Complex Number Calculator

Lets first understand, what are complex numbers and how we can perform operations on complex numbers?

Definition of a complex number

A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers, and i represents the "imaginary unit", satisfying the equation \(i^{2}=-1\). Because no real number satisfies this equation, i is called an imaginary number.

For the complex number a+bi, a is called the real part and b is called the imaginary part.

A complex number is generally denoted by \(z\).

For example, 5+8i is a complex number.

A real number a can be regarded as a complex number a+0i, whose imaginary part is zero.

A purely imaginary number bi is a complex number 0+bi, whose real part is zero.

The real part of a complex number z is denoted by \(Re(z)\) and the imaginary part of a complex number z is denoted by \(Im(z)\). For example,

$$Re(5+8i)=5\hspace{0.2cm}and\hspace{0.2cm}Im(5+8i)=8 $$

Complex Plane

A complex number z can be represented with an ordered pair \(\bigl(Re(z),Im(z)\bigr)\) of real numbers, which is interpreted as coordinates of a point in a two-dimensional space. This is called complex plane.

The definition of the complex numbers involving two arbitrary real values suggests the use of Cartesian coordinates in the complex plane. The horizontal (real) axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (imaginary) axis, with increasing values upwards.

Modulus of a complex number

The modulus (or absolute value) of a complex number z = a + bi is

$$r=|z|=\sqrt{a^2+b^2} $$

The absolute value of a complex number is the distance from the origin to the point representing the complex number in the complex plane.

The absolute value of a real number equals its absolute value as a complex number.

Example 1: Find the modulus of the complex number z=3+4i.

Solution:

$$|z|=\sqrt{3^2+4^2}$$ $$|z|=5$$

Argument of a complex number

The argument of z is the angle of the radius with the positive real axis, and is written as Arg(z).

$$\phi=Arg(a+bi)$$ $$If\hspace{0.1cm}x\ne0\hspace{0.1cm}or\hspace{0.1cm}y\ne0,$$ $$\phi= 2tan^{-1}\bigg(\frac{y}{\sqrt{x^2+y^2}+x}\bigg)\hspace{0.2cm}$$ $$Otherwise,$$ $$\phi=undefined$$

Operations on a complex number

Equality of a complex number

Two complex numbers a₁+b₁i and a₂+b₂i are equal if and only if both their real and imaginary parts are equal, that is, if a₁ = a₂ and b₁ = b₂.

Ordering of a complex number

Unlike the real numbers, there is no ordering of the complex numbers. There is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field.

Conjugate of a complex number

The complex conjugate of the complex number z = x+yi is given by x−yi. It is denoted by \(\bar{z}\).

Geometrically, \(\overline{z}\) is the "reflection" of complex number z about the real axis. Conjugating twice gives the original complex number.

$$\overline{\overline{z}}=z$$

The real and imaginary parts of a complex number z can be extracted as

$$Re(z)=\frac{z+\overline{z}}{2}\hspace{0.2cm}and$$ $$Im(z)=\frac{z-\overline{z}}{2i}$$

A complex number is real if and only if it equals its own conjugate.

Example 2: Find the conjugate of the complex number z=3+4i.

Solution:

$$\overline{z}=3-4i$$

Addition of complex numbers

Two complex numbers z₁ and z₂ are added as

$$z_1=a_1+b_1i\hspace{0.1cm}and\hspace{0.1cm}z_2=a_2+b_2i$$ $$z_1+z_2=(a_1+a_2)+(b_1+b_2)i$$

Example 3: Find the sum of the complex numbers z₁=3+4i and z₂=7+8i.

Solution:

$$z_1+z_2=(3+7)+(4+8)i$$ $$z_1+z_2=10+12i$$

Subtraction of complex numbers

Two complex numbers z₁ and z₂ are easily subtracted as

$$z_1-z_2=(a_1-a_2)+(b_1-b_2)i$$

Example 4: Find the difference between the complex numbers z₁=3+4i and z₂=7+8i.

Solution:

$$z_1-z_2=(3-7)+(4-8)i$$ $$z_1-z_2=-4-4i$$

Multiplication of complex numbers

Two complex numbers z₁ and z₂ are multiplied as

$$z_1z_2=(a_1+b_1i)(a_2+b_2i)$$ $$=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i$$

Example 5: Find the product of complex numbers z₁=3+4i and z₂=7+8i.

Solution:

$$z_1z_2=(3+4i)(7+8i)$$ $$z_1z_2=(21-32)+(24+28)i$$ $$z_1z_2=-11+52i$$

Square of a complex number

The square of the complex number z=a+bi, can be calculated as

$$z^2=(a+bi)^2$$ $$z^2=a^2+(bi)^2+(2ab)i$$ $$z^2=(a^2-b^2)+(2ab)i$$

Example 6: Find the square of the complex number z=3+4i.

Solution:

$$z^2=(3+4i)^2$$ $$z^2=3^2+(4i)^2+(2.3.4)i$$ $$z^2=(9-16)+24i$$ $$z^2=-7+24i$$

Reciprocal of complex number

Using conjugation, the reciprocal of a nonzero complex number z=a+bi can be written as

$$\frac{1}{z}=\frac{\overline{z}}{z\overline{z}}=\frac{\overline{z}}{|z|^2}$$ $$\frac{1}{z}=\frac{\overline{z}}{a^2+b^2}$$ $$\frac{1}{z}={\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}}i$$

Example 7: Find the reciprocal of the complex number z=3+4i.

Solution:

$$\frac{1}{z}=\frac{\overline{z}}{z\overline{z}}=\frac{\overline{z}}{|z|^2}$$ $$\frac{1}{z}=\frac{3-4i}{5^2}$$ $$\frac{1}{z}={\frac{3}{25}-\frac{4}{25}i}$$ $$\frac{1}{z}=0.12-0.16i$$

Division of complex numbers

Two complex numbers z₁ and z₂ are divided as

$$\frac{z_1}{z_2}=z_1.\frac{1}{z_2}$$ $$=(a_1+b_1i)\bigg(\frac{a_2}{a_2^2+b_2^2}-\frac{b_2}{a_2^2+b_2^2}i\bigg)$$

Example 8: Divide the complex number z₁=7+8i by the complex number z₂=3+4i.

Solution:

$$\frac{z_1}{z_2}=(7+8i).\frac{1}{3+4i}$$ $$\frac{z_1}{z_2}=(7+8i)\bigg(\frac{3}{3^2+4^2}-\frac{4}{3^2+4^2}i\bigg)$$ $$\frac{z_1}{z_2}=(7+8i)(0.12-0.16i)$$ $$\frac{z_1}{z_2}=2.12-0.16i$$

Square root of a complex number

The square roots of complex number z=a+bi (with \(b \ne 0\)) are another complex numbers \(\pm (\alpha+\beta i)\), where

$$\alpha=\sqrt{\frac{a+\sqrt{a^2+b^2}}{2}}\hspace{0.2cm}and$$ $$\beta=sgn(b)\sqrt{\frac{-a+\sqrt{a^2+b^2}}{2}}$$

where, sgn is the signum function.

Example 9: Find the square root of the complex number z=3+4i.

Solution: Let square root of the complex number z be another complex numbers \(\pm (\alpha+\beta i)\),

$$\alpha=\sqrt{\frac{3+\sqrt{3^2+4^2}}{2}}=2\hspace{0.2cm}and$$ $$\beta=sgn(4)\sqrt{\frac{-3+\sqrt{3^2+4^2}}{2}}=1$$ $$\sqrt{3+4i}=2+i$$

For calculating the different properties of complex numbers using above complex number calculator, you have to first enter the complex number, after that choose the operation, then you will get the resultant complex number in the Result section.