Complex Number Calculator
Add, subtract, multiply, divide and raise complex numbers to a power, or take the conjugate, reciprocal or square root of one. Enter values like 3 + 4i or 2 - i, with j accepted for i. The result is shown in rectangular form with its modulus, its argument in radians and degrees, and the polar and exponential forms, rounded to your chosen precision. Computed with the mathjs complex number engine. For engineering, physics and algebra students.
Frequently asked questions
How do I write a complex number?
Write the real part followed by the imaginary part with i, for example 3 + 4i, 2 - i, -0.5i or just 7 for a real number. Engineers can use j instead of i, as in 3 + 4j. Spaces and an explicit multiplication sign as in 4*i are allowed.
Which operations are available?
Add, subtract, multiply, divide and power use both numbers, z1 and z2. Conjugate, reciprocal and square root use only the first number and ignore the second field. The operation menu shows which inputs are used.
What are the modulus and argument?
The modulus |z| is the distance of the number from the origin, the square root of the real part squared plus the imaginary part squared. The argument is the angle from the positive real axis, between -180 and 180 degrees. For 3 + 4i the modulus is 5 and the argument is about 53.13 degrees.
What are the polar and exponential forms?
The polar form writes the number as its modulus and angle, r ∠ θ, with the angle in degrees. The exponential form writes the same number as r e^(iθ) with the angle in radians, following Euler's formula. Both describe the same point as the rectangular form a + bi.
How is complex division done?
Division multiplies numerator and denominator by the conjugate of the divisor, which turns the denominator into a real number. For example (3 + 4i) ÷ (1 - 2i) gives -1 + 2i. Dividing by 0 is undefined and the tool reports an error.
How is a complex power calculated?
z1 ^ z2 is computed as e raised to z2 times the principal logarithm of z1, the standard definition used by mathjs. Integer exponents give the familiar results, such as (1 + i)^2 = 2i. Non-integer exponents return the principal value, one of possibly several valid answers.
Which square root is returned?
The tool returns the principal square root, the one with a nonnegative real part. The other square root is its negative. For example the square root of -4 is 2i, and -2i is the second root.
What is the conjugate used for?
The conjugate of a + bi is a - bi, the reflection across the real axis. Multiplying a number by its conjugate gives the square of its modulus, a real number. It is the key step in dividing complex numbers.
How precise are the results?
The calculations use double precision floating point numbers and the results are rounded to 6 decimal places by default, adjustable from 0 to 15. Tiny leftovers from rounding, such as an imaginary part around 1e-16 next to a real part of 1, are shown as zero. Angles are rounded to the same number of places.
Can I enter numbers in polar form?
No, inputs must be in rectangular form a + bi. To convert from polar form, compute a = r cos θ and b = r sin θ, for example with the Scientific Calculator. The result here always includes the polar form of the answer.