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w = x x 2 + y 2 + - y x 2 + y 2 i .

We then have

z × w = ( x + y i ) × ( x x 2 + y 2 + - y x 2 + y 2 i = x 2 x 2 + y 2 - - y 2 x 2 + y 2 + ( x - y x 2 + y 2 + y x x 2 + y 2 ) i = x 2 + y 2 x 2 + y 2 + 0 x 2 + y 2 i = 1 + 0 i = 1 ,

as desired.

Prove parts (1) and (2) of [link] .

One might think that these kinds of improvements of the real numbers will go on and on. For instance, we might next have to create and adjoin another object j so that the number i has a square root; i.e., so that the equation i - z 2 = 0 has a solution. Fortunately and surprisingly, this is not necessary,as we will see when we finally come to the Fundamental Theorem of Algebra in [link] .

The subset of C consisting of the pairs x + 0 i is a perfect (isomorphic) copy of the real number system R . We are justifiedthen in saying that the complex number system extends the real number system, and we will say that a real number x is the same as the complex number x + 0 i . That is, real numbers are special kinds of complex numbers. The complex numbers of the form 0 + y i are called purely imaginary numbers. Obviously, the only complex number that is both real and purely imaginary is the number 0 = 0 + 0 i . The set C can also be regarded as a 2-dimensional space, a plane, and it is also helpful to realize that the complex numbers form a 2-dimensional vector space over the fieldof real numbers.

If z = x + y i , we say that the real number x is the real part of z and write x = ( z ) . We say that the real number y is the imaginary part of z and write y = ( z ) .

If z = x + y i is a complex number, define the complex conjugate z ¯ of z by z ¯ = x - y i .

The complex number i satisfies i 2 = - 1 , showing that the negative number - 1 has a square root in C , or equivalently that the equation 1 + z 2 = 0 has a solution in C . We have thus satisfied our initial goal of extending the real numbers. But what about other complex numbers?Do they have square roots, cube roots, n th roots? What about solutions to other kinds of equations than 1 + z 2 ?

  1. Prove that every complex number has a square root. HINT: Let z = a + b i . Assume w = x + y i satisfies w 2 = z , and just solve the two equations in two unknowns that arise.
  2. Prove that every quadratic equation a z 2 + b z + c = 0 , for a , b , and c complex numbers, has a solution in C . HINT: If a = 0 , it is easy to find a solution. If a 0 , we need only find a solution to the equivalent equation
    z 2 + b a z + c a = 0 .
    Justify the following algebraic manipulations, and then solve the equation.
    z 2 + b a z + c a = z 2 + b a z + b 2 4 a 2 - b 2 4 a 2 + c a = ( z + b 2 a ) 2 - b 2 4 a 2 + c a .

What about this new field C ? Does every complex number have a cube root, a fourth root, does every equation have a solution in C ? A natural instinct would be to suspect that C takes care of square roots, but that it probably does not necessarily have higher order roots.However, the content of the Fundamental Theorem of Algebra, to be proved in [link] , is that every equation of the form P ( z ) = 0 , where P is a nonconstant polynomial, has a solution in C . This immediately implies that every complex number c has an n th root, for any solution of the equation z n - c = 0 would be an n th root of c .

The fact that the Fundamental Theorem of Algebra is true is a good indication that the field C is a “good” field. But it's not perfect.

In no way can the field C be made into an ordered field. That is, there exists no subset P of C that satisfies the two positivity axioms.

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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