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1. Open calculator.

2. Enter a square number like 25.

3. Click on the square root button in the calculator.

4. Now subtract the result from the square root of the number, in this case 5.

5. Click '=' button on the calculator and note that the result is not 0.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,366

Hi;

It is not a bug.

**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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Is it a different way of expressing Zero?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,366

First congratulations, half a zillion programmers will never know that in their entire life! A zillion math types do not believe it! They suspect that big bad Mi**8cr---o$--oft is at it again. For once they are not to blame.

**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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If it not a bug, whats it?

I didn't understand what you're talking of

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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

Isn't this the precision stuff you always mention, bobbym?

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: 86,366

Isn't this the precision stuff you always mention, bobbym?

Computer math ≠ human math

Okay, first you must get the spiel.

The hardest thing for computers to do is arithmetic. People find that amazing but it is true. The problem is not a bug but it is inherent in the way computers do arithmetic.

1) To a computer the number line is not solid like they draw it in mathematics. Instead it looks like the dots and dashes of morse code. This is because some numbers do not exist for a computer. For instance there is no 1 / 3 on its number line, just a big hole.

2) (a - b)(a+b)≠(a^2 - b^2). Algebraically equal expressions are not equal to a computer.

3) Addition is not commutative.

The order of addition can drastically affect the answer!

4) A computer can not subtract or multiply without possible disastrous error.

5) You never use the quadratic formula to solve a quadratic equation because on a computer it will give inaccurate results.

6) Newton's iteration though taught is rarely the best one for the job.

7) A computer can not compare theoretically equal quantities.

and many more.

**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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So, Is it possible that human beings will eventually develop something better?

5) You never use the quadratic formula to solve a quadratic equation because on a computer it will give inaccurate results.

Would you give an example?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,366

Hi Agnishom;

Yes, we could come up with something better. A human brain! But I think that has already been made.

Here is a whole bunch of examples that I worked on for MIF:

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 remember the problem on calculating sqrt(3.0000000000000001)-sqrt(3) you posed. Everything gives out 0 when the expression is evaluated directly, even M!

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: 86,366

Yes, without taking special pains on that one, subtractive cancellation will wipe out the answer.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathgogocart****Member**- Registered: 2012-04-29
- Posts: 1,426

Agnishom wrote:

1. Open calculator.

2. Enter a square number like 25.

3. Click on the square root button in the calculator.

4. Now subtract the result from the square root of the number, in this case 5.

5. Click '=' button on the calculator and note that the result is not 0.

they have a bug

Hey.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,366

One way or another all math programs will do this, even Mathematica.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathgogocart****Member**- Registered: 2012-04-29
- Posts: 1,426

bobbym wrote:

One way or another all math programs will do this, even Mathematica.

Thats Funny.

Hey.

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And sad....

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,366

Hi Agnishom;

It is a property of how computers and mathematics mix. Once you know about it you are prepared. We will go over it when the the time comes.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**philandy****Member**- Registered: 2013-06-10
- Posts: 11

If you had to use binary, decimal, and floats all at the same time, you might come up with sqrt(4)-2=-8.1648465955514287168521180122928e-39 as well. Notice the sqrt(1)-1 is fine. Personally I prefer to use the Google Dual-Mode Calculator app in Google mode, which gives sqrt(4)-2=0.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,380

hi

It's not a bug. It is just that calculators and computers can only do arithmetic imperfectly. Numbers are represented at machine code level by binary numbers. All fractions that do not have a power of two as the denominator have a recurring binary form. Since no machine can hold an infinite number of binary digits, it is inevitable that the representation will involve some sort of truncation of the true value. This leads to tiny errors between the value calculated and the true answer.

In the early days of calculators it was much more common to find such errors arising. Two things (at least two; there must be more that I don't know about) can improve the situation. (i) Increase the number of binary digits used to represent the number; (ii) use guard digits. The latter means that more digits are used than are displayed. The guard digits are used to round off the displayed results. This eliminates a lot of the problem.

On a calculator you can reveal the guard digits by doing a sum and then subtracting the displayed answer from itself. The difference is the guard. If rounding up has occurred you need to adjust what you see displayed by one to allow for the rounding.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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