Imaginary Numbers
Equations such as x2 + 26 = 1 have no solutions in the set of real numbers.
In order to solve such equations, it is necessary to use Imaginary Numbers.
The symbol i represents an imaginary number defined as i =
Using the number i, we can solve
the equation above:
1, and i2 = -1.
x2 + 26 = 1
x2
= -25
x
= √−25
x
= √25 × √−1
x
= 5i
For any positive real number n, √−𝑛 = √𝑛∙i.
Imaginary numbers are not real numbers, so some properties of real numbers
do not apply to imaginary numbers. One such property is the Product Rule for
𝑛
𝑛
𝑛
𝑛
𝑛
Radicals: √𝑎 √𝑏 = √𝑎𝑏 applies only if √𝑎 and √𝑏 exist as real numbers.
This means the rule does not apply to √−49 ∙ √−4 because √−49 and √−4
do not define real numbers. √−49 ∙ √−4 = √49 i ∙ √4 i
= 7i ∙2i
= 14i2
= –14
Note: In this case, applying the Product Rule of Radicals would give you 14 instead of -14.
Using powers of i
i = √−1 = i
i2 = -1
i3 = i2∙i = -1i = -i
i4 = i2∙i2 = (-1)(-1) = 1
we can create this chart of
Imaginary Numbers with
exponents from 1 to 4
Imaginary Number Chart
i1
i2
i3
i4
Examine these higher powers of i….. notice a pattern?
i5 = i4∙i1 = i
= i
6
4 2
i = i ∙i = (1)(-1)
= -1
7
4 3
i = i ∙i = (1)(-i)
= -i
8
4 4
i = i ∙i = (1) (1)
=1
=
=
=
=
i
-1
-i
1
Imaginary Numbers
The pattern is the same in the Imaginary Number Chart. Notice the sequence
i, -1, - i, 1 repeats as the power of i increases. This shows us a method for
evaluating imaginary numbers. In short, we divide by 4 to get the remainder.
Here’s how:
Steps:
a) Divide the exponent by 4.
b) Look at the remainder after dividing.
c) Attach the remainder number to an " i."
d) Check the imaginary number chart to see
what corresponds to the remainder number.
Example: Evaluate i27
6
4 27
Solution:
-24
3
Imaginary Number Chart
i1
=
i
i2
=
-1
i3
=
-i
i4
The remainder is 3.
Attach it to an " i ".
i3
Referring to the
chart, i3 = – i
Therefore, i2 7 = –i.
=
1
The pattern i, -1, - i, 1 continues as the exponent on i increases by 1. To
remember this pattern, think of the phrase, “I won! I won!”