Science Forums 2

[CraigD]

Easy questions first…

Originally Posted by Don Blazys

P.S. What kind of calculator do you have that is accurate to 18 decimal places…

I’m using the built in calculator of an implementation of the MUMPS programming language (specifically, Intersystems Caché) that I usually have a terminal session connected to. If I recall correctly, the language standard requires it give at least 16 digits precision, and most do better.

I’ve written my own calculators in MUMPS for when I must be able to, for example, do precise arithmetic with million digit numbers. MUMPS is traditionally a database-focused language, used a lot in medicine and banking, so I’m a bit of an oddball for using it for math and science, although it has interesting and fairly unappreciated, IMHO, potential in these disciplines.

MUMPS is a programming languages, so a bit much to install and manage just to have a decent calculator, though very handy if you do, and learn just a bit about using it. You may have notices in some of my posts that you can write pretty useful MUMPS programs with a very small number of keystrokes.

There are several applications that include high or arbitrary precision calculators, such as Mathematica and Maple. Apps like these do a lot more than just literal calculations – I find that they make me feel rather stupid, as in a sense, they “know” much more math than I do.

Many folk find the ruby language handy for use as a calculator Lots about it, including documentation, tutorials, and a free Windows implementation of it, can be found here. If you can’t or don’t want to install it on your computer, you can use it via a browser here.

Originally Posted by Don Blazys

... or is such accuracy available only in conjunction with a computer?

Short answer, yes, you need a computer capable of running your calculator app of choice. Though in principle you can run nice calculator apps on the average cellphone these days, in practice this is pretty tricky. Mostly, I use a computer as a calculator. I’m rarely without a computer of a sort, having a PalmOS handheld in my pocket most of the time, though my calculators on it aren’t as good as on a Windows or unix machine. I hope someday soon to have about the same resources on my handheld as on a normal PC - I’m not quite there yet, but may be soon, as there are a growing number of handhelds being brought to market that run general-purpose operating systems, typically linux.

On the subject, you might find the hypo thread What is your favorite calculator? interesting.

Originally Posted by Don Blazys

Are they expensive?

Assuming you have a PC, you can get free copies of Caché or several other implementation of MUMPS (there’s also an open source version). Ruby is also free, via the links above.

Mathematica and Maple are a bit pricey, though if you’re with a school, you can usually get a good discount.

Now for a harder question…

Originally Posted by Don Blazys

Also, is it now proper to "cross out" the cancelled T's in the term on the left as you did in the second equation in post#12? I have a problem with that because once the T's are "crossed out", the term on the left appears as a^x, which can be construed as (1)a^x=(N/N)a^x where N does not equal T, and while the equation would remain true, it would no longer be an identity, and we would be either ambiguous (in the case of (1)a^x), or inconsistent (in the case of (N/N)a^x) as to the value of the cancelled factor or cancelled common factor. Again, what do you say?

Algebraically, it’s always proper to replace a term of the form with 1. When doing so, it’s important to retain any domain constraints of the replaced term (eg: )

Usually, one tries to write math canonically, replacing any terms equal to 1, so


is preferable to


. Whether you prefer


is largely aesthetic. I like it. I like even more this variation:


I don’t think it can be written much more tersely than this.

All of these are identities, with the restriction , , , and .