Activator When I take five and add six, I get eleven, but when I take six and add seven, I get one. Who/What am I?

What are imaginary and complex numbers? Graph it Solve for x: x2 + 1 = 0 ? What number when multiplied by itself gives us a negative one? parabola does not intersect x-axis - NO REAL ROOTS No such real number

i Imaginary Numbers If is not a real number, then is a non-real or Definition: A pure imaginary number is any number that can be expressed in the form bi, where b is a real number such that b ≠ 0, and i is the imaginary unit. i b = 5 In general, for any real number b, where b > 0:

i2 = i2 = i Powers of i –1 –1 i0 = 1 i1 = i i2 = –1 i3 = –i i4 = 1 If i2 = – 1, then i3 = ? = i2 • i = –1( ) = –i i3 i4 = i6 = i8 = i5 i7 i2 • i2 = (–1)(–1) = 1 = i4 • i = 1( ) = i i4 • i2 = (1)(–1) = –1 = i6 • i = -1( ) = –i i6 • i2 = (–1)(–1) = 1 What is i82 in simplest form? 82 ÷ 4 = 20 remainder 2 equivalent to i2 = –1 i82

A little saying to help you remember Once I Lost one Missing eye  

i Properties of i Addition: 4i + 3i = 7i Subtraction: 5i – 4i = i Multiplication: (6i)(2i) = 12i2 = –12 Division:

a + bi Complex Numbers A complex number is any number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Definition: a + bi real numbers pure imaginary number Any number can be expressed as a complex number: 7 + 0i = 7 a + bi 0 + 2i = 2i

The Number System Complex Numbers i -i i3 i9 Real Numbers Rational Numbers i i75 -i47 Irrational Numbers Integers Whole Numbers Counting Numbers 2 + 3i -6 – 3i 1/2 – 12i

Model Problems Express in terms of i and simplify: = 10i = 4/5i Write each given power of i in simplest terms: i49 = i i54 = -1 i300 = 1 i2001 = i Add: Multiply: Simplify:

How do we add and subtract complex numbers? Do Now: Simplify:

Adding Complex Numbers (2 + 3i) + (5 + i) = (2 + 5) + (3i + i) = 7 + 4i In general, addition of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i Combine the real parts and the imaginary parts separately. Find the sum of convert to complex numbers combine reals and imaginary parts separately

Subtracting Complex Numbers What is the additive inverse of 2 + 3i? -(2 + 3i) or -2 – 3i Subtraction is the addition of an additive inverse (1 + 3i) – (3 + 2i) = (1 + 3i) + (-3 – 2i) = -2 + i In general, subtraction of complex numbers: (a + bi) – (c + di) = (a – c) + (b – d)i Subtract change to addition problem combine reals and imaginary parts separately

Model Problems Add/Subtract and simplify: (10 + 3i) + (5 + 8i) = 15 + 11i (4 – 2i) + (-3 + 2i) = 1 Express the difference of in form a + bi

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: