Why does N not upgrade precision? [duplicate]

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• 
  A problem about function N
  
  2 answers
  
  

Precision[N[1.0, 20]]
Precision[N[1, 20]]

MachinePrecision
20.

It would be so much more intuitive and less error prone, if Precision[N[1.0, 20]] would be 20 and not MachinePrecision. Why do I have to explicitly use N[Rationalized[1.0], 20] to upgrade the precision?

If it is about the warning messages during the calculations, Mathematica could return a warning already at the Precision[N[1.0, 20]] step.

Edit
I am not attempting to use N for rounding. I also understand the difference between MachinePrecision and arbitrary precision, which is very well described in this answer.

I want to know, why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N, and why I need to wrap it with Rationalize.

Is it performance?

Is it some deep semantic meaning of N vs SetPrecision?

Is set SetPrecision any different from N@Rationalize@?

precision machine-precision arbitrary-precision

share|improve this question

edited Aug 31 at 8:02

asked Aug 30 at 17:27

Johu

2,461827

marked as duplicate by Michael E2, Carl Woll, Henrik Schumacher, eyorble, bbgodfrey Sep 1 at 0:24

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't know why and it also puzzles me frequently. Use SetPrecision instead.
  – Henrik Schumacher
  Aug 30 at 17:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Darn! SetPrecision does exactly, what I thought N should do. Why do I even bother reading tutorial/NumericalPrecision and other related guides?
  – Johu
  Aug 30 at 17:43
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Henric My simple idea about this is the following. Machine numbers are doublefloats; they form a number system by themselves. Something like precision is undefined for double floats. Therefore, when we ask for the precision of a double float, Mathematica returns MachinePrecision and we cannot approximate them with the function N with a second argument. By using SetPrecision instead of N, we actually assign a precision to a double float. In contrast, exact numbers have an infinite precision by themselves and therefore can be approximated to any precision in the second argument of N.
  – Fred Simons
  Aug 30 at 18:26
  
  

  
  
  
  

add a comment | 

up vote
5
down vote

favorite

This question already has an answer here:

• 
  A problem about function N
  
  2 answers
  
  

Precision[N[1.0, 20]]
Precision[N[1, 20]]

MachinePrecision
20.

It would be so much more intuitive and less error prone, if Precision[N[1.0, 20]] would be 20 and not MachinePrecision. Why do I have to explicitly use N[Rationalized[1.0], 20] to upgrade the precision?

If it is about the warning messages during the calculations, Mathematica could return a warning already at the Precision[N[1.0, 20]] step.

Edit
I am not attempting to use N for rounding. I also understand the difference between MachinePrecision and arbitrary precision, which is very well described in this answer.

I want to know, why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N, and why I need to wrap it with Rationalize.

Is it performance?

Is it some deep semantic meaning of N vs SetPrecision?

Is set SetPrecision any different from N@Rationalize@?

precision machine-precision arbitrary-precision

share|improve this question

edited Aug 31 at 8:02

asked Aug 30 at 17:27

Johu

2,461827

marked as duplicate by Michael E2, Carl Woll, Henrik Schumacher, eyorble, bbgodfrey Sep 1 at 0:24

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't know why and it also puzzles me frequently. Use SetPrecision instead.
  – Henrik Schumacher
  Aug 30 at 17:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Darn! SetPrecision does exactly, what I thought N should do. Why do I even bother reading tutorial/NumericalPrecision and other related guides?
  – Johu
  Aug 30 at 17:43
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Henric My simple idea about this is the following. Machine numbers are doublefloats; they form a number system by themselves. Something like precision is undefined for double floats. Therefore, when we ask for the precision of a double float, Mathematica returns MachinePrecision and we cannot approximate them with the function N with a second argument. By using SetPrecision instead of N, we actually assign a precision to a double float. In contrast, exact numbers have an infinite precision by themselves and therefore can be approximated to any precision in the second argument of N.
  – Fred Simons
  Aug 30 at 18:26
  
  

  
  
  
  

add a comment | 

up vote
5
down vote

favorite

up vote
5
down vote

favorite

This question already has an answer here:

• 
  A problem about function N
  
  2 answers
  
  

Precision[N[1.0, 20]]
Precision[N[1, 20]]

MachinePrecision
20.

It would be so much more intuitive and less error prone, if Precision[N[1.0, 20]] would be 20 and not MachinePrecision. Why do I have to explicitly use N[Rationalized[1.0], 20] to upgrade the precision?

If it is about the warning messages during the calculations, Mathematica could return a warning already at the Precision[N[1.0, 20]] step.

Edit
I am not attempting to use N for rounding. I also understand the difference between MachinePrecision and arbitrary precision, which is very well described in this answer.

I want to know, why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N, and why I need to wrap it with Rationalize.

Is it performance?

Is it some deep semantic meaning of N vs SetPrecision?

Is set SetPrecision any different from N@Rationalize@?

precision machine-precision arbitrary-precision

share|improve this question

edited Aug 31 at 8:02

asked Aug 30 at 17:27

Johu

2,461827

This question already has an answer here:

• 
  A problem about function N
  
  2 answers
  
  

Precision[N[1.0, 20]]
Precision[N[1, 20]]

MachinePrecision
20.

It would be so much more intuitive and less error prone, if Precision[N[1.0, 20]] would be 20 and not MachinePrecision. Why do I have to explicitly use N[Rationalized[1.0], 20] to upgrade the precision?

If it is about the warning messages during the calculations, Mathematica could return a warning already at the Precision[N[1.0, 20]] step.

Edit
I am not attempting to use N for rounding. I also understand the difference between MachinePrecision and arbitrary precision, which is very well described in this answer.

I want to know, why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N, and why I need to wrap it with Rationalize.

Is it performance?

Is it some deep semantic meaning of N vs SetPrecision?

Is set SetPrecision any different from N@Rationalize@?

This question already has an answer here:

• 
  A problem about function N
  
  2 answers
  
  

precision machine-precision arbitrary-precision

share|improve this question

edited Aug 31 at 8:02

asked Aug 30 at 17:27

Johu

2,461827

share|improve this question

share|improve this question

edited Aug 31 at 8:02

edited Aug 31 at 8:02

edited Aug 31 at 8:02

asked Aug 30 at 17:27

Johu

2,461827

asked Aug 30 at 17:27

Johu

2,461827

asked Aug 30 at 17:27

Johu

2,461827

2,461827

marked as duplicate by Michael E2, Carl Woll, Henrik Schumacher, eyorble, bbgodfrey Sep 1 at 0:24

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

marked as duplicate by Michael E2, Carl Woll, Henrik Schumacher, eyorble, bbgodfrey Sep 1 at 0:24

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't know why and it also puzzles me frequently. Use SetPrecision instead.
  – Henrik Schumacher
  Aug 30 at 17:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Darn! SetPrecision does exactly, what I thought N should do. Why do I even bother reading tutorial/NumericalPrecision and other related guides?
  – Johu
  Aug 30 at 17:43
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Henric My simple idea about this is the following. Machine numbers are doublefloats; they form a number system by themselves. Something like precision is undefined for double floats. Therefore, when we ask for the precision of a double float, Mathematica returns MachinePrecision and we cannot approximate them with the function N with a second argument. By using SetPrecision instead of N, we actually assign a precision to a double float. In contrast, exact numbers have an infinite precision by themselves and therefore can be approximated to any precision in the second argument of N.
  – Fred Simons
  Aug 30 at 18:26
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't know why and it also puzzles me frequently. Use SetPrecision instead.
  – Henrik Schumacher
  Aug 30 at 17:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Darn! SetPrecision does exactly, what I thought N should do. Why do I even bother reading tutorial/NumericalPrecision and other related guides?
  – Johu
  Aug 30 at 17:43
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Henric My simple idea about this is the following. Machine numbers are doublefloats; they form a number system by themselves. Something like precision is undefined for double floats. Therefore, when we ask for the precision of a double float, Mathematica returns MachinePrecision and we cannot approximate them with the function N with a second argument. By using SetPrecision instead of N, we actually assign a precision to a double float. In contrast, exact numbers have an infinite precision by themselves and therefore can be approximated to any precision in the second argument of N.
  – Fred Simons
  Aug 30 at 18:26
  
  

  
  
  
  

2

2

I don't know why and it also puzzles me frequently. Use SetPrecision instead.
– Henrik Schumacher
Aug 30 at 17:33

I don't know why and it also puzzles me frequently. Use SetPrecision instead.
– Henrik Schumacher
Aug 30 at 17:33

Darn! SetPrecision does exactly, what I thought N should do. Why do I even bother reading tutorial/NumericalPrecision and other related guides?
– Johu
Aug 30 at 17:43

Darn! SetPrecision does exactly, what I thought N should do. Why do I even bother reading tutorial/NumericalPrecision and other related guides?
– Johu
Aug 30 at 17:43

2

2

@Henric My simple idea about this is the following. Machine numbers are doublefloats; they form a number system by themselves. Something like precision is undefined for double floats. Therefore, when we ask for the precision of a double float, Mathematica returns MachinePrecision and we cannot approximate them with the function N with a second argument. By using SetPrecision instead of N, we actually assign a precision to a double float. In contrast, exact numbers have an infinite precision by themselves and therefore can be approximated to any precision in the second argument of N.
– Fred Simons
Aug 30 at 18:26

@Henric My simple idea about this is the following. Machine numbers are doublefloats; they form a number system by themselves. Something like precision is undefined for double floats. Therefore, when we ask for the precision of a double float, Mathematica returns MachinePrecision and we cannot approximate them with the function N with a second argument. By using SetPrecision instead of N, we actually assign a precision to a double float. In contrast, exact numbers have an infinite precision by themselves and therefore can be approximated to any precision in the second argument of N.
– Fred Simons
Aug 30 at 18:26

add a comment | 

2 Answers
2

active

oldest

votes

up vote
7
down vote



accepted

N can only lower precision, it cannot raise precision.

N[1.3`4, 10] //Precision

4.

Since MachinePrecision is considered to be the lowest possible precision for a number, applying N to a machine precision number does nothing.

Addendum

(Hopefully the following will resolve some of your confusion)

Let's start with your title. N does not upgrade precision because that would mean that N is making up digits so that the precision could increase. Making up digits to increase the precision of a number is a job for SetPrecision.

Think of N as taking an input with some validated amount of precision, and returning an output with a smaller validated amount of precision (approximating). By validated amount of precision, I'm referring to the precision tracking that goes on for arbitrary precision numbers. This is the value returned by Precision. N will never return a number with a larger validated amount of precision, because in order to do so, it would have to make up digits. So:

Precision[N[3`4, 10]] 

4.

If N returned a number with 10 digits of precision, those digits would be completely made up.

Therefore, you can trust the value returned by N. If the value returned by N had 10 digits of precision, than the input had enough precision so that a 10 digit approximation could be obtained. If the value returned by N had less than 10 digits of precision, than the input did not have enough precision to determine a 10 digit approximation.

Finally, let's consider machine numbers. It should be clear that machine numbers have no precision tracking, so the validated amount of precision is 0. There is no way for N to convert such numbers into arbitrary precision numbers with some validated amount of precision.

share|improve this answer

edited Aug 31 at 20:57

answered Aug 30 at 17:48

Carl Woll

56k272146

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Another great piece of information not mentioned in manual for N or in selected guides.
  – Johu
  Aug 30 at 17:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
  – Brett Champion
  Aug 30 at 18:03
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
  – mikado
  Aug 30 at 19:40
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
  – Carl Woll
  Aug 30 at 20:01
  

  
  
  
  

add a comment | 

up vote
4
down vote

Is it some deep semantic meaning of N vs SetPrecision?

Yes. SetPrecision forcibly changes the precision. N simply computes a numerical (i.e. inexact) approximation up to the given number of digits. While doing this, N does respect the rules of inexact arithmetic, and it does not prevent precision tracking from working the way it usually does. I.e. if we operate together a number with 3 digits of precision and a number with 20 digits of precision, the result will still have only ~3 digits. N won't add extra digits not present in the original number—that would "making up data", and thus should be controlled explicitly by you, the user. That's what SetPrecision is for.

share|improve this answer

answered Aug 31 at 9:17

Szabolcs

152k13415896

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
  – Daniel Lichtblau
  Aug 31 at 21:36
  

  
  
  
  

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
7
down vote



accepted

N can only lower precision, it cannot raise precision.

N[1.3`4, 10] //Precision

4.

Since MachinePrecision is considered to be the lowest possible precision for a number, applying N to a machine precision number does nothing.

Addendum

(Hopefully the following will resolve some of your confusion)

Let's start with your title. N does not upgrade precision because that would mean that N is making up digits so that the precision could increase. Making up digits to increase the precision of a number is a job for SetPrecision.

Think of N as taking an input with some validated amount of precision, and returning an output with a smaller validated amount of precision (approximating). By validated amount of precision, I'm referring to the precision tracking that goes on for arbitrary precision numbers. This is the value returned by Precision. N will never return a number with a larger validated amount of precision, because in order to do so, it would have to make up digits. So:

Precision[N[3`4, 10]] 

4.

If N returned a number with 10 digits of precision, those digits would be completely made up.

Therefore, you can trust the value returned by N. If the value returned by N had 10 digits of precision, than the input had enough precision so that a 10 digit approximation could be obtained. If the value returned by N had less than 10 digits of precision, than the input did not have enough precision to determine a 10 digit approximation.

Finally, let's consider machine numbers. It should be clear that machine numbers have no precision tracking, so the validated amount of precision is 0. There is no way for N to convert such numbers into arbitrary precision numbers with some validated amount of precision.

share|improve this answer

edited Aug 31 at 20:57

answered Aug 30 at 17:48

Carl Woll

56k272146

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Another great piece of information not mentioned in manual for N or in selected guides.
  – Johu
  Aug 30 at 17:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
  – Brett Champion
  Aug 30 at 18:03
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
  – mikado
  Aug 30 at 19:40
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
  – Carl Woll
  Aug 30 at 20:01
  

  
  
  
  

add a comment | 

up vote
7
down vote



accepted

N can only lower precision, it cannot raise precision.

N[1.3`4, 10] //Precision

4.

Since MachinePrecision is considered to be the lowest possible precision for a number, applying N to a machine precision number does nothing.

Addendum

(Hopefully the following will resolve some of your confusion)

Let's start with your title. N does not upgrade precision because that would mean that N is making up digits so that the precision could increase. Making up digits to increase the precision of a number is a job for SetPrecision.

Think of N as taking an input with some validated amount of precision, and returning an output with a smaller validated amount of precision (approximating). By validated amount of precision, I'm referring to the precision tracking that goes on for arbitrary precision numbers. This is the value returned by Precision. N will never return a number with a larger validated amount of precision, because in order to do so, it would have to make up digits. So:

Precision[N[3`4, 10]] 

4.

If N returned a number with 10 digits of precision, those digits would be completely made up.

Therefore, you can trust the value returned by N. If the value returned by N had 10 digits of precision, than the input had enough precision so that a 10 digit approximation could be obtained. If the value returned by N had less than 10 digits of precision, than the input did not have enough precision to determine a 10 digit approximation.

Finally, let's consider machine numbers. It should be clear that machine numbers have no precision tracking, so the validated amount of precision is 0. There is no way for N to convert such numbers into arbitrary precision numbers with some validated amount of precision.

share|improve this answer

edited Aug 31 at 20:57

answered Aug 30 at 17:48

Carl Woll

56k272146

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Another great piece of information not mentioned in manual for N or in selected guides.
  – Johu
  Aug 30 at 17:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
  – Brett Champion
  Aug 30 at 18:03
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
  – mikado
  Aug 30 at 19:40
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
  – Carl Woll
  Aug 30 at 20:01
  

  
  
  
  

add a comment | 

up vote
7
down vote



accepted

up vote
7
down vote



accepted

N can only lower precision, it cannot raise precision.

N[1.3`4, 10] //Precision

4.

Since MachinePrecision is considered to be the lowest possible precision for a number, applying N to a machine precision number does nothing.

Addendum

(Hopefully the following will resolve some of your confusion)

Let's start with your title. N does not upgrade precision because that would mean that N is making up digits so that the precision could increase. Making up digits to increase the precision of a number is a job for SetPrecision.

Think of N as taking an input with some validated amount of precision, and returning an output with a smaller validated amount of precision (approximating). By validated amount of precision, I'm referring to the precision tracking that goes on for arbitrary precision numbers. This is the value returned by Precision. N will never return a number with a larger validated amount of precision, because in order to do so, it would have to make up digits. So:

Precision[N[3`4, 10]] 

4.

If N returned a number with 10 digits of precision, those digits would be completely made up.

Therefore, you can trust the value returned by N. If the value returned by N had 10 digits of precision, than the input had enough precision so that a 10 digit approximation could be obtained. If the value returned by N had less than 10 digits of precision, than the input did not have enough precision to determine a 10 digit approximation.

Finally, let's consider machine numbers. It should be clear that machine numbers have no precision tracking, so the validated amount of precision is 0. There is no way for N to convert such numbers into arbitrary precision numbers with some validated amount of precision.

share|improve this answer

edited Aug 31 at 20:57

answered Aug 30 at 17:48

Carl Woll

56k272146

N can only lower precision, it cannot raise precision.

N[1.3`4, 10] //Precision

4.

Since MachinePrecision is considered to be the lowest possible precision for a number, applying N to a machine precision number does nothing.

Addendum

(Hopefully the following will resolve some of your confusion)

Let's start with your title. N does not upgrade precision because that would mean that N is making up digits so that the precision could increase. Making up digits to increase the precision of a number is a job for SetPrecision.

Think of N as taking an input with some validated amount of precision, and returning an output with a smaller validated amount of precision (approximating). By validated amount of precision, I'm referring to the precision tracking that goes on for arbitrary precision numbers. This is the value returned by Precision. N will never return a number with a larger validated amount of precision, because in order to do so, it would have to make up digits. So:

Precision[N[3`4, 10]] 

4.

If N returned a number with 10 digits of precision, those digits would be completely made up.

Therefore, you can trust the value returned by N. If the value returned by N had 10 digits of precision, than the input had enough precision so that a 10 digit approximation could be obtained. If the value returned by N had less than 10 digits of precision, than the input did not have enough precision to determine a 10 digit approximation.

Finally, let's consider machine numbers. It should be clear that machine numbers have no precision tracking, so the validated amount of precision is 0. There is no way for N to convert such numbers into arbitrary precision numbers with some validated amount of precision.

share|improve this answer

edited Aug 31 at 20:57

answered Aug 30 at 17:48

Carl Woll

56k272146

share|improve this answer

share|improve this answer

edited Aug 31 at 20:57

edited Aug 31 at 20:57

edited Aug 31 at 20:57

answered Aug 30 at 17:48

Carl Woll

56k272146

answered Aug 30 at 17:48

Carl Woll

56k272146

answered Aug 30 at 17:48

Carl Woll

56k272146

56k272146

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Another great piece of information not mentioned in manual for N or in selected guides.
  – Johu
  Aug 30 at 17:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
  – Brett Champion
  Aug 30 at 18:03
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
  – mikado
  Aug 30 at 19:40
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
  – Carl Woll
  Aug 30 at 20:01
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Another great piece of information not mentioned in manual for N or in selected guides.
  – Johu
  Aug 30 at 17:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
  – Brett Champion
  Aug 30 at 18:03
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
  – mikado
  Aug 30 at 19:40
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
  – Carl Woll
  Aug 30 at 20:01
  

  
  
  
  

1

1

Another great piece of information not mentioned in manual for N or in selected guides.
– Johu
Aug 30 at 17:55

Another great piece of information not mentioned in manual for N or in selected guides.
– Johu
Aug 30 at 17:55

2

2

I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
– Brett Champion
Aug 30 at 18:03

I think that's the last example under Scope > Machine and Adaptive Precision, although that was not the first place I looked...
– Brett Champion
Aug 30 at 18:03

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Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
– mikado
Aug 30 at 19:40

Actually, you can specify precision lower than MachinePrecision. This could be useful in error analysis.
– mikado
Aug 30 at 19:40

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@mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
– Carl Woll
Aug 30 at 20:01

@mikado You can specify a number lower than the value of MachinePrecision (i.e. $MachinePrecision). I don't think that's relevant to the question, though.
– Carl Woll
Aug 30 at 20:01

add a comment | 

up vote
4
down vote

Is it some deep semantic meaning of N vs SetPrecision?

Yes. SetPrecision forcibly changes the precision. N simply computes a numerical (i.e. inexact) approximation up to the given number of digits. While doing this, N does respect the rules of inexact arithmetic, and it does not prevent precision tracking from working the way it usually does. I.e. if we operate together a number with 3 digits of precision and a number with 20 digits of precision, the result will still have only ~3 digits. N won't add extra digits not present in the original number—that would "making up data", and thus should be controlled explicitly by you, the user. That's what SetPrecision is for.

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answered Aug 31 at 9:17

Szabolcs

152k13415896

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
  – Daniel Lichtblau
  Aug 31 at 21:36
  

  
  
  
  

add a comment | 

up vote
4
down vote

Is it some deep semantic meaning of N vs SetPrecision?

Yes. SetPrecision forcibly changes the precision. N simply computes a numerical (i.e. inexact) approximation up to the given number of digits. While doing this, N does respect the rules of inexact arithmetic, and it does not prevent precision tracking from working the way it usually does. I.e. if we operate together a number with 3 digits of precision and a number with 20 digits of precision, the result will still have only ~3 digits. N won't add extra digits not present in the original number—that would "making up data", and thus should be controlled explicitly by you, the user. That's what SetPrecision is for.

share|improve this answer

answered Aug 31 at 9:17

Szabolcs

152k13415896

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
  – Daniel Lichtblau
  Aug 31 at 21:36
  

  
  
  
  

add a comment | 

up vote
4
down vote

up vote
4
down vote

Is it some deep semantic meaning of N vs SetPrecision?

Yes. SetPrecision forcibly changes the precision. N simply computes a numerical (i.e. inexact) approximation up to the given number of digits. While doing this, N does respect the rules of inexact arithmetic, and it does not prevent precision tracking from working the way it usually does. I.e. if we operate together a number with 3 digits of precision and a number with 20 digits of precision, the result will still have only ~3 digits. N won't add extra digits not present in the original number—that would "making up data", and thus should be controlled explicitly by you, the user. That's what SetPrecision is for.

share|improve this answer

answered Aug 31 at 9:17

Szabolcs

152k13415896

Is it some deep semantic meaning of N vs SetPrecision?

Yes. SetPrecision forcibly changes the precision. N simply computes a numerical (i.e. inexact) approximation up to the given number of digits. While doing this, N does respect the rules of inexact arithmetic, and it does not prevent precision tracking from working the way it usually does. I.e. if we operate together a number with 3 digits of precision and a number with 20 digits of precision, the result will still have only ~3 digits. N won't add extra digits not present in the original number—that would "making up data", and thus should be controlled explicitly by you, the user. That's what SetPrecision is for.

share|improve this answer

answered Aug 31 at 9:17

Szabolcs

152k13415896

share|improve this answer

share|improve this answer

answered Aug 31 at 9:17

Szabolcs

152k13415896

answered Aug 31 at 9:17

Szabolcs

152k13415896

answered Aug 31 at 9:17

Szabolcs

152k13415896

152k13415896

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
  – Daniel Lichtblau
  Aug 31 at 21:36
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
  – Daniel Lichtblau
  Aug 31 at 21:36
  

  
  
  
  

Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
– Daniel Lichtblau
Aug 31 at 21:36

Very good explanation for the respective behaviors. Also answers: "why does Mathematica not consider the description of the number existing in the underling binary representation as exact if I apply N"
– Daniel Lichtblau
Aug 31 at 21:36

add a comment |