Complex Numbers and Quadratic Equations

Complex Numbers

• The square root of −1 is represented by the symbol i. It is read as iota.
  i = or i² = −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 z₁ = a + ib and z₂ = 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, z₁ = (x + 1) − 10i and z₂ = 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 i² = −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 z₁ = a + ib and z₂ = c + id are equal if a = c and b = d.

• Let's now try and solve the …