aim: dividing complex numbers course: adv. alg. & trig. aim: how do we divide complex numbers?...

$: Aim: Dividing Complex Numbers Course: Adv. Alg. & Trig. Aim: How do we divide complex numbers? Do Now: Express as an equivalent fraction with a rational$
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


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.


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


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


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


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:


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


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:

aim: dividing complex numbers course: adv. alg. & trig. aim: how do we divide complex numbers?...

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