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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Hi;

This question popped up in another thread and was answered there. Here we look at why that idea worked and the general theory.

It obvious that both square roots are going to be nearly equal and therefore that subtraction will cause much subtractive cancellation. This will cause the loss of significant digits and indeed many calculators will return 0.

The Taylor series is the main tool of numerical work. It comes to the rescue here too!

If we expand the following around 0 in terms of epsilon we get:

we can reaarange this to,

This is exactly what we need, we call x = 3 and epsilon = .000 000 000 000 000 1

We can truncate 2) to

now just plug in:

Which is quite accurate for only 2 terms of a taylor series.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Hi bobbym

What was wrong then whit my suggestion in the other thread?I just set x=3 immediately.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Did you try it and see if it got the right answer?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

http://www.wolframalpha.com/input/?i=x% … 0000000001

It is just a problem that Wolfram doesn't want to calculate it to more digits.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Before I can comment on anything I ask, why are you using Wolfram when you have mathematica?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

I don't know.It makes my computer slow.

*Last edited by anonimnystefy (2012-04-23 20:30:27)*

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Even when it is off?!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

No,when it is open.I haven't deleted it!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

So then shut it down when you are finished with it.

First, Alpha in its desire to make an engine that understands English has ruined the mathematica language! You will never learn the correct syntax by using alpha.

Here is what you should have entered

N[x/(2sqrt(3))-x^2/(24sqrt(3)),20] for x=10^(-16)

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

I have opened Mathematica and entered:

N[x/(2*Sqrt[3]) - x^2/(24*Sqrt[3]), 20] for x = 0.0000000000000001;

but it gives me:

"Set::write: Tag Times in 2.88675*10^-17 1.*10^-16 for is Protected. >>"

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

That is the problem! That code is for Alpha. They are not the same!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

I even tried setting x=... before it.It doesn't give me more digits.How do I enter the code so that I get more digits?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

You are not listening. Mathematica and alpha are very different.

Put this in alpha:

N[x/(2sqrt(3))-x^2/(24sqrt(3)),20] for x=10^(-16)

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Can't you show me what to do in Mathematica?I want to learn Mathematica language and functions.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

You use Mathematica functions for alpha too!

Mathematica allows many forms of input. This is the simplest:

N[x/(2Sqrt[3])-x^2/(24Sqrt[3]),20] /. x->10^(-16)

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

It still gives me 2.88675*10^-17

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Your display digits is set too low.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

How do I change that?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Try this first, Edit then Preferences then Appearance then Numbers. Tell me when you are there.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Actually,I found it,and I set 100,and it gives me this output:

2.8867513459481294*10^-17

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

100 is too large. Set it to 16 and then calculate Sqrt[2.], what do you get?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

1.414213562373095

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Then it is working correct. Go into preferences and get to internet connectivity. Uncheck allow mathematica to access the internet.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Why?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,435

Shutting off the internet to mathematica might speed up the machine greatly. If you want to communicate with alpha use your browser.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**