Math Is Fun Forum

Yes, you do think.  And you are quite good at it!

Hi!

And here is a little different way to look at commutativity and associativity.

Commutativity can be viewed more as a language problem than an axiom problem.  Let me
explain.  Take the example of adding four and five.  We typically write and speak in a linear array
of letters or syllables.  This forces us to say one of the numbers first and the other second.  So
we say "four plus five" or "five plus four" and write 4+5 or 5+4.  But we want these two
expressions to mean the same thing; that is, we want them to get us to "9."

As a function (binary operator in this case) we have a function "+" which maps (4,5) to 9 and
also (5,4) to 9.  If we had a way to "say" the four and five SIMULTANEOUSLY, then we would
have but one way to "speak" or "write" about the addition of four and five.  Thus we wouldn't
have to "invoke" a commutativity rule.

Actually we could define addition of 4 and 5 as a mapping from the SET {4,5} to 9.  Since the
set is unordered this gives us only one way to express the combination of the 4 and 5 to get
the 9.  Then "4+5" and "5+4" could be viewed as ways of saying that {4,5} maps to 9 or that
these are ways of expressing the result of mapping {4,5} to 9.  But then the notation +{4,5}=9
is a bit awkward compared to 4+5=9 or 5+4=9.  On the other hand as the sets get larger, for
example {3,4,5}, +{3,4,5}=12 is not so bad.  And here with the three numbers using this
approach we don't have to get into associativity either.

If we define 3, 4, and 5 by 3={ooo}, 4={oooo} and 5={ooooo} (multisets) then adding these
three sets together is just in essence dumping them altogether to get {oooooooooooo}.  How
we dump them together (all at once, one at a time, etc.) is immaterial.  We get the same result.

So commutativity and associativity can be view more as a language problem than an axiom
problem.  On the other hand, the additive and multiplicative identity and inverse axioms are
axiom problems in the sense that they do not just involve ways of "saying the same thing".
They inject the notion of "existence" into the system.

All this being said, viewing the way our systems of numbers have evolved, it is perhaps easier
to just introduce commutativity and associativity as axioms and be done with it.

Have a very blessed day!

Or another way to look at it:

0th power of 12357 ends in 1.
1st power ends in 7.   2nd power ends in 9.   3rd power ends in 3.   4th power ends in 1
5th power ends in 7    6th power ends in 9    7th power ends in 3.    8th power ends in 1
9         ...            7    10      ...               9   11       ...              3    12    ...                  1, etc.

There are only four digits that occur in the powers and these repeat in the sequence 1, 7, 9, 3, 1, ...
Divide the power (655 in this case) by four and see what the remainder is.  It is three.  So the 655th
power ends in the same digit that the 3rd power does, namely 3. 

But this is a bit confusing since we get 3 for the remainder and for the last digit.
Try raising to the 656 power.  Then dividing 656 by 4 we get a remainder of zero.  So the 656
power ends in 1 same as the 0th power.

For the 654th power dividing 654 by 4 has remainder 2.  So the 654th power ends in 9 the same
as the second power.

For the 653rd power dividing 653 by 4 has remainder 1 so the units digit of the 653rd power is 7 the
same as the 1st, 5th, 9th, ... powers.

Be very blesses!

Maybe we could use a "Klein bottle" for the robot and put the load on the side.

Hi Al-Allo,

Webster's dictionary defines it as "a specialist or expert in mathematics."
There are several hundred areas of mathematics where one could spend the rest of their
life doing research in mathematics.  PhD's in math typically know MUCH LESS than
1% of the mathematics known worldwide today.  I'd say that anyone who is willing to call
themselves a mathematician probably is one.

Now if you are talking about a "working" statistician, they go through a battery of tests that
would make diamond wilt.  It's not an easy road, but usually quite lucrative.  They are
mathematicians, but usually only expert in a small area of mathematics.

And today some of the most active mathematicians use computers to bounce their ideas
off of often.  Wonderful tools for exploring patterns, testing formulas, etc.

And age doesn't necessarily have a lot to do with it.  Gauss could have probably been called
a mathematician before he was a teenager.  He taught himself to add columns of numbers
by the age two and a half.  I was still trying to get my walking down about that age.

Have a super day!

Considering just positive integers, there are 6 factorizations of 495 into two factors:
1*495, 3*165, 5*99, 9*55, 11*45 and 15*33.  Each of these corresponds to one of
the six (x,y) pairs bobbym listed in post #4.  For example: 1*495=248^2-247^2 and
24^2-9^2 = (24+9)(24-9) = 33*15.

In general for an odd composite number each of its unique factorizations (other than a perfect
square factorization) corresponds to a difference of squares.  For 9 = 1*9 we get 5^2-4^2
= (5+4)(5-4) = 9*1 but 3*3 has no difference of squares representation unless we allow zero:
3^2-0^2 = (3+0)(3-0) = 3*3.

But we were talking about POSITIVE integers.

If M is an odd composite number and M=n*m where n and m are different, we get

If I recall correctly this was involved in one of Fermat's methods of factoring odd composites.

Have a grrreeeeaaaaaaat day!

Wasn't there a guy that bc (before computers) calculated pi out to 700 places by hand.  Then ac (after
computers) it was discovered that he made a mistake in the 200th digit and all the rest was trash
since each digit depended on the previous digit.  Probably spent the great bulk of his 15 year project
on the last 500 digits.  Mercy me!  But 200 digits of pi ain't bad, eh? 

Sometimes mathematicians play "fast and loose" with definitions.  We probably will not find total
agreement about what a "polynomial" is.  Some books will define "a polynomial in x" as opposed
to just a polynomial.  And what about a "polynomial in two variables?"  If y=2+x is x+y a binomial?

And what does "quotient" refer to?  If we consider 7÷2 as a fraction, is 7/2 a quotient?  Or is it
3.5?  Or is the quotient 3 (with remainder 1)?

And what is a fraction?  Which among the following are fractions?

     x, 2/x,  x/y,  2/3, 2÷3, x÷y, 2*(1/x), x/1, x/3, xy where y=1/z, z where z=1/y, etc.

     Let x=1/y,  y=1/3 and z=1/x.   Which are fractions?  x, y, 1/x, 1/y, y/3, xy, xz, yz, 1/yz, etc.

What is an arithmetic fraction, an algebraic fraction, etc.?

And in geometry is an equilateral triangle also isoseles?  There has been disagreement on whether
to make the set of equilateral triangles a subclass of isoseles triangles or a separate category of
triangles.

The best we find at times is a "local" definition where an author defines a term precisely for his
following discussion.  And often other mathematicians may find fault with this.

Mathematics is a LANGUAGE and has not been (nor ever is likely to be) "nailed down" so as to be
without ambiguity or disagreement even for the most common and "simple" concepts.

Sometimes we just have to "roll with the punches" and at times ask for clarification of what the
author intends (as is often the case in this forum).

Often pushing for the exact meaning of all the terms we encounter may result in returning to the
undefined terms of a system.  But then the expression we obtain for a "higher level" concept's
definition may be so long and involved with the elementary undefined terms as to be basically
incomprehensible.

As an example consider the "Sheffer stroke" or "Dagger" in T/F two valued logic.  Each of these
can be used to define the usual AND, OR, if..then, if and only if, NOT, exclusive OR.  But the
expressions for some of these are quite long and complicated using just the one stroke or dagger.
It is interesting that one "operation" can be used to define all the usual stuff, but it is way too
unwieldly to want to use it.  As humans we work better with the AND, OR, etc.

As another example consider binary vs hexadecimal.  Working with hexadecimal is much easier
for us humans than working with binary, especially when numbers are fairly large.  The strings
of 1's and 0's just get too long and difficult to deal with.

1/2(grateDAY )!

(3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)^(3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)
where 3!! means (3!)!, 3!!! means ((3!)!)! etc.
Note that (2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)^(2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!) = 4 so
there is a bit of a leap from using 2 to using 3 in these expressions.

Hi therussequilibrium! 

You didn't waste my time.  I enjoyed looking up and reading about the Riemann Zeta function.
Quite interesting (and mostly over my head).  Gauss at age 15 came up with the n/ln(n) formula!
He was quite the genius, eh?

Have a blessed day!

Thanks!  Works great!  Handles enough digits to do my bank account in binary

How hard would it be to allow the user to input a base of their own choosing?

Have a great day!

(B)  When a=c we obtain ax²+bx+a= 0 which has roots

       -b±√(b²-4a²) 
                2a

And multiplying these two roots together gives b²-(b²-4a²) = 4a²/4a² = 1.
                                                                          4a²

So the two roots are reciprocals of each other,  BUT the Discriminant is not negative unless
b²<4a².  So |b|<|2a| is required for the roots to be complex (not real roots).

So the first condition would be true if it were |b|<|2a| instead of |b|<|a|.

From this problem we see that there are infinitely many complex number pairs that are complex
conjugates AND reciprocals at the same time.  But there is only ONE pair of complex numbers that
are both OPPOSITES AND RECIPROCALS at the same time.  And that would be ...