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**Imaginary number**

Solutions of this quadratic equation x^{2} - 4 = 0 are 2 or -2 .

The solutions are real.

Solutions of the quadratic equation x^{2} + 4 = 0 are $\xc2\pm \sqrt{\text{-4}}$

The square of a real number is always non-negative.

Solutions of the quadratic equation x^{2} + 4 = 0 are not real.

$\xc2\pm \sqrt{\text{-4}}$ is written as $\xc2\pm \sqrt{\text{-4}}$ x $\sqrt{\text{-1}}$.

A number whose square is negative is called an imaginary number.

**Complex number**

Let $\sqrt{\text{-1}}$ be denoted by the letter "i", i.e. i^{2} = -1.

â‡’ x = Â± 2 x i

â‡’ x = 2i or - 2i

Therefore, the solutions of the quadratic equation x^{2} + 4 = 0 are 2i or - 2i.

These numbers are called complex numbers.

The solution of the quadratic equation ax^{2}+bx+c = 0, where D = b^{2} - 4ac < 0 , is not possible in the real number system. Such equations will always have complex roots.

A number of the form a+ib, where a and b are real numbers, is called a complex number. Such numbers are denoted by the letter 'z'.

z = a + ib

Where 'a' is called the real part of the complex number, denoted by **Re z** and, 'b' is called the imaginary part, denoted by

Ex: z = 3 + 2i, z = 5 + âˆš2i, z = -6 + ($\frac{\text{1}}{\text{3}}$)i

In z = 3 + 2i, 3 is the real part and 2 is the imaginary part.

If the imaginary part of a complex number is zero, then the number is called a purely real number.

Ex: z = 5+0i = 5

If the real part of a complex number is zero, then the number is called a purely imaginary number.

Ex: z = 0 + 7i = 7i

**Equality of complex numbers**

Consider two complex numbers z_{1 }= a+ib, z_{2} = x+iy

Two complex numbers are said to be equal, if their corresponding real parts and imaginary parts are equal.

z_{1} = z_{2}, if a = x and b = y

Ex: Find the values of x and y, if the complex numbers 3x+(2x+y)i, 9+7i are equal.

Sol:

3x+(2x+y)i = 9+7i

Real parts are equal â‡’ 3x = 9

âˆ´ x = 3

Imaginary parts are equal â‡’ 2x+y = 7

Putting, x = 3

(2x3)+y = 7

âˆ´ y = 1

**Imaginary number**

Solutions of this quadratic equation x^{2} - 4 = 0 are 2 or -2 .

The solutions are real.

Solutions of the quadratic equation x^{2} + 4 = 0 are $\xc2\pm \sqrt{\text{-4}}$

The square of a real number is always non-negative.

Solutions of the quadratic equation x^{2} + 4 = 0 are not real.

$\xc2\pm \sqrt{\text{-4}}$ is written as $\xc2\pm \sqrt{\text{-4}}$ x $\sqrt{\text{-1}}$.

A number whose square is negative is called an imaginary number.

**Complex number**

Let $\sqrt{\text{-1}}$ be denoted by the letter "i", i.e. i^{2} = -1.

â‡’ x = Â± 2 x i

â‡’ x = 2i or - 2i

Therefore, the solutions of the quadratic equation x^{2} + 4 = 0 are 2i or - 2i.

These numbers are called complex numbers.

The solution of the quadratic equation ax^{2}+bx+c = 0, where D = b^{2} - 4ac < 0 , is not possible in the real number system. Such equations will always have complex roots.

A number of the form a+ib, where a and b are real numbers, is called a complex number. Such numbers are denoted by the letter 'z'.

z = a + ib

Where 'a' is called the real part of the complex number, denoted by **Re z** and, 'b' is called the imaginary part, denoted by

Ex: z = 3 + 2i, z = 5 + âˆš2i, z = -6 + ($\frac{\text{1}}{\text{3}}$)i

In z = 3 + 2i, 3 is the real part and 2 is the imaginary part.

If the imaginary part of a complex number is zero, then the number is called a purely real number.

Ex: z = 5+0i = 5

If the real part of a complex number is zero, then the number is called a purely imaginary number.

Ex: z = 0 + 7i = 7i

**Equality of complex numbers**

Consider two complex numbers z_{1 }= a+ib, z_{2} = x+iy

Two complex numbers are said to be equal, if their corresponding real parts and imaginary parts are equal.

z_{1} = z_{2}, if a = x and b = y

Ex: Find the values of x and y, if the complex numbers 3x+(2x+y)i, 9+7i are equal.

Sol:

3x+(2x+y)i = 9+7i

Real parts are equal â‡’ 3x = 9

âˆ´ x = 3

Imaginary parts are equal â‡’ 2x+y = 7

Putting, x = 3

(2x3)+y = 7

âˆ´ y = 1