Here you will learn what is the complex number class 11 and equality of complex numbers with examples.
Let’s begin –
Complex Number Class 11
If a, b are two real numbers, then a number of the form a + ib is called aa complex number.
Example : 7 + 2i, -1 + i, 3 – 2i, 0 + 2i, 1 + 0i etc. are complex numbers
Real and imaginary parts of a complex number : If z = a + ib is a complex number, then ‘a’ is called the real part of z and ‘b’ is known as the imaginary part of z.
The real part of z is denoted Re (z) and the imaginary part by Im (z).
Example : If z = 3 – 4i, then Re (z) = 3 and Im (z) = -4.
Purely real and purely imaginary complex numbers : A complex number z is purely real if its imaginary part is zero i.e. Im (z) = 0 and purely imaginary if its real part is zero i.e. Re (z) = 0.
Set of Complex Numbers : The set of all complex numbers is denoted by C i.e. C = {a + ib : a, b \(\in\) R}.
Since a real number ‘a’ can be written as a + 0i. Therefore, every real number is a complex number number. Hence, R \(\subset\) C, where R is the set of all real numbers.
Equality of Complex Numbers
Two Complex numbers \(z_1\) = \(a_1 + ib_1\) and \(z_2\) = \(a_2 + ib_2\) are equal if
\(a_1\) = \(a_2\) and \(b_1\) = \(b_2\)
i.e. \(Re(z_1)\) = \(Re(z_2)\) and \(Im(z_1)\) = \(Im(z_2)\)
Example : If \(z_1\) = 2 – iy and \(z_2\) = x + 3i are equal, find x and y.
Solution : We have,
\(z_1\) = \(z_2\)
\(\implies\) 2 – iy = x + 3i \(\implies\) 2 = x and -y = 3 \(\implies\) x = 2 and y = -3
