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Old 02-10-2009   #132 (permalink)
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Post The need to distinguish typographical transformations from algebraic ones

Quote:
Originally Posted by Don Blazys View Post
"Crossing out" the T's on both sides of the equation:

T\left(\frac{a}{T}\right)^{\left(\frac{\frac{x\ln(a)}{\ln(T)}-1}{\frac{\ln(a)}{\ln(T)}-1}\right)}=\left(\frac{T}{T}\right)a^x

would result in:

\left(a\right)^{\left(\frac{\frac{x\ln(a)}{\ln()}-1}{\frac{\ln(a)}{\ln()}-1}\right)}=a^x
If by “crossing out the Ts”, one means “removing every occurrence of the character “T”, this is true. However, this typographical is not a valid algebraic transformation, any more than “crossing out the ‘+’s” is a valid transformation of
“1+2+3=6” into “123=6”.

In algebra, the correct meaning “crossing out” like terms is “summing the exponents of like terms”. So
T \left( \frac{a}{T} \right)^b
is rewritten
T^1 a^b T^{-b}
summing the exponents of like terms gives
T^{1-b} a^b
which may be rewritten
\frac{a^b}{T^{b-1}}

I don’t doubt that there are many “mathematical crises” in the form of schoolteachers who understand Math so poorly they can’t distinguish between typographical and algebraic transformations. The solution, however, is IMHO better education of teachers and more careful selection of teachers by schools to prevent people who don’t understand fundamental math from teaching it, not new, informal, and unsupported ideas about algebra, like “Blayzs terms”.
Quote:
Originally Posted by Don Blazys View Post
In my heart of hearts, I know that I am nothing but a tired old fuddy duddy with a cantankerous disposition and a fading memory, which at the very least qualifies me as being "fundamentally humble".
Ah, but you’re a very fun tired, cantankerous old fuddy duddy, and a valued member of hypography.

Humility, in my experience, is an easier character trait to claim than to attain. It’s easy to express humility by acknowledging ones shortcomings. It’s harder to express humility by recognizing that criticism of a cherished idea is correct, and the idea is wrong.


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