1. 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?

3. 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

4. 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:

5. 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

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

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

8. 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

9. 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

10. 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:

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

12. 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

13. Subtracting Complex Numbers
What is the additive inverse of i?
-(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

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

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

16. 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

17. 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

18. 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.

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

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

21. 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

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

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

24. 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

25. 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

26. 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.

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

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

29. 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

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