Convert a polar complex number to its rectangular form (a + bi) in seconds. Enter the modulus (r) and argument (φ) — in degrees or radians — and get back the real part (a) and imaginary part (b), along with the full rectangular expression.
Rectangular Form (a + bi)
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Real Part (a)
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Imaginary Part (b)
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Modulus |z|
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Argument (degrees)
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Argument (radians)
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The rectangular form expresses a complex number as z = a + bi, where 'a' is the real part and 'b' is the imaginary part. It represents the number as a point (a, b) on the complex plane, with 'a' along the horizontal (real) axis and 'b' along the vertical (imaginary) axis.
Given a complex number with modulus r and argument φ, the rectangular form is found using: a = r × cos(φ) and b = r × sin(φ). The resulting rectangular form is z = a + bi. Make sure your angle is in the correct unit (degrees or radians) before computing.
exp(iπ/3) has modulus r = 1 and argument φ = π/3 radians (60°). Using the conversion formulas: a = cos(π/3) = 0.5 and b = sin(π/3) ≈ 0.8660. So the rectangular form is 0.5 + 0.8660i.
Polar form represents a complex number using its distance from the origin (modulus r) and its angle from the positive real axis (argument φ), written as r∠φ or r·e^(iφ). Rectangular form uses the Cartesian coordinates (a + bi). Both represent the same number — just in different ways.
By convention, the modulus of a complex number is always a non-negative real number, as it represents the distance from the origin. If a negative value is entered for r, it can be interpreted by flipping the argument by 180°, but standard usage requires r ≥ 0.
The argument φ is the angle that the line connecting the origin to the complex number on the complex plane makes with the positive real axis. It can be expressed in degrees or radians. The principal argument is typically given in the range (−180°, 180°] or (−π, π].
The cos and sin functions behave differently depending on whether the input is in degrees or radians. For example, cos(90°) = 0, but cos(90 radians) ≈ −0.4480. Always specify which unit your argument is in to get the correct rectangular form.
In AC circuit analysis, impedances and phasors are often expressed in polar form (magnitude and phase angle). Converting to rectangular form (resistance + j·reactance) is essential for adding or subtracting complex impedances, making this conversion a core skill in circuits courses.