Let z = 8 + 3i and w = 7 + 2i, find z/w and z.w

The sum of a real number and an imaginary number is termed a Complex number. These are the numbers that can be written in the form of a+ib, where a and b both are real numbers. It is denoted by z. Here in complex number form the value ‘a’ is called the real part which is denoted by Re(z), and ‘b’ is called the imaginary part Im(z). It is also called an imaginary number. In complex numbers form a +bi, ‘i’ is an imaginary number called “iota”.

The value of i is (√-1) or we can write as i² = -1.

For example

• 3 + 4i is a complex number, where 3 is a real number (Re) and 4i is an imaginary number (Im).
• 2 + 5i is a complex number where 2 is a real number (Re) and  5i is an imaginary number (im)

The Combination of real number and imaginary number is called a Complex number.

For z = 8 + 3 i and w = 7 + 2 i, find z/w. That is, determine (8 + 3i) (7 + 2i) and simplify as much as possible, writing the result in the form a + bi, where a and b are real numbers. 

Solution:

Given: z = 8 + 3 i

w = 7 + 2i

To find z/w,

= (8 + 3i )/ (7 + 2i)

Multiplying the numerator and denominator with the conjugate of denominators.

= {(8 + 3i) / (7 + 2i) } × {(7 – 2i) / (7 – 2i)}

= {(8 + 3i)(7 – 2i)} / {(7 + 2i)(7 – 2i)}

= {56 – 16i + 21i – 6i²} / {(7)² – (2i)²}

= {56 + 5i – 6(-1)} / {49 – 4(i² )}

= {56 + 5i +6} / {49 +4}

= (62 +5i) / 53

z/w = 62/53 + 5/53i

Now, (8 + 3i) (7 + 2i)

= {56 + 16i + 21i + 6i²}

= {56 + 37i + 6(-1)}

= {56 + 5i – 6}

= 50 + 5i

Similar problems

Question 1: Express (2 – i)/(1 + i) in standard form?

Solution:

Given: (2 – i)/(1 + i)

Multiplying the numerator and denominator with the conjugate of denominators,

= {(2 – i)/(1 + i) × (1 – i)/(1 – i)}

= {(2 – i)(1 – i)} / {(1)² – (i)²}

= {2 – 2i – i – i²} / (1-i²)

= {2 – 3i – (-1)} / (1+1)

= ( 3 – 3i) / 2

= 3/2 – 3/ 2 i

Question 2:  Simplify in form of a + ib, (-5i)(2/8i)

Solution:

Given: (-5i)(2/8i)

= (- 10/8 )i²

= (- 10/8 )(-1)

= 10/8 + 0i

=  5/4 + 0i

Question 3:  For z = 3 + 3 i and w = 5 + 2 i, find z/w. That is, determine (3 + 3i) (5 + 2i) and simplify as much as possible, writing the result in the form a+bi, where a and b are real numbers.

Solution:

Given: z = 3 + 3 i

w = 5 + 2i

To find z/w

= (3 + 3i )/ (5 + 2i)

Multiplying the numerator and denominator with the conjugate of denominators.

= {(3 + 3i) / (5 + 2i)} × {(5 – 2i) / (5 – 2i)}

= {(3 + 3i)(5 – 2i)} / {(5 + 2i)(5 – 2i)}

= {15 – 6i +15i – 6i²} / {(5)² – (2i)²}

= {15 + 9i – 6(-1)} / {49 – 4(i² )}

= {15 + 9i + 6} / {49 + 4}

= (21 + 9i) / 53

 z/w = 21/53 + 9/53i

Now (3 + 3i) (5 + 2i)

= {15 + 6i +15i + 6i²}

= {15 + 21i + 6(-1)}

= {15 + 21i – 6}

= 9 + 21i

Question 4: Perform the indicated operation and write the answer in standard form: (2 – 14i)(2 + 14i)

Solution: 

Given: (2 – 14i)(2 + 14i)

= {(4 + 28i – 28i -196i² )}

= ( 4 + 196)

= 200 + 0i

Question 5: Perform the indicated operation and write the answer in standard form: (7 + 2i) × (5 – 4i)

Solution: 

(7 + 2i) × (5 – 4i)

= (35 – 28i + 10i – 8i²)

= {35 – 18i – 8(-1)} 

= 35 – 18i + 8

= 43 – 18i

Question 6: Simplify -3 + 8i?

Solution:

The multiplicative inverse of a complex number z is simply 1/z.

It is denoted as: 1 / z  or  z – 1 (Inverse of z)

Here z = -3 + 8i

Therefore z = 1/z

= 1 / (-3 + 8i)

Now rationalizing,

= 1/(-3 + 8i) × (-3 – 8i)/(-3 – 8i)

= (-3 – 8i ) / {(-3)² – 8²i²}

= (-3 – 8i) / {9 + 64}

= (-3 – 8i)/ (73)

= -3/73 – 8i/73

Question 7: What is the answer to the following problem, (-3i)(9i)(-1).

Solution: 

Given: (-3i)(9i)(-1)

= -3i × 9i × (-1)

= -27i² × -1 {i² = -1}

= -27 (-1) × -1

= 27 × -1

= -27 + 0i