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Suppose C were an ordered field, and write P for its set of positive elements. Then, since every square in an ordered field must be in P (part (e) of [link] ), we must have that - 1 = i 2 must be in P . But, by part (a) of [link] , we also must have that 1 is in P , and this leads to a contradiction of the law of tricotomy. We can't have both 1 and - 1 in P . Therefore, C is not an ordered field.

Although we may not define when one complex number is smaller than another, we can define the absolute value of a complex number and the distance between two of them.

If z = x + y i is in C , we define the absolute value of z by

| z | = x 2 + y 2 .

We define the distance d ( z , w ) between two complex numbers z and w by

d ( z , w ) = | z - w | .

If c C and r > 0 , we define the open disk of radius r around c , and denote it by B r ( c ) , by

B r ( c ) = { z C : | z - c | < r } .

The closed disk of radius r around c is denoted by B ¯ r ( c ) and is defined by

B ¯ r ( c ) = { z C : | z - c | r } .

We also define open and closed punctured disks B r ' ( c ) and B ¯ r ' ( c ) around c by

B r ' ( c ) = { z : 0 < | z - c | < r }

and

B ¯ r ' ( c ) = { z : 0 < | z - c r } .

These punctured disks are just like the regular disks, except that they do not contain the central point c .

More generally, if S is any subset of C , we define the open neighborhood of radius r around S , denoted by N r ( S ) , to be the set of all z such that there exists a w S for which | z - w | < r . That is, N r ( S ) is the set of all complex numbers that are within a distance of r of the set S . We define the closed neighborhood of radius r around S , and denote it by N ¯ r ( S ) , to be the set of all z C for which there exists a w S such that | z - w | r .

  1. Prove that the absolute value of a complex number z is a nonnegative real number. Show in addition that | z | 2 = z z ¯ .
  2. Let x be a real number. Show that the absolute value of x is the same whether we think of x as a real number or as a complex number.
  3. Prove that max ( | ( z ) | , | ( z ) | ) | z | | ( z ) | + | ( z ) | . Note that this just amounts to verifying that
    max ( | x | , | y | ) x 2 + y 2 | x | + | y |
    for any two real numbers x and y .
  4. For any complex numbers z and w , show that z + w ¯ = z ¯ + w ¯ , and that z ¯ ¯ = z .
  5. Show that z + z ¯ = 2 ( z ) and z - z ¯ = 2 i ( z ) .
  6. If z = a + b i and w = a ' + b ' i , prove that | z w | = | z | | w | . HINT: Just compute | ( a + b i ) ( a ' + b ' i ) | 2 .

The next theorem is in a true sense the most often used inequality of mathematical analysis. We have already proved the triangle inequality for the absolute value of real numbers, and the proof wasnot very difficult in that case. For complex numbers, it is not at all simple, and this should be taken as a good indication that it is a deep result.

Triangle inequality

If z and z ' are two complex numbers, then

| z + z ' | | z | + | z ' |

and

| z - z ' | | | z | - | z ' | | .

We use the results contained in [link] .

| z + z ' | 2 = ( z + z ' ) ( z + z ' ) ¯ = ( z + z ' ) ( z ¯ + z ' ¯ ) = z z ¯ + z ' z ¯ + z z ' ¯ + z ' z ' ¯ = | z | 2 + z ' z ¯ + z ' z ¯ ¯ + | z ' | 2 = | z | 2 + 2 ( z ' z ¯ ) + | z ' | 2 | z | 2 + 2 | ( z ' z ¯ ) | + | z ' | 2 | z | 2 + 2 | z ' z ¯ | + | z ' | 2 = | z | 2 + 2 | z ' | | z | + | z ' | 2 = ( | z | + | z ' | ) 2 .

The Triangle Inequality follows now by taking square roots.

REMARK The Triangle Inequality is often used in conjunction with what's called the “add and subtract trick.”Frequently we want to estimate the size of a quantity like | z - w | , and we can often accomplish this estimation by adding and subtracting the same thing within the absolute value bars:

| z - w | = | z - v + v - w | | z - v | + | v - w | .

The point is that we have replaced the estimation problem of the possibly unknown quantity | z - w | by the estimation problems of two other quantities | z - v | and | v - w | . It is often easier to estimate these latter two quantities, usually by an ingenious choice of v of course.

  1. Prove the second assertion of the preceding theorem.
  2. Prove the Triangle Inequality for the distance function. That is, prove that
    d ( z , w ) d ( z , v ) + d ( v , w )
    for all z , w , v C .
  3. Use mathematical induction to prove that
    | i = 1 n a i | i = 1 n | a i | .

It may not be necessary to point out that part (b) of the preceding exercise provides a justification for the name “triangle inequality.”Indeed, part (b) of that exercise is just the assertion that the length of one side of a triangle in the plane is less than or equalto the sum of the lengths of the other two sides. Plot the three points z , w , and v , and see that this interpretation is correct.

A subset S of C is called Bounded if there exists a real number M such that | z | M for every z in S .

  1. Let S be a subset of C . Let S 1 be the subset of R consisting of the real parts of the complex numbers in S , and let S 2 be the subset of R consisting of the imaginary parts of the elements of S . Prove that S is bounded if and only if S 1 and S 2 are both bounded.

    HINT: Use Part (c) of [link] ..

  2. Let S be the unit circle in the plane, i.e., the set of all complex numbers z = x + i y for which | z | = 1 . Compute the sets S 1 and S 2 of part (a).

  1. Verify that the formulas for the sum of a geometric progression and the binomial theorem ( [link] and [link] ) are valid for complex numbers z and z ' . HINT: Check that, as claimed, the proofs of those theorems work in any field.
  2. Prove [link] for complex numbers z and z ' .

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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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