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Complex Numbers

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(yx) 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 yx = −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 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(yx) 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 yx = −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 z1 = a + ib and z2 = c + id is defined as

z1 + z2 = (a + c) + i(b + d)
For example: (4 + 3i) + (…

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