1. How do we divide complex numbers?
Do Now:
Express as an equivalent fraction
with a rational denominator.

2. Multiplicative identity: real numbers - 1 1 + 0i complex numbers - = 1
Identities & Inverses
Multiplicative identity:
real numbers - 1
1 + 0i
complex numbers -
= 1
ex. (2 + 3i)(1 + 0i) = 2 + 3i
Multiplicative inverse:
real numbers -
1/n
complex numbers -
1/(a + bi)
ex.
(n)(1/n) = 1
real numbers
(3)(1/3) = 1
complex numbers
(a + bi)(1/(a + bi) = 1

3. Rationalizing the Denominator
means to remove
the complex number (i)
from the denominator
recall:
rational number
irrational number
Multiply fraction by a form of the identity element 1.
Multiply fraction by a form of the identity element 1.
Simplify if possible
Simplify if possible

4. Rationalizing the Denominator (binomial)
the reciprocal of 2 + 3i is not
in complex number form
We need to change the fraction and
remove the imaginary number
from the denominator; we need to
rationalize the denominator: how?
Use the conjugate of the complex number
(a + bi)(a – bi) = a2 + b2
The product of two complex numbers
that are conjugates is a real number.

5. Rationalizing the Denominator
multiplicative inverse
unrationalized denominator
rationalized denominator
Show that (3 – i) and are inverses.

6. Dividing Complex Numbers
Divide 8 + i by 2 – i
write in fractional form
rationalize the fraction
by multiplying by
conjugate of denom.
simplify
check:

7. Model Problems
Write the multiplicative inverse of 2 + 4i
in the form of a + bi and simplify.
write inverse as fraction
rationalize by
multiplying by conjugate
simplify

8. Model Problems
Divide and check: (3 + 12i) ÷ (4 – i)
write in fractional form
rationalize the fraction
by multiplying by
conjugate of denom.
simplify
check:

9. How do we divide complex numbers?
Do Now:
Express as an equivalent fraction
with a rational denominator.

10. Multiplicative identity: real numbers - 1 1 + 0i complex numbers - = 1
Identities & Inverses
Multiplicative identity:
real numbers - 1
1 + 0i
complex numbers -
= 1
ex. (2 + 3i)(1 + 0i) = 2 + 3i
Multiplicative inverse:
real numbers -
1/n
complex numbers -
1/(a + bi)
ex.
(n)(1/n) = 1
real numbers
(3)(1/3) = 1
complex numbers
(a + bi)(1/(a + bi) = 1

11. Rationalizing the Denominator
means to remove
the complex number (i)
from the denominator
recall:
rational number
irrational number
Multiply fraction by a form of the identity element 1.
Multiply fraction by a form of the identity element 1.
Simplify if possible
Simplify if possible

12. Rationalizing the Denominator (binomial)
the reciprocal of 2 + 3i is not
in complex number form
We need to change the fraction and
remove the imaginary number
from the denominator; we need to
rationalize the denominator: how?
Use the conjugate of the complex number
(a + bi)(a – bi) = a2 + b2
The product of two complex numbers
that are conjugates is a real number.

13. Rationalizing the Denominator
multiplicative inverse
unrationalized denominator
rationalized denominator
Show that (3 – i) and are inverses.

14. Dividing Complex Numbers
Divide 8 + i by 2 – i
write in fractional form
rationalize the fraction
by multiplying by
conjugate of denom.
simplify
check:

15. Model Problems
Write the multiplicative inverse of 2 + 4i
in the form of a + bi and simplify.
write inverse as fraction
rationalize by
multiplying by conjugate
simplify

16. Model Problems
Divide and check: (3 + 12i) ÷ (4 – i)
write in fractional form
rationalize the fraction
by multiplying by
conjugate of denom.
simplify
check: