Switch seamlessly among a plus bi polar and cis representations with clear computation steps selectable degree or radian angles and flexible rounding. Validate branch cuts confirm principal argument and copy outputs instantly for homework circuits signals and control systems where complex arithmetic clarity and consistency truly matter. Useful for math engineering physics and robotics.
Euler form expresses a complex number using magnitude and angle rather than separate real and imaginary parts. Any value a plus bi can be written as r times e to the i theta where r equals the modulus square root of a squared plus b squared and theta equals the argument given by arctangent two of b and a. This representation compresses rotation and scaling into two intuitive parameters. Engineers prefer it when multiplying dividing or raising complex numbers to powers because magnitudes multiply and angles simply add or subtract. It also links directly to sine and cosine through the identity e to the i theta equals cosine theta plus i sine theta often shortened as cis theta. With Euler form you can switch effortlessly between time domain sinusoids and phasors in circuit analysis compute stable poles in control design and interpret frequency response. Care is needed choosing the principal argument often restricted to minus pi to pi or zero to two pi and selecting degrees or radians consistently. Rectangular inputs convert cleanly to polar using consistent branch rules defined.
Use Rectangular → Polar when you have a plus bi. Use Polar → Rectangular when you have r and an angle in degrees or radians. The tool outputs both views.
The principal argument is a chosen standard range for angles. Common ranges are (−π, π] and [0, 2π). Selecting one ensures consistent results across calculations and documentation.
Only the numeric representation of the angle changes. Internally the angle is normalized and conversions between units are handled exactly to avoid ambiguity.
If the magnitude is zero the angle is undefined. The calculator sets the angle to zero for display while noting that any angle represents the same zero vector.
Rounding changes the displayed digits but internal calculations use full precision from your inputs. Increase decimals if you need tighter agreement in back conversions.
Cis is shorthand for cos theta plus i sin theta. It is convenient in teaching and matches the Euler form while remaining readable in plain text outputs.
Atan2 uses both a and b to place the angle in the correct quadrant. The principal range setting then wraps the angle consistently for display and further work.
Yes. The magnitude and phase map directly to phasors. For control systems Euler form helps express poles and zeros and simplifies multiplication and exponentiation rules.
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