Introduction
Earlier in the text we discussed about the set of real numbers. We have also seen how real numbers can be represented by points in a straight line, called the real line or the number line. Every real number corresponds to a point in the real line and conversely. One important fact about real number is
“The square of a real number is never negative.”
A problem occurs when we solve the equation of the type x² + 4 = 0 Because there is no value of x in the real number system that the square of x is negative. In order to overcome this difficulty, we have to extend the system of numbers. This is done by introducing new set of numbers called complex numbers. It is believed that Cardon used complex numbers in 1546.
Complex Numbers
An ordered pair of real numbers is defined to be a Complex Number. Thus, if a and b are real numbers, the ordered pair written as (a, b) is a complex number. The first number, a is called the real part and the second number, b the imaginary part of the complex number.
Since each ordered pair can be located by a point in the plane so, the complex number (a, b) can be located by a point in the plane. The real part a taken along the x-axis is known as the real axis and the imaginary part b, taken along the y-axis is known as the imaginary axis. The plane in which the complex number is plotted is known as the complex plane.
A complex number is usually denoted by a single letter such as z, w etc. If z = (a, b) is a complex number, the real and the imaginary parts of z are respectively denoted by Re(z) and Im(z) . Thus Re(z) = a and Im(z) = b. Two complex numbers (a, b) and (c, d) are said to be equal if and only if a = c and b = d.
The Imaginary Unit
We have seen that the complex number (a, 0) plays the same role as the real number a Hence the real number a can be treated as the complex number (a, 0) with the imaginary part 0. Thus the complex number (1, 0) is same as the real number 1. The complex number (0, 1) with the real part zero and the imaginary part 1 is denoted by the letter ‘7’ (the iota in the Greek alphabet) and it is called the imaginary unit. The symbol i was introduced by Euler in 1777. The complex number (0, 1) is denoted by i and is called the imaginary unit.
Theorems
Theorem I
i² = – 1
Proof:-
i² = i.i = (0, 1)(0, 1)
= (0 – 1, 0 + 0)
= (-1,0)
= – (1,0)
= -1
Theorem II
If a and b are real numbers then the complex number (a, b) can be written as (a + ib) where i = (0, 1), and i² = -1
Proof:-
a+b = (a,0) + (0, 1) (b, 0)
= (a,0) + (0,b)
= (a, b)
So, if a and b are real numbers then (a + ib) is said to be a complex number where i² = -1.
Exercise – 7.1
Introduction
Earlier in the text we discussed about the set of real numbers. We have also seen how real numbers can be represented by points in a straight line, called the real line or the number line. Every real number corresponds to a point in the real line and conversely. One important fact about real number is
“The square of a real number is never negative.”
A problem occurs when we solve the equation of the type x² + 4 = 0 Because there is no value of x in the real number system that the square of x is negative. In order to overcome this difficulty, we have to extend the system of numbers. This is done by introducing new set of numbers called complex numbers. It is believed that Cardon used complex numbers in 1546.
Conjugate of a Complex Number
Let z = a + ib be a complex number. Then the conjugate of the complex number z denoted by ˉz = a – ib =(a – b) Similarly if z = a – ib be the given complex number, then its conjugate ˉz is given by ˉz = a + ib.
Absolute Value of a Complex Number
The absolute value of a complex number z = a + ib is defined as the non-negative real number (a² + b²)½. It is denoted by |z| or |a + ib |. The absolute value of a complex number is also known as the modulus of the complex number.
Powers of i
We have defined i as that number whose square is -1, and it is the imaginary unit. Any real multiple of i is an imaginary number. Since,
i² = – 1 we have
i³ = i²×i = (- 1)× i = – i
i⁴ = (i²)² = (- 1)² = 1
i⁵ = (i⁴)× i = (1)× i = i
i⁶ = (i²)³ = (- 1)³ = – 1
i¹⁰ = (i²)⁵ = (- 1)⁵ = – 1
