[Don Blazys]

In this topic, I will be making an extraordinarily "strange claim", almost unbelievable in fact, and it will be up to you, the reader, to decide if the properties of logarithms, strange as they may be, are telling us the truth!

So hold on to your hats and buckle your saftey belts, because you are about to embark on the strangest, wildest, and perhaps most wonderfull mathematical ride of your life!

Let all variables herein represent non-negative integers. Then, the recently discovered identity:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

shows that it is algebraically impossible to "cross out" the cancelled T's. What does this mean to you? Should we stop teaching students to "cross out" cancelled factors and common factors? I say yes, and will present my reasons for doing so as this thread continues. What do you say?

Don.

[Jay-qu]

Have you heard of latex? Try giving that a go and it will make your maths more legible. At the same time you may as well try and give a more mathematical formalism of what it is you are trying to prove/disprove. Let me know how you get on.

[CraigD]

Originally Posted by Don Blazys

Let all variables herein represent non-negative integers. Then, the recently discovered identity:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

If the above expression assumes the usual precedence of operations (eg: exponentiation before multiplication), then the given equation is provably false.

As Jay notes, math is much more readable using hypography’s available LaTeX package, so I’ll translate Don’s text to that for this very short proof:

removing identities (ones),

expanding and combining common coefficients,

dividing both sides of equation by ,

we find

so,

except for special cases and .

Originally Posted by Don Blazys

… shows that it is algebraically impossible to "cross out" the cancelled T's. What does this mean to you?

Because the first given is false, no further comments are formally meaningful.

Originally Posted by Don Blazys

Should we stop teaching students to "cross out" cancelled factors and common factors? I say yes, and will present my reasons for doing so as this thread continues. What do you say?

I say yes, too, but with some qualification.

From several years of tutoring college Math, a combined couple of years of teaching remedial college and GED-track high school Math, and the common parental experience of helping several now adult children (my own and others) through grades K-12, I think there’s a great need to acquaint students with the idea of mathematical formalism – that is, the idea that, as someone put it, “algebra is a collection of exact rules for manipulating marks on paper” – at nearly the earliest age possible, 5 or 6 years.

Reliance on phrases like “cross out” and “cancel”, in my experience, usually show a lack of grasp of the fundamentals of formalism, a lack that can handicap students in Math and a large collection of similar disciplines for the rest of their lives. Unfortunately, in the present-day US, and, I suspect, many other states’, schools, formalism is usually introduced only in the last years of high school or later years of college (ages 16 to 21), and then only to students with specializing in Math-intense education paths.

IMHO, a strong grasp of formalism needs to be as essential as knowing ones alphabet, and taught as early and as well. My opinion is by no means original or untried, dating back at least to suggestions from turn-of-the-20th-century mathematicians and educators, and having made sporatic widespread appearances in curriculum under monikers such as “new math” for over 50 years, as well as being taught in isolated instances in many schools and by many teachers within and outside of traditional schools. My experience with attempting to promote this approach, however, has taught me that it’s not an easy task. It’s necessary to first win the hearts and minds of professional educators, who are often themselves math-adverse to the extent of near innumeracy. Learned early and well, formalism is, in my experience, an easy concept to understand and apply, but studied late in ones education, very difficult to really grasp and appreciate.

PS: Don, you can pick up the essentials of hypography’s implementation of the LaTeX math package and markup code tags by clicking “quote” on this post and inspecting the text. More in depth documentation on LaTeX can be found in many places, including a couple of my favorites, the wikimedia article “Help: Displaying a formula” and this easy-to-use tutorial. For documentation on hypography's supported markup tags like [math], see the BB Code List FAQ page.

In math, presentation – pretty handwriting, or its computerized equivalent – can be very effective, and is IMHO worth making a strong effort to learn well in whatever medium you find yourself.

[Don Blazys]

To: Jay-qu. Due to circumstances beyond my control, I can't post in LaTex at this time. However, if you keep in mind that operations are to be performed from the intermost parenthesis outward, then you should have no trouble understanding the identity:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)).

You know, when I was in high school back in the 60's, we didn't have personal computers, calculators and the like. We used pencil, paper and slide rule to solve our math problems, and although my generation developed much of the technology in use today, we never became overly dependent on it. Just for the fun of it, why don't you give it a go and see if you can verify the above identity without me putting it into LaTex?

Don.

[Don Blazys]

To: CraigD. The equation:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

is both true and correct. It is not false!
You made a mistake when you put it into LaTex.
Note that the expression: (ln(T)) is a divisor, not a multiplier.
Please try putting it into LaTex again, but with the correct operations this time.You will then find that it is an identity, and that it is indeed algebraically impossible to "cross out" the cancelled T's.

Don.

[Jay-qu]

Originally Posted by Don Blazys

To: Jay-qu. Due to circumstances beyond my control, I can't post in LaTex at this time.

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)).

Just for the fun of it, why don't you give it a go and see if you can verify the above identity without me putting it into LaTex?

Don.

LateX only helps the output look nice, it doesnt make the identity any easier to solve. Its just nicer to read as craig has shown - and no its not like doing things with pencil and paper is harder, in fact I much rather it for maths. Displaying maths on the computer is much slower and cumbersome than on paper.

[Qfwfq]

And I did not need to work it onto paper, nor reach Craig's post, to see it is not an actual identity, and when I got to Craig's post I noticed some slips which Don himself didn't point out.

Don I see no way for the exponent to identically equal 1 independently of x and if there's any way of the logarithms in the exponent compensating the T's in the base, go ahead and show us. You will not increase your audience by harping about pencil and paper in the good ole days instead of making the effort to use LaTeX.

[CraigD]

Originally Posted by Don Blazys

To: CraigD. The equation:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

is both true and correct. It is not false!

As I trivially prove in post #2, the equation is true only for the special cases of or

The flaw with the proposed identity is that exponentiation isn’t associative with multiplication, that is .

Originally Posted by Don Blazys

You made a mistake when you put it into LaTex.

I did. My apologies – I’ve corrected it.

Since the exponent in question still equate to , however, the difference has no impact on the proof.

Originally Posted by Don Blazys

Due to circumstances beyond my control, I can't post in LaTex at this time.

No special software is needed to use the LaTeX math package in hypography posts, just the post entry box, or whatever text editor you use to make posts. Once you’ve learned a few basic functions, such as \frac{a}{b} as an equivalent of a/b, you’ll find LaTeX text it’s very similar to the text you’ve already written. For example, the LaTeX for
(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))
is
\left( \frac{T}{T}\right)a^x = T \left( \frac{a}{T} \right) ^{\left( \frac{\left( \frac{x \ln(a)}{\left( \ln(T)-1\right)} \right)}{\left( \frac{\ln(a)}{\left( \ln(T)-1\right)} \right)} \right)}
If the parenthesis made unnecessary by the other markups are omitted, it’s an even simpler
\frac{T}{T}a^x = T \left(\frac{a}{T} \right)^{ \frac{\frac{x \ln(a)}{\ln(T)-1}}{\frac{\ln(a)}{\ln(T)-1}} }

Originally Posted by Don Blazys

You know, when I was in high school back in the 60's, we didn't have personal computers, calculators and the like. We used pencil, paper and slide rule to solve our math problems, and although my generation developed much of the technology in use today, we never became overly dependent on it.

We didn’t have PCs when I was in high school in the ‘70s, either. Though there are some input devices that manage to make a computer fairly equivalent to paper and pencil, I, and nearly everyone I know, still use paper and pencil for most math and graphical sketching. Only when “neatening up” for posting do we transcribe into LaTeX or another computer displayable form.

An alternative to using LaTeX or similar math markup schemes is to scan paper documents and upload them as images or thumbnails in your posts. You’ll notice that members such as Turtle (who writes a lot of complicated, pretty things that can’t be rendered in LaTeX) do this quite a bit. For ordinary math expressions, however, most people find it easier to use LaTeX, especially folk who don’t have easy access to a scanner.

[Don Blazys]

To:Qfwfq. The identity:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

is true! It is however, quite counterintuitive, and most people have a very hard time believing it. One professor even went so far as to graph both sides on a supercomputer, then called me the next day to express his utter amazement. So please keep at it. Once you understand it, I guarantee that you will also be amazed. You can find the formal derivation of the above equation on my website (donblazys.com), as well as a lively discussion about it on the Marilyn Vos Savant general discussion forum.

Don.

[Don Blazys]

To:CraigD. The identity:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

is true. Please keep in mind that the logarithms can be "crossed out" if and only if x=1. For all other values of x, the logarithmic exponent must remain exactly as it is. In other words, don't confuse the above logarithmic exponent, which is:

((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)) for

x((ln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)).

There is also a lot more information about this equation, including its derivation, on my website (donblazys.com).

As for putting the above equation in LaTex, I really appreciate all the help that you have given me so far. However, the problem that I am having is this:
While on the "reply to thread" board, after I type in the "commands":
[math] \left(\frac{T}{T}\right)a^x....etc.,
I can't get those "commands" to convert my equation into LaTex.
Any suggestions?

Gotta go now, but I will be back on Monday. Have a great weekend!

Don.