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

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

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

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.

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

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

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

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

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

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

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

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.

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

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

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

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