Math Is Fun Forum

True!  I should have asked the question in a separate paragraph.  I was just trying to point out the easy way to do integration by parts in case he hadn't seen it.  It's really cool!

Dr. Guy passed away several years ago.  However I did have the pleasure of taking several courses from him.  Dr. Guy and a high school teacher R. J. Wood (Edison High in San Antonio) were my mentors.  My style of teaching was pattered after Dr. Guy's.

Top of the mornin' to you!   

yup! you got it right except for adding in the constant of integration.  Have you seen the tabular technique of integration by parts?  As I understand it was created by Dr. William T. Guy at the University of Texas in the 1960's or so.  Some calculus textbooks introduce this tabular technique.  It makes integration by parts a snap and useful instead of something to be avoided.

Don,
I've viewed some of the axioms as "shoddy" myself over the years.  For example the axioms of commutativity state that for each x and y in field F, x+y=y+x and xy=yx.  To me this is not as much a property of addition and multiplication as it is a property of the way we write these expressions.
We write in a linear left to right manner.  So we have to choose one of the variables to write first and the other is written second.  The axiom states that it doesn't matter which way we write it.  The result is supposed to be the same.  But suppose we superimpose 1 and 3 instead of writing 1+3 or 3+1.  If someone doesn't see us write it,  they wouldn't know which we wrote first and which second or whether we wrote part of each and then the rest of each, etc.  Hence it is more of a concession to our method of writing sums than it is to the sum itself.

We could perhaps view the symmetry axiom similarly.  If we wish to indicate the equality of a and b then we can write a=b or b=a.  But our linear language makes us do one or the other.  So the axiom says that it doesn't matter. 

Yes.  I see nothing wrong with that equality even if x=0.  It's the step
x^(1-1) = x^1 * x^(-1) that I have trouble with when x=0.  It appears to be applying the law of exponents x^(a+b) = x^a * x^b.  When x=0 and b=-1 the second factor is 0^(-1) which is what I have trouble with.  To me this is indicating the reciprocal of 0 (equivalent notationally to 1/0) which the multiplicative inverse axiom precludes. 

I'm afraid there is a fallacy in the argument that 0^0 = 0/0.

Let y be the reciprocal of x.  Then xy=1.  Now x^0 = x^(1-1) = x^1*x^(-1) = x*y = 1 (not x/y Oops)
This works fine for non-zero x.  But when x=0 the y in the formula does not exist since zero has no reciprocal.  Hence the quantities  x^(-1) and y in the third and fourth expressions do not exist.  Hence we cannot conclude that 0^0 = 0/0.

It is not valid to apply the law of exponents x^(1-1) = x^1*x^(-1) when x=0.
So 0^0 is not equal to 0/0 via this argument.  We are then free to define 0^0 unless we can find some valid reason that it is not permissible; that is, some valid reason that it must remain undefined.  So if we want to claim that 0^0 is undefined we need to find a valid proof of this.

0^0 can be defined nicely immediately after introducing the field axioms which give us multiplication and the multiplicative identity 1. 

There are a number of formulas in mathematics that are much nicer if 0^0 is defined to be 1.
You have probably seen several of them, so I won't go into them.

I don't see any reason to go up to calculus to try to make 0^0 work.  The indeterminant form
0^0 in calculus is a form that limits may tend to BUT they never actually become 0^0.  Hence using these types of limits seems inappropriate.

Let's go back and define exponentiation in a field (we will use the real number field).  The usual definition is something like:  ... x^3=x*x*x,  x^2=x*x,  x^1=x and following this pattern we would get x^0 to be a blank space.  Hence x^0 is usually defined separately as x^0=1 if x is not equal to zero.  And then 0^0 is usually said to be one or undefined.

To do it a little differently, let's define exponentiation as follows:
..., x^3=1*x*x*x.  x^2=1*x*x.  x^1=1*x.  so following this pattern x^0=1.  We don't need a separate definition for x^0.  And this definition automatically makes 0^0=1 since it doesn't matter what the x is.  Just drop the last x off of the right side of the definition for x^1=1*x.

This definition has been around a while, but it is still not widely known.
0*x=0 comes from a similar definition: ...3*x=0+x+x+x, 2*x=0+x+x, 1*x=0+x, 0*x=0.

This kind of definition works for any group.  If e is the identity for the group and # is the symbol for the operation and x is an element of the group then we have
   ... x^3=e#x#x#x,  x^2=e#x#x,  x^1=e#x,  x^0=e.
And this could be done with coefficients (like 3x, 2x, 1x, 0x above but in groups repeating the operation with a fixed element is usually represented by exponentiation).

This also works for union and intersection in topology where the identity for union is the null set and the identity for intersection is the whole space X.  This eliminates the need for vacuous arguments for unions and intersections over a null family of sets.

How we define things in mathematics is extremely important.  Also the symbolism we use can "make or break" a concept. 

In the field axioms the multiplicative inverse axiom does not allow for zero to have a multiplicative inverse:

For each non-zero x in F there is a y in F such that x*y=1.  Typically we write y as 1/x.

If x were zero then y would be "1/0"  so x*(1/x) = 0*(1/0) = 0/0. 
If zero had a multiplicative inverse then this would yield one. 
On the other hand it has been shown many times that zero times any number is zero.
Thus we could conclude that 1=0.  This makes any number equal to zero.  Hence our
number system is reduced to a single number. 
Such a system would be nice to work with, but not very useful.

The field axioms and the order axioms for the real number system do not involve limits.
Hence I have trouble "hooking up" with arguments about the real number system that
resort to limits from calculus.  Calculus has lots of "strange things" due to pushing things
to infinity.  But that's a totally different story.

You're right!  Even further back than I thought.  You'd think a mathematician could count to 5 and stop there when supposed to!

You just wrote the sequence of letters "jkhkj."  If you consider this a set then it is a multiset, namely {k,k,j,j,h} with the order being unimportant.  If you consider this a set from classical
set theory corresponding to T/F logic then the duplicates would be eliminated and it would be
equal to {k,j,h} with the order unimportant.  {k,j,h} can be considered the Typeset of the
original multiset {k,k,j,j,h}; that is, T{k,k,j,j,h}={k,j,h} where T is an operator that eliminates
the duplicates from a set leaving one of each type.

That being said, is there a question somehow encoded in your sequence "jkhkj"?

bobbym and stefy,
Thanks for your replys.  Indeed I am a rank novice at navigating this site.  Been teaching math for
40 some odd years, but am not very saavy at navigating on the internet.  I'll try a simple one line
quote and see if that will stick.  By the way, the first two times I signed up for this site the password
given to me would not work when I tried to log in.  Third time was a charm.  Now I can log on.
Thanks

I must admit the "and you are looking at it" (meaning the second number) didn't register in my mind
to begin with.  So now what does that mean.  Is it part of the given number, a permutation of it, or
what?  Or could we be looking at it somewhere in the three columns?  Somewhere within the three
columns I am seeing every digit 0 through 9.  Can I then create any number I want?  This opens up
a whole new ball game in my mind.