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Introduction to Complex Numbers

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Introduction to Complex Numbers - Lesson 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 ± -4

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

Solutions of the quadratic equation x2 + 4 = 0 are not real.

± -4  is written as ± -4   x -1 .

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

Complex number

Let   -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 + ( 1 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.


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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