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mesa
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I need to write, "(k)1/2= a finite term recursive expression of 'k' where 'k' is the set of all Reals where 'k' is greater than 0 and 'k' for all complex values" in math.
Thanks!
Thanks!
-- seems to be incomplete.and 'k' for all complex values
Simon Bridge said:Basic setbuilding notation:
http://www.mathwords.com/s/set_builder_notation.htm
...where do you get stuck?
Personally I'm having trouble following the English description of what k is supposed to be.
By "k is the set of... " do you mean that k is a member of the set, or is the label assigned to the set?
Considering that the set of reals is a subset of complex numbers, are you saying that k is any complex number that is not a real number 0 or less?
Mark44 said:I too am confused by what you wrote as the description for k. The first part is pretty clear; namely, {k ##\in## R | k > 0}, but this part --
-- seems to be incomplete.
mesa said:Considering these conditions, is this right?
$$\{k\in C\mid k>0\}$$
jbriggs444 said:The Complex numbers have no natural order. So it makes little sense to write "k>0" when k is complex.
I think that what you are trying to get at is that √k is well defined when k is both real and positive.
For complex k there are always two solutions for x2=k except when k=0.
For k positive and real, one solution is positive and one is negative. √k is taken to denote the positive one.
For k negative and real, the two solutions are complex conjugates whose real part is zero. One is equal to 0+√(-k) i. The other is equal to 0-√(-k)i. Neither one is preferred.
For k complex with a non-zero imaginary part the two solutions will be 180 degrees apart on the complex plane. Neither one is preferred.
Simon Bridge said:Personally I'm having trouble following the English description of what k is supposed to be.
By "k is the set of... " do you mean that k is a member of the set, or is the label assigned to the set?
I would write the set this way:Simon Bridge said:##\mathbb C \cup \{x\in\mathbb R:x>0\}## is just the set of complex numbers.
<puzzled> which set would you write that way?Mark44 said:I would write the set this way:me said:\mathbb C \cup \{x\in \mathbb R : x>0\}
{z ##\in## C | Re(z) > 0}. This gives you the right half of the complex plane.
Simon Bridge said:##\mathbb C \cup \{x\in\mathbb R:x>0\}## is just the set of complex numbers.
Remember the reals are a subset of complex numbers.
The union of a set and it's subset is itself.
Similarly, the intersection of a set and it's subset is the subset.
You want to exclude the subset.
k is a complex number that is not a negative real number.
(Is k allowed to be zero? Proceeding as if k cannot be zero either.)
i.e. The real part of k can be negative, but only if the imaginary part is not zero.
Something like:$$k\in\mathbb C : k\notin \{x\in\mathbb R: x\leq 0\}$$... ie. k is a member of the relative compliment of complex numbers and negative real numbers.$$k\in\mathbb C \backslash \{x\in\mathbb R: x\leq 0\}$$
Simon Bridge said:Aside: If ##x,y\in\mathbb R## and ##i=\sqrt{-1}## then ##z=x+iy\in\mathbb C##.
Since y=0 is a real number, it follows that ##z=x\in\mathbb C##,
i.e. all real numbers are members of the set of complex numbers.
Simon Bridge said:Are there numbers that are not in the set of complex numbers that you don't want k to be?
Simon Bridge said:Notes: we have all been using "blackboard bold" for special sets.
Some people advise against it i.e.
Krantz, S., Handbook of Typography for the Mathematical Sciences, Chapman & Hall/CRC, Boca Raton, Florida, 2001, p. 35.
See http://en.wikipedia.org/wiki/Complement_(set_theory ) for the use of compliments in set theory.mesa said:Okay, this makes sense. All that is needed is the $$\notin$$ notation to exclude negative Reals from the complex plane. Or for the second example the '\' in place of $$: k\notin$$
Very nice!
There's quaternions etc.This part I am a little confused on, what numbers are left that are outside the set of complex numbers since it also contains all Reals when bi=0?
On another note, in order to improve with these types of things do you have another (preferably more frugal [than blackboard bold]) recommendation for mathematical typography?/QUOTE]Krantz (2001) [with Donald Knuth] suggests plain old bold-face. Upper-case italics is also used.
i.e. http://en.wikipedia.org/wiki/Set_notation ,
http://en.wikipedia.org/wiki/Set_(mathematics )
... examples of each.
-------------------------------
Krantz, S. (2001), Handbook of Typography for the Mathematical Sciences, Chapman & Hall/CRC, Boca Raton, Florida, p. 35.
Simon Bridge said:See http://en.wikipedia.org/wiki/Complement_(set_theory ) for the use of compliments in set theory.
I think the most succinct way is: ##\{k\in\mathbb C : \text{Arg}(k)\neq \pi\}##
... although I think that includes k=0, since it's argument is undefined. Easy to fix.
Simon Bridge said:What is the context of this?
I'm assuming it's not an arbitrary homework question because of how tricky the wording was to pharse. It would have been quite tricky to guide you to the compliment or argument solution too.
Simon Bridge said:There's quaternions etc.
You'll notice when you go from reals to complex numbers, you lose the property of being ordered?
Quaternions (a 4-dimensional normed R-algebra) lose commutativity, but are associative.
Octonions (an 8-dimensional normed R-algebra) lose associativity, but are alternative.
Sedenions (a 16-dimensional R-algebra) lose alternativity, and are also not a normed R-algebra, because there are zero-divisors.
More:
http://en.wikipedia.org/wiki/Cayley–Dickson_construction
http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(normed_division_algebras )
Neat result of what?I enjoy finding identities and exploring math in general. This particular function I thought was a neat result but lacking the proper mathematical vernacular makes it difficult to share with people who really know these things.
... quaternions have application in computer science for their ability to efficiently represent rotations.I haven't studied these, but have heard of them. I will look through your links!
Simon Bridge said:Neat result of what?
Simon Bridge said:... quaternions have application in computer science for their ability to efficiently represent rotations.
Basically, define basis elements i,j,k (not the cartesian unit vectors) so that ##i^2=j^2=k^2=ijk=-1##
A quaternion is q=a+bi+cj+dk.
http://en.wikipedia.org/wiki/Quaternion
... you can similarly look up the others.
However, the reason I brought it up is because sometimes an overarching set can be left implicit ... i.e. ##\{a:a<0\}## would usually be interpreted as the set of negative reals, and ##\{a:\Re(a)>0\}## would imply complex numbers are intended.
So you could write ##\{k:\text{Arg}(k)\neq \pi \}## and be confident that people would realize you meant to define k over the complex plane.
It's a language - so there are lots of ways to express yourself. The exact approach you choose depends on what you want to draw the readers attention to.
Oh right.I was looking at the golden ratio and trying to work a general form, after the task was completed (and some new insights gained) it appeared it may be possible to build a general form function for radicals (our f(k)=√k). The result I thought was 'neat'.
Simon Bridge said:Oh right.
You were thinking that k can be any complex not a negative real?
Positive reals can have real square-roots fersure, but I don't see how excluding negative reals from k helps considering you are allowing any complex number otherwise.
Normally the radical indicates a positive square root so that if ##y=x^2## then ##x\in\{\pm\sqrt{y}\}## ... that can mess things up. The result is that the negative reals are excluded, by definition, from the solution set.
This what you are thinking of?
Simon Bridge said:Don't know what you mean by "general form function" in this context.
Anyway - that would be for another thread ;)
"Math" is short for mathematics, which is the study of numbers, quantities, and shapes.
The word "math" can be written as "math" or "maths" in British English. It is also sometimes written as "mathematics".
The best way to write "math" in English is to use the correct spelling and grammar. This includes using capitalization and punctuation when necessary.
Yes, an example of writing "math" in a sentence could be: "I love solving math problems in my spare time."
There is no specific format for writing "math" in English, as it can be used in various contexts and forms. However, it is important to use proper spelling and grammar when writing "math".