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A teacher's guide to the square root of the imaginary number.

This is arguably the most advanced, difficult thing we do all year. But I like it because it contains absolutely nothing they haven’t already done. It’s not here because it’s terribly important to know i size 12{ sqrt {i} } {} , or even because it’s terribly important to know that all numbers can be written in a + b i format. It is here because it reinforces certain skills—squaring out a binomial (always a good thing to practice), working with variables and numbers together, setting two complex numbers equal by setting the real part on the left equal to the real part on the right and ditto for the imaginary parts, and solving simultaneous equations.

Explain the problem we’re going to solve, hand it out, and let them go. Hopefully, by the end of class, they have all reached the point where they know that 1 2 size 12{ { {1} over { sqrt {2} } } } {} + 1 2 size 12{ { {1} over { sqrt {2} } } } {} i and 1 2 size 12{ { {1} over { sqrt {2} } } } {} 1 2 size 12{ { {1} over { sqrt {2} } } } {} i are the two answers, and have tested them.

Note that right after this in the workbook comes a more advanced version of the same thing, where they find 1 3 size 12{ nroot { size 8{3} } { - 1} } {} (all three answers: –1 , 1 2 + 3 2 size 12{ { { sqrt {3} } over {2} } } {} i , and 1 2 3 2 size 12{ { { sqrt {3} } over {2} } } {} i ). I tried using this for the whole class, and it was just a bridge too far. But you could give it to some very advanced students—either as an alternative to the i size 12{ sqrt {i} } {} exercise, or as an extra credit follow-up to it.


They should finish the worksheet if they haven’t done so, including #7. Then they should also do the “Homework: Quadratic Equations and Complex Numbers.” It’s a good opportunity to review quadratic equations, and to bring in something new! (It’s also a pretty short homework.)

When going over the homework, make sure they did #3 by completing the square—again, it’s just a good review, and they can see how the complex answers emerge either way you do it. #4 is back to the discriminant, of course: if b 2 - 4 a c < 0 then you will have two complex roots. The answer to #6 is no. The only way to have only one root is if that root is 0. (OK, 0 is technically complex…but that’s obviously not what the question meant, right?)

The fun is seeing if anyone got #5. The answer, of course, is that the two roots are complex conjugates of each other—real part the same, imaginary part different sign. This is obvious if you rewrite the quadratic formula like this:

x = b 2a size 12{ { { - b} over {2a} } } {} ± b 2 4 ac 2a size 12{ { { sqrt {b rSup { size 8{2} } - 4 ital "ac"} } over {2a} } } {}

and realize that the part on the left is always real, and the part on the right is where you get your i from.

Time for another test!

Not much to say here, except that you may want to reuse this extra credit on your own test—if they learn it from the sample (by asking you) and then get it right on your test, they learned something valuable.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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
it is a goid question and i want to know the answer as well
characteristics of micro business
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
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