Imaginary Numbers

Today in class we learn about imaginary numbers. Imaginary numbers are number that you can write down, but they're not possible to calculate.

Examples of imaginary numbers: √-2, √-4, 0.999999...., and etc...

Here are some things that you need to know:
Properties of " i ": i = √-1 and i² = -1 (note: i is not a variable like x or y, it never changes)

How to simplify:
ex. 1)
i² x i³ = (-1)i³ = (-1)(i²)i¹
^ here you change the i²s into (-1)
= (-1)(-1)i
^ and next you multiply the (-1)s
= i

ex.2)
5 + √-8 = 5 + √(-1)(8)
^ first you factor the (-1) out of the (-8)
= 5 + √-1 √8
^ next you square root the (-1), since the √-1 is i (reminder i = √-1)
= 5 + i√4 x 2
^ finally you square root the 8
= 5 + 2i√2

ex.3)
√25 + √-16 = 5 + √(-1)(26)
^ basically you do the same like ex.2
= 5 + √-1 √16
= 5 + 4i

ex.4)
(2i + 3)(i² - 5)
^ first you distribute the factors
= 2i³ - 10i + 3i² - 15
^ now you change the i²s into -1 (note: doing this first makes it easier)
= 2(i²)i - 10i = 3(-1) - 15
= 2(-1)i - 10i -3 -15
= -12i - 18

Trial questions:
1. 2i + 5 / i + 1 = (2i + 5) /(i + 1) x ( i - 1) /( i - 1)
^ in order to get rid of the i in the denominator you need to multiply it by the difference of squares
= 2(i²) - 2i + 5i - 5 /(i²) - i - 1 + 1
^ the (- i) and the (+ i) cancel each other
= 2(-1)- 2i + 5i - 5 / (-1) - 1
= -7 + 3i /-2

2. (3i² - 4)⁴ = (3(-1) - 4)⁴
^ the easiest thing to do is to change the i² into -1 and do the rest
= (-7)⁴
= 2401

That is all we did for half the class and the other half is our homework, well that is basically what we did all class.

Homework: Complex Numbers worksheets (all of the first page and the rest are even numbers questions)
Due: Thursday, March 15, 2007

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