Download complex number - Miami Beach Senior High School

Bell Work #1
Solve the quadratic equation x2 + 1 = 0.
Solving for x , gives x2 = – 1
x2   1
x  1
We make the following definition:
i  1
Complex Numbers
i  1
2
Note that squaring both sides yields: i  1
therefore i 3  i 2 * i1  1* i  i
and i 4  i 2 * i 2  (1) * (1)  1
so
and
i  i * i  1* i  i
5
4
i  i * i  1* i  1
6
4
And so on…
2
2
Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Real Numbers
Imaginary
Numbers
Complex Numbers
Definition of a Complex Number
If a and b are real numbers, the number a + bi is a
complex number, and it is said to be written in
standard form.
If b = 0, the number a + bi = a is a real number.
If a = 0, the number a + bi is called an imaginary
number.
Write the complex number in standard form
1   8  1  i 8  1  i 4  2  1  2i 2
Addition and Subtraction of Complex
Numbers
If a + bi and c +di are two complex numbers written
in standard form
Sum:
Difference:
(a  bi )  (c  di )
(a  bi )  (c  di)
Perform the subtraction and write the answer
in standard form.
( 3 + 2i ) – ( 6 + 13i )
3 + 2i – 6 – 13i
–3 – 11i
8   18  4  3i 2 
8  i 9  2  4  3i 2 
8  3i 2  4  3i 2
4
Multiplying Complex Numbers
Multiplying complex numbers is similar to
multiplying polynomials and combining like terms.
Perform the operation and write the result in
standard form.( 6 – 2i )( 2 – 3i )
F
O
I
L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i
Consider ( 3 + 2i )( 3 – 2i )
9 – 6i + 6i – 4i2
9 – 4( -1 )
9+4
13
This is a real number. The product of two
complex numbers can be a real number.