If Z1 2 3i and Z2 4 5i find Z1 Z2 Z1 Z2 Z1Z2 Z1Z2Solut

If Z_1 = 2 + 3i, and Z_2 = 4 - 5i, find: Z_1 + Z_2, Z_1 - Z_2, Z_1Z_2, Z_1/Z_2.

Solution

z₁ = 2 + 3i , z₂ = 4 - 5i

a) z₁ + z₂ = (2 + 3i) + (4 - 5i)

==> z₁ + z₂ = (2 + 4) + (3 - 5)i

==> z₁ + z₂ = 6 - 2i

b) z₁ - z₂ = (2 + 3i) - (4 - 5i)

==> z₁ - z₂ = (2 - 4) + (3 -(-5)) i

==> z₁ + z₂ = -2 + 8i

c) z₁z₂ = (2 + 3i)(4 - 5i)

==> 2(4) + 2(-5i) + 3i(4) + 3i(-5i)

==> 8 - 10i + 12i - 15i²

==> 8 + 2i - 15(-1)              since i² = -1

==> 8 + 2i + 15

==> 23 + 2i

Hence z₁z₂ = 23 + 2i

d) z₁/z₂ = (2 + 3i)/(4 - 5i)

Rationalising denominator i.e, multiplying and dividing by (4 + 5i)

==> [(2 + 3i)/(4 - 5i)] [ (4 + 5i) /(4 + 5i)]

==> (2 + 3i)(4 + 5i) / [ (4 - 5i)(4 + 5i) ]

==> [ 2(4) + 2(5i) + 3i(4) + 3i(5i)] /[ (4)² - (5i)²]           since (a - b)(a + b) = a² - b²

==> [8 + 10i + 12i + 15i²]/[16 - 25i²]

==> [8 + 22i + 15(-1)]/[16 - 25(-1)]

==> (-7 + 22i) /(41)

==> -7/41 + 22i/41

Hence z₁/z₂ = -7/41 + 22i/41