polar form compared to repeated multiplication in rectangular form. Similarly, multiplication of two complex numbers in polar form simply multiplies their moduli and adds their arguments: $$r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1+\theta_2)}$$.
The polar form connects to Euler's formula $$e^{i\theta} = \cos\theta + i\sin\theta$$, one of the most celebrated results in mathematics. This formula bridges exponential functions, trigonometric functions, and complex numbers in a single elegant equation. Setting $$\theta = \pi$$ yields Euler's identity $$e^{i\pi} + 1 = 0$$, which links five fundamental mathematical constants.
In engineering and physics, polar form naturally represents oscillating quantities. Alternating current circuits use phasors — complex numbers in polar form — where the modulus represents amplitude and the argument represents phase. In control theory, Bode plots display transfer function magnitude and phase as functions of frequency, both quantities derived from the polar form. Signal processing uses polar representation for spectral analysis, and antenna engineering applies it to radiation pattern characterization.
The calculator uses the two-argument arctangent function $$\text{atan2}(b, a)$$ rather than $$\arctan(b/a)$$ to correctly determine the quadrant of the angle. This function returns values in $$(-\pi, \pi]$$, covering all four quadrants of the complex plane, whereas the simple arctangent only distinguishes two quadrants and fails for purely imaginary numbers.
Conversion from rectangular to polar form uses the modulus and argument.
Given:
$$z = a + bi$$
Step 1: Compute the modulus (r):
$$r = |z| = \sqrt{a^2 + b^2}$$
Step 2: Compute the argument (θ):
$$\theta = \text{atan2}(b, a)$$
This gives the angle in radians, converted to degrees by:
$$\theta_{\text{deg}} = \theta \times \frac{180}{\pi}$$
Step 3: Write polar form:
$$z = r(\cos\theta + i\sin\theta)$$
Verification:
$$a = r\cos\theta, \quad b = r\sin\theta$$
De Moivre's Theorem:
$$z^n = r^n(\cos n\theta + i\sin n\theta)$$
The polar form tells you two things: how far the number is from the origin (modulus r) and in which direction it points (angle θ). Numbers on the positive real axis have θ = 0°, on the positive imaginary axis θ = 90°, on the negative real axis θ = 180° (or -180°), and on the negative imaginary axis θ = -90°. The cos(θ) and sin(θ) values allow you to verify the conversion: a = r×cos(θ) and b = r×sin(θ).
Inputs
Results
r = √(1+1) = √2 ≈ 1.4142. θ = atan2(1,1) = π/4 = 45°. So 1+i = √2(cos 45° + i sin 45°).
Inputs
Results
r = √(9+16) = 5. θ = atan2(4,-3) ≈ 126.87° (second quadrant). Verify: 5×cos(126.87°) = -3, 5×sin(126.87°) = 4.
The polar form writes a complex number as $$z = r(\cos\theta + i\sin\theta)$$, where $$r = |z|$$ is the modulus (distance from origin) and $$\theta = \arg(z)$$ is the argument (angle from positive real axis). It provides a geometric representation equivalent to the rectangular form $$a + bi$$, with $$a = r\cos\theta$$ and $$b = r\sin\theta$$.
Polar form dramatically simplifies multiplication, division, and exponentiation. To multiply, you multiply moduli and add angles. To divide, you divide moduli and subtract angles. To raise to a power $$n$$, you raise the modulus to $$n$$ and multiply the angle by $$n$$ (De Moivre's theorem). These operations are much simpler than performing them in rectangular form.
The principal argument is the unique value of $$\theta$$ in the interval $$(-\pi, \pi]$$ (or equivalently $$(-180°, 180°]$$). Since angles differing by $$2\pi$$ represent the same direction, we choose this convention to have a unique representation. The function $$\text{atan2}(b, a)$$ returns the principal argument, correctly handling all four quadrants.
Euler's formula states $$e^{i\theta} = \cos\theta + i\sin\theta$$. This means the polar form $$r(\cos\theta + i\sin\theta)$$ can be written compactly as $$re^{i\theta}$$, called the exponential form. This connection unifies trigonometry and exponential functions through complex numbers and is foundational to modern mathematics and physics.
De Moivre's theorem states that $$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$$ for any integer $$n$$. This means raising a complex number to a power simply raises the modulus to that power and multiplies the argument by $$n$$. It extends to fractional powers for finding nth roots of complex numbers.
To convert from polar form $$r(\cos\theta + i\sin\theta)$$ back to rectangular form $$a + bi$$, compute $$a = r\cos\theta$$ and $$b = r\sin\theta$$. This uses the basic trigonometric relationships between the hypotenuse (modulus), adjacent side (real part), and opposite side (imaginary part) of the right triangle in the complex plane.
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