Re: (sin X)^2?
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Re: (sin X)^2?
Ricardo Navarro writes:
> I don't know if this has been answered here before so here's my question:
> does anyone know why TI decided to make you put in (sin X)^2 instead of
> sin^2 X? that would make typing in equations like that simpler...
The notation sin^2 (x) that is common in mathematical writings is really
what mathematicians call "a convenient abuse of notation". Strictly speaking,
it makes little sense, but everyone knows what you mean when you write it.
Combining functions f and g by multiplication, you can get a new function f*g
that is defined by (f*g) (x) = f(x) * g(x). The values at a point are multiplied
to get the value of the new function at the point. You could also multiply
f by itself to get f*f, which it would be possible to call f^2, and this
would be consistent with the notational convention being discussed. But,
except for the other circular functions (and maybe the hyperbolic functions)
this convention is not used elsewhere in traditional mathematical notation.
It is also possible to _compose_ two functions f and g to get a new function
fg defined by (fg) (x) = f(g(x)). Composing f with itself, you would get ff,
which is also sometimes written f^2 ! This would suggest that sin^2 (x) should
mean sin(sin(x)), though it never does.
Traditional mathematical notation is a hodepodge of good ideas that are not
necessarily consistent with each other. (We might also look at the use of f^-1
as a name for the inverse of a function, when the letter f printed upside down
would be more reasonable if less practical). When computer languages came along
a golden opportunity arose to create a new, better, more logical notational
system. Alas, again many ideas fought with each other and none won. To my own
taste,the notation used in APL is superior to anything else I have ever run
across, but adoption of the conventions established by APL runs enough counter
to
solidly-established habit that this will never happen. But a computer language
needs in any case to be consistent within itself, to allow unambiguous parsing
of entered commands.
Calculator syntax, like traditional mathematics, has mostly "just grown". As
with computer languages there is a need to be unambiguous. But there is also a
need for _convenience_ that far outweighs issues of logical consistency, and
many compromises have been made along the way. TI has generally done a pretty
good job, as their product line matured, with maintaining balance between
mathematical convention and logical clarity (the language used with the TI-89
and TI-92 is not bad, for example). But the particular abuse used in printed
mathematics involving powers of the sine and related functions just can't be
swallowed by a parser -- even though it is possible to "pretty print" this
convention back at the user in the window display. There is no substitute,
at the elementary level, for unambiguously calling for squaring the _value_
of the function. It might, howver, be worth pondering the fact that with the
TI-83 the expression sin(x)^2 will give the desired result!
I might say that, as an educator, I think this tension about notation is a
good thing, forcing users to think about the actual _meaning_ of the expressions
they enter. I always encourage my students to search for alternative ways to
enter given expressions, rather than just copying what is printed on the page.
Sometimes surprising economy is possible with a little thought (using the
reciprocal key in interesting ways, for example). It never hurts to keep an
attitude that perhaps there is a better way...
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<