Complex Numbers
Complex Numbers

The square root of −1 is represented by the symbol i. It is read as iota.
i = or i^{2} = −1 
Any number of the form a + ib, where a and b are real numbers, is known as a complex number. A complex number is denoted by z.
z = a + ib 
For the complex number z = a + ib, a is the real part and b is the imaginary part. The real and imaginary parts of a complex number are denoted by Re z and Im z respectively.

For complex number z = a + ib, Re z = a and Im z = b


A complex number is said to be purely real if its imaginary part is equal to zero, while a complex number is said to be purely imaginary if its real part is equal to zero.

For e.g., 2 is a purely real number and 3i is a purely imaginary number.


Two complex numbers are equal if their corresponding real and imaginary parts are equal.

Complex numbers z_{1} = a + ib and z_{2} = c + id are equal if a = c and b = d.

 Let's now try and solve the following puzzle to check whether we have understood this concept.
Solved Examples
Example 1:
Verify that each of the following numbers is a complex number.
Solution:
can be written as, which is of the form a + ib. Thus, is a complex number.
is not of the form a + ib. But it is known that every real number is a complex number.
Thus, is a complex number.
1 − 5i is of the form a + ib. Thus, 1 − 5i is a complex number.
Example 2:
What are the real and imaginary parts of the complex number?
Solution:
The complex number can be written as, which is of the form a + ib.
Re z = a = and Im z = b = –
Example 3:
For what values of x and y, z_{1} = (x + 1) − 10i and z_{2} = 19 + i(y − x) represent equal complex numbers?
Solution:
Two complex numbers are equal if their corresponding real and imaginary parts are equal.
For the given complex numbers,
x + 1 = 19 and y − x = −10
⇒ x = 18 and y − 18 = −10
⇒ x = 18 and y = 8
Thus, the values of x and y are 18 and 8 respectively.

The square root of −1 is represented by the symbol i. It is read as iota.
i = or i^{2} = −1 
Any number of the form a + ib, where a and b are real numbers, is known as a complex number. A complex number is denoted by z.
z = a + ib 
For the complex number z = a + ib, a is the real part and b is the imaginary part. The real and imaginary parts of a complex number are denoted by Re z and Im z respectively.

For complex number z = a + ib, Re z = a and Im z = b


A complex number is said to be purely real if its imaginary part is equal to zero, while a complex number is said to be purely imaginary if its real part is equal to zero.

For e.g., 2 is a purely real number and 3i is a purely imaginary number.


Two complex numbers are equal if their corresponding real and imaginary parts are equal.

Complex numbers z_{1} = a + ib and z_{2} = c + id are equal if a = c and b = d.

 Let's now try and solve the following puzzle to check whether we have understood this concept.
Solved Examples
Example 1:
Verify that each of the following numbers is a complex number.
Solution:
can be written as, which is of the form a + ib. Thus, is a complex number.
is not of the form a + ib. But it is known that every real number is a complex number.
Thus, is a complex number.
1 − 5i is of the form a + ib. Thus, 1 − 5i is a complex number.
Example 2:
What are the real and imaginary parts of the complex number?
Solution:
The complex number can be written as, which is of the form a + ib.
Re z = a = and Im z = b = –
Example 3:
For what values of x and y, z_{1} = (x + 1) − 10i and z_{2} = 19 + i(y − x) represent equal complex numbers?
Solution:
Two complex numbers are equal if their corresponding real and imaginary parts are equal.
For the given complex numbers,
x + 1 = 19 and y − x = −10
⇒ x = 18 and y − 18 = −10
⇒ x = 18 and y = 8
Thus, the values of x and y are 18 and 8 respectively.
 The addition of two complex numbers z_{1} = a + ib and z_{2} = c + id is defined as
z_{1} + z_{2} = (a + c) + i(b + d)
For example: (4 + 3i) + (…
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