Calculators can do it, why not us?

Calculators can do it, why not us?

 

Go right ahead. No one is stopping you.

Go right ahead. No one is stopping you.

 

Did you not read the title? I'm asking how to do it.

Edited October 23, 201114 yr by questionposter

Seriously, never heard of Google? http://mathforum.org/library/drmath/view/52469.html If you were a decent mathematician in the 1600-1700-1800s but not a great one, you probably spent at least some significant amount of your time helping expand the scope of the tables. Also, the series expansions are pretty well known today.

Edited October 23, 201114 yr by Bignose

As well as the series expansion that bignose linked to (which will give you an arbitrarily precise value if you are patient enough).

If you just want a very approximate value, it's useful to know that [math]e^3[/math] is very nearly 20.

So if we have some random number:

7052443

 

We can compare it to powers of 20

20

400

8000

160000

3200000

64000000

 

And see that It's about double fifth power of 20 [math](2\times e^{15})[/math].

e is about 3, so we can say it'll be fairly close to [math]e^{16}[/math]. Or that the natural log of this number is about 16.

 

Doing it on a calculator gives me 15.769, so I was reasonably close. Not really close enough for precise work, but if you just want an order of magnitude then it's good enough.

With a bit of practise (and maybe memorising a few more numbers) you can get one or two sig figs doing it in your head.

Edited October 23, 201114 yr by Schrödinger's hat

Seriously, never heard of Google? http://mathforum.org...view/52469.html If you were a decent mathematician in the 1600-1700-1800s but not a great one, you probably spent at least some significant amount of your time helping expand the scope of the tables. Also, the series expansions are pretty well known today.

 

The reason I didn't want to rely on google is cause I'm still not seeing exactly how its done. I can see some approximation techniques and that people made tables along with the trigonometric function tables, but how can a calculator do it and not us if we programmed only our knowledge into a calculator?

I think a calculator would do it pretty much the same way as it would be done by hand, by approximating the function with another function, which by design can be truncated for any desired accuracy. the desired accuracy is part of the programming.

The reason I didn't want to rely on google is cause I'm still not seeing exactly how its done. I can see some approximation techniques and that people made tables along with the trigonometric function tables, but how can a calculator do it and not us if we programmed only our knowledge into a calculator?

 

The calculator just uses the same approximation methods but does them very quickly. If it gets the approximation accurate to the number of digits it can display (plus a few extra for rounding error), you don't notice that it's approximate.

There are also things like computer algebra systems that work a little bit differently (more like you or I would treat a logarithm, only evaluating it if asked for a number).