# Complex Numbers

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

$i^{-1}=\frac{1}{i}$

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, $\sqrt{a}\times \sqrt{b} \neq \sqrt{ab}$.

## Identities:

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

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.

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 $27x^{2}-10x+1=0$ Solution- Given $27x^{2}-10x+1=0$ The roots of the equation is given as $\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$ $\large x= \frac{-(-10)\pm \sqrt{(-10)^{2}-4\times 27 \times 1}}{2 \times 27}$ $\large x= \frac{10\pm \sqrt{-8}}{54}$ $\large x= \frac{5}{27} \pm \frac{\sqrt{2}i}{27}$<