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This module contains some example problems involving the manipulation i, the imaginary number.

Let’s begin with a few very simple exercises designed to show how we apply the normal rules of algebra to this new, abnormal number.

A few very simple examples of expressions involving i
Simplify: i 5
Answer: 5i
Simplify: i + 5i
Answer: 6i (Add anything to 5 of itself, and you get 6 of it. Or, you can think of this as “pulling out” an i as follows: i + 5i = i ( 1 + 5 ) = 6i )
Simplify: 2i + 3
Answer: You can't simplify it.

Now let's try something a little more involved.

Example: Simplify the expression (3+2i)2
( 3 + 2i ) 2 = 3 2 + 2 ( 3 ) ( 2i ) + ( 2i ) 2 because ( x + a ) 2 = x 2 + 2 ax + a 2 as always
= 9 + 12i 4 (2i) 2 = (2i) (2i) = (2) (2) (i) (i) = 4i 2 = –4
= 5 + 12i we can combine the 9 and –4, but not the 12i .

It is vital to remember that i is not a variable, and this is not an algebraic generalization. You cannot plug i = 3 into that equation and expect anything valid to come out. The equation (3+2i) 2 = 5 + 12i has been shown to be true for only one number: that number is i , the square root of –1.

In the next example, we simplify a radical using exactly the same technique that we used in the unit on radicals , except that a 1 is thrown into the picture.

Example: Simplify 20 size 12{ sqrt { - "20"} } {}
20 size 12{ sqrt { - "20"} } {} = ( 4 ) ( 5 ) ( 1 ) size 12{ sqrt { \( 4 \) \( 5 \) \( - 1 \) } } {} as always, factor out the perfect squares
= 4 size 12{ sqrt {4} } {} 5 size 12{ sqrt {5} } {} 1 size 12{ sqrt { - 1} } {} then split it, because ab size 12{ sqrt { ital "ab"} } {} = a size 12{ sqrt {a} } {} b size 12{ sqrt {b} } {}
= 2i 5 size 12{ sqrt {5} } {} 4 size 12{ sqrt {4} } {} =2, 1 size 12{ sqrt { - 1} } {} = i , and 5 size 12{ sqrt {5} } {} is just 5 size 12{ sqrt {5} } {}
Is 2i 5 size 12{ sqrt {5} } {} really the square root of –20? If it is, then when we square it, we should get –20.
( 2i 5 ) 2 = 2 2 i 2 5 2 = 4 * -1 * 5 = -20 It works!

The problem above has a very important consequence. We began by saying “You can’t take the square root of any negative number.” Then we defined i as the square root of –1. But we see that, using i , we can now take the square root of any negative number.

Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. Jun 17, 2014 Download for free at http://cnx.org/content/col11666/1.1
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