Approximations

Z

Zelda22

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True or False?
0.075≈0.07500

I think is true, 0.075 rounded to the nearest hundred thousandths is 0.07500.

Am I missing anything here? Seems too simple. Thanks
 
Zelda22 said:
True or False?
0.075≈0.07500

I think is true, 0.075 rounded to the nearest hundred thousandths is 0.07500.

Am I missing anything here? Seems too simple. Thanks
Click to expand...
It isn't even a matter of rounding; they are the very same number, because the zeros at the end have no effect.

Most likely this was among other questions that look similar but aren't nearly equal, while this one might lead to the opposite error.

Can you show us the context (e.g. a few questions near it)? What makes me curious is why they are asking about approximation rather than equality.
 
Dr.Peterson said:
It isn't even a matter of rounding; they are the very same number, because the zeros at the end have no effect.

Most likely this was among other questions that look similar but aren't nearly equal, while this one might lead to the opposite error.

Can you show us the context (e.g. a few questions near it)? What makes me curious is why they are asking about approximation rather than equality.
Click to expand...
Thank you. Here is the original question.
 

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Like Dr. Peterson, I’d like to know more about the context because, in certain contexts, the number of decimal points specifies information about the precision of estimates.

Absent some convention about precision, [imath]a \approx b[/imath] does not entail that [imath]a \ne b.[/imath]

That is, I interpret

[math]a \approx b \iff 0 \le |a - b| < \epsilon, \text { where}\\ \epsilon \text { is a positive number that, under the circumstances, is not material.}[/math]
 
JeffM said:
Like Dr. Peterson, I’d like to know more about the context because, in certain contexts, the number of decimal points specifies information about the precision of estimates.

Absent some convention about precision, [imath]a \approx b[/imath] does not entail that [imath]a \ne b.[/imath]

That is, I interpret

[math]a \approx b \iff 0 \le |a - b| < \epsilon, \text { where}\\ \epsilon \text { is a positive number that, under the circumstances, is not material.}[/math]
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0.075 is in D3 and 0.07500 is in D5

I understand there are some rational numbers that their approximation to D3 is 0.075 and D5 is not 0.07500.
But I'm not sure if that's what this question is asking.

For example,

r=0.075160
Approximation to D3=0.075
Approximation to D5=0.07516

Would be ok to say that 0.075 is approximately equal to 0.07500?
 
Zelda22 said:
a decimal is any rational number that has a simple fractional name whose denominator is 1 or a power of 10. Such a fractional name is called a decimal name.
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r ∈ D
if and only if, r ∈ D0, or r ∈ D1, or r ∈ D2...etc
 
So [imath]\dfrac{5}{6} \notin D[/imath] So would it be in [imath]\mathbb{Q} - D[/imath]?

-Dan
 
It would truly help us help you if you would give us the formal definition of D, particularly if there are different kinds of D.

This looks more and more as if we are talking about some exact definition for the approximation symbol.
 
lookagain said:
The statement translates to "0.075 is almost equal to 0.07500."

As they are actually equal to each other, this statement is then false.
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In "measurement" sciences those are not equal - because those numbers will have implied accuracy statement.

It is really important to know the reference (context) of this question.
 
Dr.Peterson said:
It isn't even a matter of rounding; they are the very same number, because the zeros at the end have no effect.

Most likely this was among other questions that look similar but aren't nearly equal, while this one might lead to the opposite error.

Can you show us the context (e.g. a few questions near it)? What makes me curious is why they are asking about approximation rather than equality.
Click to expand...
x≈y it means "x" is the best approximation for "y" to the number of decimal places displayed in "x"
 
JeffM said:
It would truly help us help you if you would give us the formal definition of D, particularly if there are different kinds of D.

This looks more and more as if we are talking about some exact definition for the approximation symbol.
Click to expand...
JeffM said:
It would truly help us help you if you would give us the formal definition of D, particularly if there are different kinds of D.

This looks more and more as if we are talking about some exact definition for the approximation symbol.
Click to expand...
D=D0 U D1 U D2 U D3…….

DO=Z

D1=Z/10

D2=Z/100

D3=Z/1000

…

a/b if b is divisible by a prime number other than 2 or 5 is not en element of any D

For example 1/3 is not in D,
( There is not an equivalent fraction with 1 or power of 10 as denominator )

And 1/8 is in D because 1/8 =125/1000 (D3)



These are my notes
 
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