45° argument
- Input
- 1+i
- Expected output
- √2∠45° = √2·e^(iπ/4)
Modulus √2, argument π/4 rad.
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complex number to polar form
On the Forms tab, type the number in any form and see the polar form r∠θ, the trigonometric form r(cos θ + i sin θ) and the exponential form r·e^(iθ), with modulus, principal argument and the position on the Argand plane.
Modulus √2, argument π/4 rad.
Equivalent to 225° on the 0° to 360° range.
Modulus 2 and a right angle.
The principal argument of a negative real is 180°.
θ = arctan(4/3), about 53.13°.
a = 5·cos 90° = 0 and b = 5·sin 90° = 5.
Multiply numerator and denominator by the conjugate of the denominator, so the denominator becomes a real number. In (3 + 4i) ÷ (1 − 2i), the conjugate of 1 − 2i is 1 + 2i. The numerator is (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i, and the denominator is 1² + 2² = 5. The result is −1 + 2i.
Any θ + 360°·k describes the same point, so a number has infinitely many arguments. The principal one is the only one in (−180°, 180°]. The tool also shows the equivalent between 0° and 360° when the principal is negative, because some textbooks use that range.
Use π in the angle, as in 2∠π/3 or 2e^(iπ/3), or switch the Angle without a unit option to radians. Angles with the degree symbol are always in degrees. The tool writes radian results as a fraction of π when the angle is a simple multiple of it.
Use Euler’s formula, e^(iθ) = cos θ + i sin θ, with the modulus and the argument in radians: 1 + i = √2·e^(iπ/4). With r = 1 and θ = π you get the identity e^(iπ) = −1.
= 1 − 2i
Step by step
Argand plane
An angle typed without a symbol, as in 5∠30, is read in the chosen unit; angles with ° or π carry their own unit. Choose j for electrical circuits.
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