Complex number calculator
Rectangular form (standard form):
z = 3+4i
Angle notation (phasor, module and argument):
z = 5 ∠ 53°7'48″
Polar form:
z = 5 × (cos 53°7'48″ + i sin 53°7'48″)
Exponential form:
z = 5 × ei 0.9272952 = 5 × ei 0.2951672 π
Polar coordinates:
r = |z| = 5 ... magnitude (modulus, absolute value)
θ = arg z = 0.9272952 rad = 53.1301° = 53°7'48″ = 0.2951672π rad ... angle (argument or phase)
Cartesian coordinates:
Cartesian form of imaginary number: z = 3+4i
Real part: x = Re z = 3
Imaginary part: y = Im z = 4
z = 3+4i
Angle notation (phasor, module and argument):
z = 5 ∠ 53°7'48″
Polar form:
z = 5 × (cos 53°7'48″ + i sin 53°7'48″)
Exponential form:
z = 5 × ei 0.9272952 = 5 × ei 0.2951672 π
Polar coordinates:
r = |z| = 5 ... magnitude (modulus, absolute value)
θ = arg z = 0.9272952 rad = 53.1301° = 53°7'48″ = 0.2951672π rad ... angle (argument or phase)
Cartesian coordinates:
Cartesian form of imaginary number: z = 3+4i
Real part: x = Re z = 3
Imaginary part: y = Im z = 4
Calculation steps
- Complex number: 3+4i
This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers.
As an imaginary unit, use i or j (in electrical engineering), which satisfies the basic equation i2 = −1 or j2 = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2
Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.
For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.
For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.
Basic operations with complex numbers
We hope that working with the complex number is quite easy because you can work with imaginary unit i as a variable. And use the definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.Addition
It is very simple: add up the real parts (without i) and add up the imaginary parts (with i):This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i
(1+i) + (6-5i) = 7-4i
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5
Subtraction
Again it is very simple: subtract the real parts and subtract the imaginary parts (with i):This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i
(1+i) - (3-5i) = -2+6i
-1/2 - (6-5i) = -6.5+5i
(10-5i) - (-5+5i) = 15-10i
Multiplication
To multiply two complex numbers, use distributive law, avoid binomials, and apply i2 = -1.This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i
(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i
-1/2 * (6-5i) = -3+2.5i
(10-5i) * (-5+5i) = -25+75i
Division
The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the denominator's complex conjugate. This approach avoids imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it real), multiply with conjugate c-di:(c+di)(c-di) = c2+d2
c+dia+bi=(c+di)(c−di)(a+bi)(c−di)=c2+d2ac+bd+i(bc−ad)=c2+d2ac+bd+c2+d2bc−adi
(10-5i) / (1+i) = 2.5-7.5i
-3 / (2-i) = -1.2-0.6i
6i / (4+3i) = 0.72+0.96i
Absolute value or modulus
The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5|1-i| = 1.4142136
|6i| = 6
abs(2+5i) = 5.3851648
Square root
The square root of a complex number (a+bi) is z, if z2 = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is to use De Moivre's formula. Our calculator is on edge because the square root is not a well-defined function on a complex number. We calculate all complex roots from any number - even in expressions:sqrt(9i) = 2.1213203+2.1213203i
sqrt(10-6i) = 3.2910412-0.9115656i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225i
pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303i
Square, power, complex exponentiation
Our calculator can power any complex number to an integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...Famous example:
ii=e−π/2
i^2 = -1i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224i
i^i = 0.2078795764
pow(1+i,3) = -2+2i
Functions
- sqrt
- Square Root of a value or expression.
- sin
- the sine of a value or expression. Autodetect radians/degrees.
- cos
- the cosine of a value or expression. Autodetect radians/degrees.
- tan
- tangent of a value or expression. Autodetect radians/degrees.
- exp
- e (the Euler Constant) raised to the power of a value or expression
- pow
- Power one complex number to another integer/real/complex number
- ln
- The natural logarithm of a value or expression
- log
- The base-10 logarithm of a value or expression
- abs or |1+i|
- The absolute value of a value or expression
- phase
- Phase (angle) of a complex number
- cis
- is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i
- conj
- the conjugate of a complex number - example: conj(4i+5) = 5-4i
Examples:
• cube root: cuberoot(1 - 27i)• roots of Complex Numbers: pow(1 + i,1/7)
• phase, complex number angle: phase(1 + i)
• cis form complex numbers: 5 * cis(45°)
• The polar form of complex numbers: 10L60
• complex conjugate calculator: conj(4 + 5i)
• equation with complex numbers: (z + i/2 )/(1 - i) = 4z + 5i
• system of equations with imaginary numbers: x - y = 4 + 6i; 3ix + 7y=x + iy
• De Moivre's theorem - equation: z ^ 4=1
• multiplication of three complex numbers: (1 + 3i)(3 + 4i)(−5 + 3i)
• Find the product of 3-4i and its conjugate.: (3 - 4i) * conj(3 - 4i)
• operations with complex numbers: (3 - i) ^ 3
Complex numbers in word problems:
- Complex number z
Let z = -5 + i and w = 4 - 2i. Compute z - w and write your final answer in a + bi form. x = z - w - Complex conjugate
What is the conjugate of the expression 5√6 + 6√5 i? A.) -5√6 + 6√5 i B.) 5√6 - 6√5 i C.) -5√6 - 6√5 i D.) 6√5 - 5√6i - Evaluate 18
Evaluate the expression (-4-7i)-(-6-9i) and write the result in the form a+bi (Real + i* Imaginary). - ReIm notation
Let z = 6 + 5i and w = 3 - i. Compute the following and express your answer in a + bi form. w + 3z
- Complex number coordinates
Which coordinates show the location of -2+3i - Complex equation
If c - d = 2 and c = 12 + 7 i, find d. Write the result in the form of a + bi. - The expression 2
The expression (3+i)(1+2i) can be written in the form a+bi, where a and b are integers. What are the values of a and b?
more math problems »