Complex Numbers

Complex Number

Complex Number

While solving the equation of the form \(ax^{2}+bx+c = 0\), the roots of the equations can take three forms which are as follows:

Two Distinct Real Roots

Similar Root

No Real roots (Complex Roots)

The introduction of complex numbers in the 16th century made it possible to solve the equation \(x^{2}+1 = 0\). The roots of the equation are of the form \(x = \pm \sqrt{-1}\) and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.
We denote \(\sqrt{-1}\) with the symbol i, where i denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b.

Algebraic Operation on Complex numbers:

  • Addition of two complex numbers
  • Subtraction of two complex number
  • Multiplication of two complex number
  • Division of two complex number

Power of Iota (i) –

Depending upon the power of i, it can take the following values

\(i^{4k+1} = i\)

\(i^{4k+2} = -1\)

\(i^{4k+3} = -i\)

\(i^{4k} = 1\)

where k can have any integral value (positive or negative).

Similarly we can find for the negative power of i, which are as follows


Multiplying and dividing the above term with i, we have

\(i^{-1}=\frac{1}{i} \times \frac{i}{i}\)

\(i^{-1}=\frac{i}{i^{2}} = \frac{i}{-1} = -i\)

Note: \(\sqrt{-1}\times \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1} = 1 \) contradicts to the fact that \(i^{2}= -1\).

Therefore for an imaginary number, \(\sqrt{a}\times \sqrt{b} \neq \sqrt{ab}\).


(i) \(\left ( z_{1}+z_{2} \right )^{2} =z_{1}^{2}+z_{2}^{2}+2z_{1}z_{2}\)

(ii) \(\left ( z_{1}-z_{2} \right )^{2} =z_{1}^{2}+z_{2}^{2}-2z_{1}z_{2}\)

(iii) \(z_{1}^{2}-z_{2}^{2} = \left ( z_{1} + z_{2} \right )\left ( z_{1} – z_{2} \right )\)

(iv) \(\left ( z_{1}+z_{2} \right )^{3} = z_{1}^{3}+3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}+z_{2}^{3}\)

(v) \(\left ( z_{1}-z_{2} \right )^{3} = z_{1}^{3}-3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}-z_{2}^{3}\)

Modulus and Conjugate of a Complex number:

Let z=a+ib be a complex number.

The Modulus of z is represented by \(\left | z \right |\).

Mathematically, \(\left | z \right |= \sqrt{a^{2}+b^{2}}\)

The conjugate of z is denoted by \(\bar{z}\).

Mathematically, \(\bar{z}= a – ib\)

Example- Express the following in a+ib form.

\(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}\).

And then find the Modulus and Conjugate of the complex number.]

Solution- Given \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}\)

z= \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i} \times \frac{1+\sqrt{3}i}{1+\sqrt{3}i} = \frac{5-3+6\sqrt{3}i}{1+3}=\frac{1}{2} + \frac{3}{2}i\)

Modulus, \(\bar{z} = \sqrt{\left ( \frac{1}{2}\right )^{2} + \left (\frac{3}{2}\right )^{2}}\)

\(\bar{z} = \sqrt{\left ( \frac{10}{4}\right )}\)

\(\bar{z} = \frac{\sqrt{10}}{2}\)

Conjugate, \(\bar{z} = \frac{1}{2} – \frac{3}{2}i\)

Argand Plane & Polar Representation of Complex Number

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis.

Quadratic Equation:

The Quadratic equation is of the form \(ax^{2}+bx+c=0\)

where the roots of the equation are given as

\(\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

or, \(\large x= \frac{-b \pm \sqrt{D}}{2a}\)

where Discriminant (D)\( = b^{2}-4ac\)

For an equation having complex roots, D<0

Example- Find the roots of the given quadratic equation


Solution- Given \(27x^{2}-10x+1=0\)

The roots of the equation is given as

\(\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

\(\large x= \frac{-(-10)\pm \sqrt{(-10)^{2}-4\times 27 \times 1}}{2 \times 27}\)

\(\large x= \frac{10\pm \sqrt{-8}}{54}\)

\(\large x= \frac{5}{27} \pm \frac{\sqrt{2}i}{27}\)<

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Practise This Question

If three complex numbers are in A.P., then they lie on