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#26 2013-07-13 07:47:05

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

bobbym wrote:

x and y have different errors. You are going way past what is required. You only need to apply the definition in post #4 to understand this.

What about x=2.0≤2.05, y=2.05≤2.055, and x≤2.055?

If x ≤ 2.05 then of course x ≤ 2.055. What does that have to do with what we are doing?

If so, then x ≤ 2.055555555... That's why I asked whether 2.05 was an ordinary number or a number with three significant figures.

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#27 2013-07-13 10:14:06

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

What is an ordinary number?

When establishing a bound you pick the tightest one.

If x < 10, you do not care that it is less than 117 too.

1.95 <= x <= 2.05 is the tightest bound.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#28 2013-07-14 05:15:20

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

Sorry for asking too many questions.

Say x=2.0, then (1.95 ≤ x ≤ 2.05). If the two bounds are not exact, then they can be expressed further as (1.945 ≤ 1.95 ≤ 1.955) and (2.045 ≤ 2.05 ≤ 2.055), and continuing indefinitely results, 1.944444444... ≤ 1.95 ≤ x ≤ 2.05 ≤ 2.055555555... => 1.944444444... ≤ x ≤ 2.055555555...

And I find no reason why the bounds should be inexact, i.e. consisting of three significant digits.

Last edited by atran (2013-07-14 05:16:07)

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#29 2013-07-14 05:21:45

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

Nearest I can understand it is like this.

The error, written as e is .005. I would have to think that one is exact. It is not a fp number it is an interval.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#30 2013-08-02 03:00:47

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

Hi,

Why do so many books and web-pages use these two rules for addition:
1. To add two numbers with the same sign, you add their absolute values and attach their common sign to the sum.
2. To add two numbers with unlike signs, you find the difference between their two absolute values and attach the sign of the number with the larger absolute value to the difference.

Isn't it easier and better to think this way: Say (a) and (b) are two positive real numbers and a>b, then,
a+b is just simple addition
-a+-b = -(a+b), simpler to think this way
a-b is a positive number
b-a = -(a-b), in other words the opposite of the difference

Last edited by atran (2013-08-02 03:01:04)

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#31 2013-08-02 03:18:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

Hi;

Looks okay to me but you are just missing the subtraction of a negative and a positive.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#32 2013-08-02 03:35:20

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

Do you mean a-(-b)?

Last edited by atran (2013-08-02 03:36:35)

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#33 2013-08-02 03:38:41

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

Yes, it is not obvious how your rules cover that while their rule 2 does.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#34 2013-08-02 05:06:27

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

I don't see how a-(-b) is related to the second rule. First I have to turn it into an addition problem, and since subtracting a number is the same as adding its additive inverse then: a-(-b) = a+b

Last edited by atran (2013-08-02 05:06:59)

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#35 2013-08-02 07:15:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

I do not see how either now that I am looking at it again. Just a hallucination on my part.

What you have done is algebraically correct. If yours is clearer for you than the others then use yours.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#36 2013-08-02 07:36:27

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

Say -(a+b): Is the minus sign a unary operator or is the expression equivalent to -1*(a+b)?

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#37 2013-08-02 20:11:29

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

I would say that is one way of looking at it.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#38 2013-08-03 00:32:14

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

I'm really feeling lost... Is there a book you can recommend which explains (with proofs) the rules behind arithmetic and algebra? Should I search for real analysis, abstract algebra, or number theory books?

I've found this book (amazon.com/books/dp/082182693X), but its level seems to be above mine.

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#39 2013-08-03 01:47:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

I would not buy a book until I held it in my hands first and leafed through the pages. I was not able to find one but you could look for an analysis book that was a little simpler.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#40 2013-08-03 06:16:20

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

What about Hardy's A Course of Pure Mathematics (gutenberg.org/ebooks/38769)?
I couldn't find a book which derives the rules by a set of axioms.

Now really, is the minus sign as a prefix a unary operator or is it equivalent to (-1)?
If it's a unary operator, then what is the explanation for [-(a+b)=-a+-b]?

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#41 2013-08-03 06:36:19

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Arithmetic and Significant Digits

Wiki says it is a unary operator. But isn't that a programming rather than a math term?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#42 2013-08-04 07:37:58

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

I'm thinking like this:
If (a+b)+-(a+b)=0 then, a+b+-(a+b)=0
a+(b + -(a+b))=0 => b + -(a+b) = -a, therefore -(a+b)=-a+-b

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#43 2013-08-05 07:34:52

atran
Member
Registered: 2013-07-12
Posts: 91

Re: Arithmetic and Significant Digits

bob bundy wrote:

Of course, you'll still have to justify -1 x -1 = + 1  smile

Is it right to think this way: Think of - as a unary operator and then,
(-a)b= a(-b)=-(ab) => -1*-1 = -(-(1*1)) = 1
(-a)b= a(-b)=-(ab) and -a = (-1)*a => -3*-5 = -(-(3*5)) = 15

Is there a book or a source which clarifies and explains this?

Last edited by atran (2013-08-05 07:35:13)

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