Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Aim: How do we divide complex numbers?
Do Now:
6
3 5Express as an equivalent fractionwith a rational denominator.
6
3 5
3 5
3 5
6(3 5)
4
3(3 5)
2
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Identities & Inverses
Multiplicative identity:real numbers -complex numbers -
Multiplicative inverse:real numbers -complex numbers -
ex. (2 + 3i)(1 + 0i) = 2 + 3i
= 1
(n)(1/n) = 1
real numbers
(3)(1/3) = 1
ex . 2 3i 1
2 3i
1
complex numbers
(a + bi)(1/(a + bi) = 1
11 + 0i
1/n1/(a + bi)
ex.
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
10
8
7
2i
Rationalizing the Denominator
7
2i
10
8
8
8
10 8
( 8)2
10 8
8
Multiply fraction by a form of the identity
element 1.
Simplify if possible
10 8
810 2 4
810 2 2
8
5 2
2
10
3rational number
irrational number
7
2i
i
i
7i
2i 2
Multiply fraction by a form of the identity
element 1.
Simplify if possible
means to removethe complex number (i) from the denominator
i
i
7i
2
recall:
8
8
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Rationalizing the Denominator (binomial)
1
2 3i
the reciprocal of 2 + 3i is not in complex number form
We need to change the fraction andremove the imaginary number
from the denominator; we need torationalize the denominator: how?
2 3i
4 9
2 3i
13
Use the conjugate of the complex number
The product of two complex numbersthat are conjugates is a real number.
(a + bi)(a – bi) = a2 + b2
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Rationalizing the Denominator
2 3i 1
2 3i
1
multiplicative inverse
unrationalized denominator
rationalized denominator
Show that (3 – i) and are inverses.
3
10
1
10i
3 i 3
10
1
10i
33
10
1
10i
i
3
10
1
10i
9
10
3
10i
3
10i
i2
10
9
10
1
10
1
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Dividing Complex Numbers
Divide 8 + i by 2 – i
write in fractional form
rationalize the fractionby multiplying byconjugate of denom.
8 i
2 – i
2 i
2 i
simplify
15 10i
5
3 2i
8 i
2 – i
16 10i i2
4 1
check:
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Model Problems
Write the multiplicative inverse of 2 + 4iin the form of a + bi and simplify.
write inverse as fraction
1
2 4i
rationalize by multiplying by conjugate
1
2 4i
2 4i
2 4i
simplify
2 4i
4 16
1
10
1
5i
Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig.
Model Problems
Divide and check: (3 + 12i) ÷ (4 – i)
write in fractional form
rationalize the fractionby multiplying byconjugate of denom.
3 12i
4 – i
4 i
4 i
simplify
51i
17
3i
3 12i
4 – i
12 51i 12i 2
16 1
check: