Notes On Introduction to Complex Numbers - CBSE Class 11 Maths
Imaginary number Solutions of this quadratic equation  x2 - 4 = 0 are 2 or -2 . The solutions are real. Solutions of the quadratic equation x2 + 4 = 0 are $±\sqrt{\text{-4}}$ The square of a real number is always non-negative. Solutions of the quadratic equation x2 + 4 = 0 are not real. $±\sqrt{\text{-4}}$ is written as 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. i2 = -1. ⇒ x = ± 2 x i ⇒ x = 2i or - 2i Therefore, the solutions of the quadratic equation x2 + 4 = 0 are 2i or - 2i. These numbers are called complex numbers. The solution of the quadratic equation ax2+bx+c = 0, where D = b2 - 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 Im z. Ex: z = 3 + 2i, z = 5 + √2i, 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 z1 = a+ib, z2 = x+iy Two complex numbers are said to be equal, if their corresponding real parts and imaginary parts are equal. z1 = z2, 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 ⇒ 3x = 9 ∴ x = 3 Imaginary parts are equal ⇒ 2x+y = 7 Putting, x = 3 (2x3)+y = 7 ∴ y = 1

#### Summary

Imaginary number Solutions of this quadratic equation  x2 - 4 = 0 are 2 or -2 . The solutions are real. Solutions of the quadratic equation x2 + 4 = 0 are $±\sqrt{\text{-4}}$ The square of a real number is always non-negative. Solutions of the quadratic equation x2 + 4 = 0 are not real. $±\sqrt{\text{-4}}$ is written as 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. i2 = -1. ⇒ x = ± 2 x i ⇒ x = 2i or - 2i Therefore, the solutions of the quadratic equation x2 + 4 = 0 are 2i or - 2i. These numbers are called complex numbers. The solution of the quadratic equation ax2+bx+c = 0, where D = b2 - 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 Im z. Ex: z = 3 + 2i, z = 5 + √2i, 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 z1 = a+ib, z2 = x+iy Two complex numbers are said to be equal, if their corresponding real parts and imaginary parts are equal. z1 = z2, 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 ⇒ 3x = 9 ∴ x = 3 Imaginary parts are equal ⇒ 2x+y = 7 Putting, x = 3 (2x3)+y = 7 ∴ y = 1

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