Arithmetic and Significant Digits

atran · 2013-07-12 08:36:57

Hi... Subtraction from a negative number means to add the opposite of that negative number, for example:

Instead of thinking that way, is it right to think like this:
Because the former way does not really explain, for example, 5-(3+x) = 5 + -(3+x), but the second reasoning does well.

I have another issue with significant digits: Say we have a measured number x=2.0 with two significant digits. So basically, does this say that x is any value between 1.95 and 2.05: 1.95≤x<2.05 or 1.95≤x≤2.04?

Thanks for help.

bobbym · 2013-07-12 09:41:45

Hi atran;

Welcome to the forum.

For numerical work when you say 2.0 you are implying 1.95 ≤ x ≤ 2.05. This is because the error is implied to be .05 That is what I would do.

Bob · 2013-07-12 09:43:38

hi atran

Welcome to the forum.

Your explanation for negatives works well for me. Many people get in a 'right twist' trying to justify the rules.

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

if a number is given as 2.0 that means it could have any value that rounds to 2.0

So as low as 1.95 and as high as 2.05

(Because 2.05 is equidistant from 2.0 and 2.1, it is permissible to round it either way so 1.95 ≤ N ≤ 2.05 is correct. It is conventional to round 5s up but it is not compulsory.

Definitely not ≤ 2.04 as this would exclude 2.04999

Bob

bobbym · 2013-07-12 09:53:53

Hi;

The definition of significant digits as defined in Numerical Analysis is:

If y is any approximation to x then we say the kth decimal place is significant iff

Notice the less than or equal sign.

atran · 2013-07-12 22:05:57

Thank you. So x=2.0 means that 1.95 ≤ x ≤ 2.05, while 1.95 and 2.05 being exact numbers.
An exact number is a number with an infinite number of significant digits. Am I thinking right?

Last edited by atran (2013-07-12 22:06:21)

bobbym · 2013-07-12 22:10:11

2.05 is no more exact then 1.95. They both have 3 significant digits.

4 is an example of an exact number.

atran · 2013-07-12 22:14:54

What does being exact mean then? The more significant digits, the more exact a number is? In this context, is exact a synonym to accurate?
In pure mathematics, say x=2.01, isn't x an exact number?

bobbym · 2013-07-12 22:18:27

Mathematics and computation are not the same. Actually, significant digits are a measure of how stable a number is for further computation on it. 2.05 suggests an error of ± .005.

5 is exact because it is an integer. 5.0 is not because it is a floating point number.

atran · 2013-07-12 22:25:15

I've many gaps in mathematics, therefore I want, starting from the bottom, to re-learn some topics. Can you recommend me a book which covers many different topics and gives proofs for some rules? I have thought of buying "Finite Math and Applied Calculus" by Stefan Waner and Steven Costenoble.

bobbym · 2013-07-12 22:34:02

What type of math are you studying?

atran · 2013-07-12 22:38:14

I want to self-study a range of topics, from pre-algebra to calculus.

bobbym · 2013-07-12 22:42:39

I do not have that book.

Have you looked at the book, I mean held it in your hands.

atran · 2013-07-12 22:51:07

No I haven't, but the table of contents is:

0 Precalculus Review
1 Functions and Applications
2 The Mathematics of Finance
3 System of Linear Equations and Matrices
4 Matrix Algebra and Applications
5 Linear Programming
6 Sets and Counting
7 Probability
8 Random Variables and Statistics
9 Nonlinear Functions and Models
10 Introduction to the Derivative
11 Techniques of Differentiation with Applications
12 Further Applications of the Derivative
13 The Integral
14 Further Integration Techniques and Applications of the Integral
15 Functions of Several Variables
16 Trigonometric Models

bobbym · 2013-07-12 23:00:15

Looks like it covers lots of stuff. Books today are expensive. That is why I recommend you look through the book first. Sometimes google will have a copy online or you can go to a college bookstore and leaf through it. Lastly you can usually find a copy in a university library.

Check to see that you like the way the book looks. Make sure it has plenty of problems and solutions for all of them.

If you can not get your hands on it beforehand the purchase will be iffy at best. You will just have to hope for the best.

atran · 2013-07-13 00:40:12

bobbym wrote:

2.05 is no more exact then 1.95. They both have 3 significant digits.
4 is an example of an exact number.

If each of 1.95 and 2.05 has three significant figures, then 1.95 is any value between 1.945 and 1.955, and for x: 1.945 ≤ x, which is wrong since 1.95 ≤ x.

bobbym · 2013-07-13 01:26:23

then 1.95 is any value between 1.945 and 1.955

That is correct.

atran · 2013-07-13 03:11:04

So even though 1.95 is any value between 1.945 and 1.955, x remains (1.950 ≤ x).
Or is this (1.945 ≤ x) correct?

I'm a bit confused...

bobbym · 2013-07-13 03:30:17

Let's start again.

1.95 implies an error of ± .005

1.945 ≤ 1.95 ≤ 1.955

atran · 2013-07-13 03:51:31

Sorry, I meant x=2.0

bobbym · 2013-07-13 03:54:11

If you say x = 2, then you are dealing with an integer, an exact quantity.

The error is 0.

When you say x = 2.0 then you are dealing with a floating point number. That could be from a measurement or it has an implied error built in of .05

atran · 2013-07-13 04:02:36

I mean if x=2.0, then (1.95 ≤ x ≤ 2.05). But is it the case (a) or (b)?
a) 1.950000000000 ≤ x ≤ 2.050000000000 (with infinite number of significant digits)
or b) 1.945 ≤ x ≤ 2.055

bobbym · 2013-07-13 04:08:13

1.950000000000 has an implied error of .0000000000005 which is much smaller than .05.

b) I would say is just wrong.

atran · 2013-07-13 05:09:11

I still don't get it. If 1.95≤x=2.0, and if 1.945≤y=1.95, I know that y≤x, but why not 1.945≤x?
Is it because if 1.945 is rounded to two significant figures, it's 1.9 instead of (2.0)?

Last edited by atran (2013-07-13 05:09:23)

atran · 2013-07-13 05:13:38

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

bobbym · 2013-07-13 05:15:23

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?

Math Is Fun Forum

#1 2013-07-12 08:36:57

Arithmetic and Significant Digits

#2 2013-07-12 09:41:45

Re: Arithmetic and Significant Digits

#3 2013-07-12 09:43:38

Re: Arithmetic and Significant Digits

#4 2013-07-12 09:53:53

Re: Arithmetic and Significant Digits

#5 2013-07-12 22:05:57

Re: Arithmetic and Significant Digits

#6 2013-07-12 22:10:11

Re: Arithmetic and Significant Digits

#7 2013-07-12 22:14:54

Re: Arithmetic and Significant Digits

#8 2013-07-12 22:18:27

Re: Arithmetic and Significant Digits

#9 2013-07-12 22:25:15

Re: Arithmetic and Significant Digits

#10 2013-07-12 22:34:02

Re: Arithmetic and Significant Digits

#11 2013-07-12 22:38:14

Re: Arithmetic and Significant Digits

#12 2013-07-12 22:42:39

Re: Arithmetic and Significant Digits

#13 2013-07-12 22:51:07

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#14 2013-07-12 23:00:15

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#15 2013-07-13 00:40:12

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#16 2013-07-13 01:26:23

Re: Arithmetic and Significant Digits

#17 2013-07-13 03:11:04

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#18 2013-07-13 03:30:17

Re: Arithmetic and Significant Digits

#19 2013-07-13 03:51:31

Re: Arithmetic and Significant Digits

#20 2013-07-13 03:54:11

Re: Arithmetic and Significant Digits

#21 2013-07-13 04:02:36

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#22 2013-07-13 04:08:13

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#23 2013-07-13 05:09:11

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#24 2013-07-13 05:13:38

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#25 2013-07-13 05:15:23

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