Download Section 7.7

Section 7.7 (Complex Numbers)


Thus far we have mainly dealt with the real number system which excludes some roots of negative
numbers (square root of -4 or -9, for example, do not exist in the real number system)
This section covers a number system that contains these roots (in addition to the subset of real
numbers) and allows us to solve equations such as x2 + 1 = 0 (called the complex number system)
o
The imaginary unit i is used to denote the square root of -1 ( i2 = -1 and
1 = i )
Example: Write the following using i notation (complex number system)
 25   1 25 
(note: since

3
3i
  50
can be easily confused with
3 i we will write 3 i as i 3 )
Keep in mind that the product rule for radicals doesn’t necessarily hold true for imaginary numbers
(when multiplying, always write numbers in terms of i first)
 4  9  (4)(9)  36  6
NO
 4  9  2i(3i)  6i 2  6(1)  6 YES
Examples: Multiply / Divide
 25   1 
27   3 


2 7 
8
2

A complex number is a number that can be written in the form a + bi where a and b are real numbers
(note that the real numbers are a subset => 7.2 = 7.2 + 0i as an example)
Complex numbers can be manipulated as follows (given a + bi and c + di as complex numbers)
o a+bi = c+di if and only if the real parts are equal and the imaginary parts are equal (a=b & c=d)
o sum => add the real parts and then add the imaginary parts => (a+bi) + (c+di) = (a+c) + (b+d)i
o difference => same as sum but with subtraction => (a+bi) – (c+di) = (a-c) + (b-d)i
o product => multiply as though they are binomials and use i2 = -1 to simplify
Examples: Add / Subtract / Multiply
(5 + 2i) + (4 – 3i) =
6i – (2 – i) =
(-2 – 4i) – (-3) =
-5i * 3i =
-2i (6 – 2i) =
(3 – 4i)(6 + i) =
(1 – 2i)2 =
(6 + 5i)(6 – 5i) =


Complex numbers (a + bi) and (a – bi) are called complex conjugates and (a + bi)(a – bi) = a2 + b2 (real)
We use complex conjugates to divide a complex number (multiply top and bottom by the conjugate of
the denominator to eliminate imaginary number in the denominator)
Examples: Divide (write in the form a + bi) => (recall that division can be checked by multiplication)
3i

2  3i
6

5i
Example: Using i2 = -1, find the following higher powers of i (you might also use i4)
i1 =
i5 =
i9 =
i40 =
i2 =
i6 =
i10 =
i50 =
i3 =
i7 =
i11 =
4
i =
8
i =
12
i =
i-10 =
Bonus (+0 points): How is it possible to trace this design in one continuous movement without crossing a line
on the way?