Complex numbers

Do you need a calculator to convert from polar to binomial form or from binomial to polar? Here you have a tool dedicated to complex numbers along with much more information for you to know the properties, operations and much more.

To change a number from polar to binomial or vice versa just select the change you want to make in our calculator, write the values that correspond in each case and press the calculate button to obtain the equivalence automatically. 

What is a complex number?

Representation of a complex number

A complex number is a extension of real numbers and are composed of a real part and an imaginary part. Complexes can be represented in binomial or polar form:

  • Binomial Formare written in the form z = a + bi, where a is the real part and b is the imaginary part.
  • Polar shapeare represented as z= rαwhere r is the modulus and α is the argument.

And what are complex numbers used for? The answer is long but as a summary we can say that we use them to solve operations which have no real solution (as the square root of a negative number).

This is how the imaginary unit represented by the letter i y que equivale a la raíz cuadrada de - 1, es decir:

i = √-1

Change from polar to binomial form

For to convert a complex number from polar to binomial formIf we do not use the following mathematical formula, we have to apply the following mathematical formula:

z = rα = r (cos α + i sen α)

For example, if we want to go to binomial the complex 3240ºIf we do not use the formula, we will apply the formula as follows:

z = 3240º = 3 (cos 240º + i sen 240º) = 3 (-0,5 - 0,866i) = -1,5 - 2,598i

If you find it useful, here we leave you the corresponding calculators to obtain the sine: and the cosine.

Convert from binomial to polar

Modulus of a complex number

If we want to do the inverse conversion, i.e., pass a complex number of Cartesian to polar coordinatesthen we have to perform the following process:

  1. The first thing will be calculate the modulus of the complex number (z = a + bi), for which we apply this formula:

|z| = √(a2 + b2)

  1. Now we have to obtain the argument of polar coordinateswhich is obtained with this formula:

α = arctg (b/a)

We leave you with our calculator of arcotangent to solve the above operation immediately.

We will propose a exercise solved in which we are going to polarize the complex 1 + 2i.

|z| = √(12 + 22) = √(1 + 4) = √5

α = arctg (b/a) = arctg (2/1) = 63.43º.

Finally we have that our complex number 1 + 2i in Cartesian coordinates is expressed as √563,43º in polar coordinates


Complex number calculator

Here we will show you which are the main ones. operations that can be performed with complex numbers.

Please note that some of these operations only can be made in binomial or polar formso you will need to switch from one format to another depending on how the complex number you have is expressed.

Addition and subtraction of complex numbers

The addition or subtraction of complex numbers in binomial form is very simple. We only have to add the real parts and the imaginary parts as you can see in the following expression:

(a+bi)+(c+di) = (a+c)+(b+d)i
(a+bi)-(c+di) = (a-c)+(b-d)i

For example:

(3 + 2i) + (1 + 5i) = (3 + 1) + (2 + 5)i = 4 + 7i
(4 + 3i) - (2 + 1i) = (4 - 2) + (3 - 1)i = 2 + 2i

Properties of the sum

  • Commutative property: z1 + z2 = z2 + z1
  • Associative property (z1 + z2) + z3 = z1 + (z2 + z3)
  • 0 is the neutral element

Multiplication of complex numbers

Multiplying complex numbers can be done either in polar form as well as in binomial form. Let's see how it is done in each case:

Product of complex numbers in binomial form:

(a+bi)-(c+di) = (a-c - b-d)+(a-d+b-c)i

Exercise solved:

(1 + 3i)·(2+1i) = 1·2 - 3i2 + 1i + 6i = 2 - 3 + i + 6i = -1 + 7i

Remember that i2 is equal to -1

Product of complex numbers in polar form:


Solved example:

230º - 535º = (2 - 5)30º+35º. = 1065º

Properties of multiplication:

  • Commutative property: z1 - z2 = z2 - z1
  • Associative property: (z1 - z2) - z3 = z1 - (z2 - z3)
  • Distributive property: z1 - (z2 + z3) = z1 - z2 + z1 - z3
  • 1 is the neutral element (1 + 0i)

Division of complex numbers

Division is another of the operations that allows us to work in both binomial and polar form.. Let's see how to proceed in each of them:

Division of complex numbers in binomial form

If we have two complex numbers expressed in binomial form, the way to divide them is as follows:

Division of complex numbers in binomial form

If you are interested in knowing how this expression is arrived at, here is the demonstration in which the numerator and denominator have been multiplied by the conjugate of the denominator, so we obtain a real number in that part of the fraction:

Demonstration of the division of complexes in binomial form.

Division of complex numbers in polar form

For divide complexes in polar form simply divide the modulus and subtract the arguments as indicated in the following formula:

Division of complexes in polar form

Powers of complex numbers

In the event that we have to calculate the power of a complex numberIn this section we explain how to calculate it in each case:

Power of a complex number in binomial form

This is the formula to be developed n times to calculate the power of a complex number in binomial form:

Power of a complex number in binomial form

Depending on the complexity, it may be easier for you to convert complex number to polar form and calculate power as described in the following section.

Powers of a complex number in polar form

In my opinion, calculating powers in polar form is much easier to do than in binomial form since we only have to apply the following formula:

(mα)n = mn

As you can see, you just have to raise the modulus to the power n and multiply the argument by the value of n.

Modulus of a complex number

If you would like more information on how the modulus of a complex numberClick on the link we have just left you.

Conjugate of a complex number

The conjugate of a complex number in binomial form is that which has the same real part and changed sign the imaginary part, so that if our complex is Z = a + bi, its conjugate will be:

Conjugate of a complex number


Properties of the conjugate of a complex number

These are the properties that a complex number has:

  • The conjugate of a real number is itself. For example, the conjugate of 3 +0i is equal to 3 -0i, in short, it is 3 in both cases.
  • The conjugate of an imaginary number is its opposite. For example: the conjugate of 5i is -5i
  • The conjugate of conjugate is the initial imaginary number.

Conjugate of conjugate

  • The sum of a complex number plus its conjugate is equal to twice its real part:

Sum of the complex and its conjugate

  • The subtraction of a complex minus its conjugate is equal to twice its imaginary part:

Subtraction of a complex minus its conjugate

  • The product of a complex by its conjugate is equal to the square of the real part plus the square of the imaginary part:

Product of a complex by its conjugate

  • For the sum and product of several conjugates it holds that:

Sum and product of two conjugates

Complex numbers in Excel

Excel has numerous functions to operate with complex numbers. Below is a table that will serve as a summary to see what they are and what they do.

If you don't know how to use it or what it's for, feel free to leave us a comment and we'll help you with anything to do with complex numbers.

COMPLEXConvert real and imaginary coefficients to a complex number.
IM.ABSCalculates the modulus of a complex.
IM.ANGULOCalculate the angle of the argument (in radians)
IMAGINARYGives us the coefficient of the imaginary part
IM.CONJUGATEDCalculates the conjugate of a complex
IM.COSIt gives us the cosine of a complex number.
IM.DIVWe will use it to divide two complex numbers
IM.EXPAllows us to calculate the exponential form of a complex
IM.LNReturns the neperian logarithm of a complex number.
IM.LOG10Gives us the logarithm in base 10 of the complex
IM.LOG2This function calculates the logarithm in base 2.
IM.POTHelps us to calculate the power of a complex
IM.PRODUCTAllows us to multiply up to 29 complex numbers
IM.ROOT2We will use it to get the square root of the complex
IM.REALGives us the real part of the complex number
IM.SENOWith this function we calculate the sine of a complex number
IM.SUMThe result will be the sum of two complex numbers
IM.SUSTRThis function is used to subtract two complex numbers

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