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This module introduces the concept of complex numbers in Algebra.

A “complex number” is the sum of two parts: a real number by itself, and a real number multiplied by i . It can therefore be written as a + bi , where a and b are real numbers.

The first part, a , is referred to as the real part . The second part, bi , is referred to as the imaginary part .

Examples of complex numbers a + bi ( a is the “real part”; bi is the “imaginary part”)
3 + 2i a = 3 , b = 2
π a = π , b = 0 (no imaginary part: a “pure real number”)
-i a = 0 , b = -1 (no real part: a “pure imaginary number”)

Some numbers are not obviously in the form a + bi . However, any number can be put in this form.

Example 1: Putting a fraction into a + bi form ( i in the numerator)
3 4i 5 size 12{ { {3 - 4i} over {5} } } {} is a valid complex number. But it is not in the form a + bi , and we cannot immediately see what the real and imaginary parts are.
To see the parts, we rewrite it like this:
3 4i 5 size 12{ { {3 - 4i} over {5} } } {} = 3 5 size 12{ { {3} over {5} } } {} 4 5 size 12{ { {4} over {5} } } {} i
Why does that work? It’s just the ordinary rules of fractions, applied backward. (Try multiplying and then subtracting on the right to confirm this.) But now we have a form we can use:
3 4i 5 size 12{ { {3 - 4i} over {5} } } {} a = 3 5 size 12{ { {3} over {5} } } {} , b = 4 5 size 12{ { {4} over {5} } } {}
So we see that fractions are very easy to break up, if the i is in the numerator. An i in the denominator is a bit trickier to deal with.
Example 2: Putting a fraction into a + bi form ( i in the denominator)
1 i size 12{ { {1} over {i} } } {} = 1 i i i size 12{ { {1 cdot i} over {i cdot i} } } {} Multiplying the top and bottom of a fraction by the same number never changes the value of the fraction: it just rewrites it in a different form.
= i 1 size 12{ { {i} over { - 1} } } {} Because i i is i 2 , or –1.
= -i This is not a property of i , but of –1. Similarly, 5 1 size 12{ { {5} over { - 1} } } {} = –5 .
1 i size 12{ { {1} over {i} } } {} : a = 0 , b = -1 since we rewrote it as -i , or 0 - 1i

Finally, what if the denominator is a more complicated complex number? The trick in this case is similar to the trick we used for rationalizing the denominator: we multiply by a quantity known as the complex conjugate of the denominator .

Definition of complex conjugate

The complex conjugate of the number a + bi is a - bi . In words, you leave the real part alone, and change the sign of the imaginary part.

Here is how we can use the “complex conjugate” to simplify a fraction.

Example: Using the Complex Conjugate to put a fraction into a + bi form
5 3 4i size 12{ { {5} over {3 - 4i} } } {} The fraction: a complex number not currently in the form a + b i
= 5 ( 3 + 4i ) ( 3 4i ) ( 3 + 4i ) size 12{ { {5` \( 3+4i \) } over { \( 3 - 4i \) \( 3+4i \) } } } {} Multiply the top and bottom by the complex conjugate of the denominator
= 15 + 20 i 3 2 ( 4i ) 2 size 12{ { {"15"+"20"i} over {3 rSup { size 8{2} } - \( 4i \) rSup { size 8{2} } } } } {} Remember, ( x + a ) ( x a ) = x 2 –a 2
= 15 + 20 i 9 + 16 size 12{ { {"15"+"20"i} over {9+"16"} } } {} ( 4i ) 2 = 4 2 i 2 = 16 ( –1 ) = –16 , which we are subtracting from 9
= 15 + 20 i 25 size 12{ { {"15"+"20"i} over {"25"} } } {} Success! The top has i , but the bottom doesn’t. This is easy to deal with.
= 15 25 size 12{ { {"15"} over {"25"} } } {} + 20 i 25 size 12{ { {"20"i} over {"25"} } } {} Break the fraction up, just as we did in a previous example.
= 3 5 size 12{ { {3} over {5} } } {} + 4 5 size 12{ { {4} over {5} } } {} i So we’re there! a = 3 5 size 12{ { {3} over {5} } } {} and b = 4 5 size 12{ { {4} over {5} } } {}

Any number of any kind can be written as a + bi . The above examples show how to rewrite fractions in this form. In the text, you go through a worksheet designed to rewrite 1 3 size 12{ nroot { size 8{3} } { - 1} } {} as three different complex numbers. Once you understand this exercise, you can rewrite other radicals, such as i size 12{ sqrt {i} } {} , in a + bi form.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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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
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what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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.
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for screen printed electrodes ?
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s. Reply
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Graphene has a hexagonal structure
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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