if z=2+3i and w=5-4i show that zw bar= z bar w bar
Answers
Answer:
1 Complex Numbers
1.1 Algebra of Complex Numbers
The number i is declared by law to satisfy the equation
i
2 = −1.
A complex number is an expression of the form x + yi, with x and y real
numbers. Complex numbers are added, subtracted, and multiplied as with
polynomials. Examples
(2 + 3i) + (5 − 6i) = 7 − 3i.
(2 + 3i) − (5 − 6i) = −3 + 9i.
(2 + 3i)(5 − 6i) = 10 + 15i − 12i − 18i
2 = (10 + 18) + i(15 − 12) = 28 + 3i.
Division is trickier. Observe that
(2 − 3i)(2 + 3i) = 4 + 9 + 6i − 6i = 13.
That means that
13
2 + 3i
= 2 − 3i.
Divide both sides by 13:
1
2 + 3i
=
2 − 3i
13
.
In general,
1
a + bi =
a − bi
a
2 + b
2
.
Now for division
5 − 6i
2 + 3i
= (5 − 6i) ×
1
2 + 3i
=
(5 − 6i) ×
2 − 3i
13
=
(5 − 6i)(2 − 3i)
13
= −8/13 − 27/13i.
Problems: Compute A + B and A − B and AB and A/B for
• A = 2i and B = 1 − i.
• A = 1 + 2i and B = 4 − 3i
• A = 2 − 5i and B = −2 + 7i
Step-by-step explanation: