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Proving that 1/3 has no finite decimal representation
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There is a problem where i need to prove that 1/3 has no finite decimal representation
Here's my proof, can someone tell me if its valid?
Proof
Lets assume there is a decimal representation for $frac13$, Therefore:
$ exists n,b in mathbbN $ : $ (fracb10^n=frac13$) $ land (sum_k=1^n fraca_k10^k=frac13)$
By the theorem: $frac13 = fracb10^n $
b = $frac10^n3$ = $frac(2 times 5)^n3$
Thats a contradiction ($b notin mathbbN$), Since that fraction is irreducible (Both 2,5,3 are prime numbers).
Is my proof valid? If not, Can someone explain what's wrong with it?
Thanks.
algebra-precalculus proof-verification decimal-expansion
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
asked Jul 30 at 15:43
Dvir Peretz
526
add a comment |Â
up vote
5
down vote
favorite
There is a problem where i need to prove that 1/3 has no finite decimal representation
Here's my proof, can someone tell me if its valid?
Proof
Lets assume there is a decimal representation for $frac13$, Therefore:
$ exists n,b in mathbbN $ : $ (fracb10^n=frac13$) $ land (sum_k=1^n fraca_k10^k=frac13)$
By the theorem: $frac13 = fracb10^n $
b = $frac10^n3$ = $frac(2 times 5)^n3$
Thats a contradiction ($b notin mathbbN$), Since that fraction is irreducible (Both 2,5,3 are prime numbers).
Is my proof valid? If not, Can someone explain what's wrong with it?
Thanks.
algebra-precalculus proof-verification decimal-expansion
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
asked Jul 30 at 15:43
Dvir Peretz
526
I'd say $0.33333...$ is a perfectly fine decimal representation of $1/3$.
– Henning Makholm
Jul 30 at 15:45
Yes it looks Ok. But you should maybe write "cannot be finitely represented by decimal number system" or something.
– mathreadler
Jul 30 at 15:45
It is fine, though you can immediately write $1/3=b/10^n$ where $n$ is the number of decimals.
– Yves Daoust
Jul 30 at 15:47
1
Yeah, I meant finite decimal representation, And 0.333... isn't a finite representation.
– Dvir Peretz
Jul 30 at 15:48
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
There is a problem where i need to prove that 1/3 has no finite decimal representation
Here's my proof, can someone tell me if its valid?
Proof
Lets assume there is a decimal representation for $frac13$, Therefore:
$ exists n,b in mathbbN $ : $ (fracb10^n=frac13$) $ land (sum_k=1^n fraca_k10^k=frac13)$
By the theorem: $frac13 = fracb10^n $
b = $frac10^n3$ = $frac(2 times 5)^n3$
Thats a contradiction ($b notin mathbbN$), Since that fraction is irreducible (Both 2,5,3 are prime numbers).
Is my proof valid? If not, Can someone explain what's wrong with it?
Thanks.
algebra-precalculus proof-verification decimal-expansion
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
asked Jul 30 at 15:43
Dvir Peretz
526
There is a problem where i need to prove that 1/3 has no finite decimal representation
Here's my proof, can someone tell me if its valid?
Proof
Lets assume there is a decimal representation for $frac13$, Therefore:
$ exists n,b in mathbbN $ : $ (fracb10^n=frac13$) $ land (sum_k=1^n fraca_k10^k=frac13)$
By the theorem: $frac13 = fracb10^n $
b = $frac10^n3$ = $frac(2 times 5)^n3$
Thats a contradiction ($b notin mathbbN$), Since that fraction is irreducible (Both 2,5,3 are prime numbers).
Is my proof valid? If not, Can someone explain what's wrong with it?
Thanks.
algebra-precalculus proof-verification decimal-expansion
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
asked Jul 30 at 15:43
Dvir Peretz
526
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
edited Jul 30 at 15:53
José Carlos Santos
112k1696172
112k1696172
asked Jul 30 at 15:43
Dvir Peretz
526
asked Jul 30 at 15:43
Dvir Peretz
526
asked Jul 30 at 15:43
Dvir Peretz
526
526
I'd say $0.33333...$ is a perfectly fine decimal representation of $1/3$.
– Henning Makholm
Jul 30 at 15:45
Yes it looks Ok. But you should maybe write "cannot be finitely represented by decimal number system" or something.
– mathreadler
Jul 30 at 15:45
It is fine, though you can immediately write $1/3=b/10^n$ where $n$ is the number of decimals.
– Yves Daoust
Jul 30 at 15:47
1
Yeah, I meant finite decimal representation, And 0.333... isn't a finite representation.
– Dvir Peretz
Jul 30 at 15:48
add a comment |Â
I'd say $0.33333...$ is a perfectly fine decimal representation of $1/3$.
– Henning Makholm
Jul 30 at 15:45
Yes it looks Ok. But you should maybe write "cannot be finitely represented by decimal number system" or something.
– mathreadler
Jul 30 at 15:45
It is fine, though you can immediately write $1/3=b/10^n$ where $n$ is the number of decimals.
– Yves Daoust
Jul 30 at 15:47
1
Yeah, I meant finite decimal representation, And 0.333... isn't a finite representation.
– Dvir Peretz
Jul 30 at 15:48
I'd say $0.33333...$ is a perfectly fine decimal representation of $1/3$.
– Henning Makholm
Jul 30 at 15:45
I'd say $0.33333...$ is a perfectly fine decimal representation of $1/3$.
– Henning Makholm
Jul 30 at 15:45
Yes it looks Ok. But you should maybe write "cannot be finitely represented by decimal number system" or something.
– mathreadler
Jul 30 at 15:45
Yes it looks Ok. But you should maybe write "cannot be finitely represented by decimal number system" or something.
– mathreadler
Jul 30 at 15:45
It is fine, though you can immediately write $1/3=b/10^n$ where $n$ is the number of decimals.
– Yves Daoust
Jul 30 at 15:47
It is fine, though you can immediately write $1/3=b/10^n$ where $n$ is the number of decimals.
– Yves Daoust
Jul 30 at 15:47
1
1
Yeah, I meant finite decimal representation, And 0.333... isn't a finite representation.
– Dvir Peretz
Jul 30 at 15:48
Yeah, I meant finite decimal representation, And 0.333... isn't a finite representation.
– Dvir Peretz
Jul 30 at 15:48
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
accepted
It is almost correct, but you should not write $2.5$ when what you mean is $2times5$.
And, yes, $frac(2times5)^n3=frac2^n5^n3$, which is indeed an irreducible fraction. You didn't say why it is irreducible, but it is easy: since $3$ is prime and $3nmid2^n5^n$, $3$ and $2^n5^n$ are coprime and therefore, yes, the fraction is irreducible.
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
1
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,*is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such ascdotandtimes, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.
– JMoravitz
Jul 30 at 16:16
As for the pros and cons of usingcdotversustimes, again both are quite common in my experience.timesruns the risk of being confused for a variable $x$ depending on the handwriting or font, whilecdotruns the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefertimeswhen it is purely arithmetic such as $2times 5$ while I tend to prefercdotwhen it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.
– JMoravitz
Jul 30 at 16:23
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
It is almost correct, but you should not write $2.5$ when what you mean is $2times5$.
And, yes, $frac(2times5)^n3=frac2^n5^n3$, which is indeed an irreducible fraction. You didn't say why it is irreducible, but it is easy: since $3$ is prime and $3nmid2^n5^n$, $3$ and $2^n5^n$ are coprime and therefore, yes, the fraction is irreducible.
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
1
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,*is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such ascdotandtimes, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.
– JMoravitz
Jul 30 at 16:16
As for the pros and cons of usingcdotversustimes, again both are quite common in my experience.timesruns the risk of being confused for a variable $x$ depending on the handwriting or font, whilecdotruns the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefertimeswhen it is purely arithmetic such as $2times 5$ while I tend to prefercdotwhen it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.
– JMoravitz
Jul 30 at 16:23
add a comment |Â
up vote
5
down vote
accepted
It is almost correct, but you should not write $2.5$ when what you mean is $2times5$.
And, yes, $frac(2times5)^n3=frac2^n5^n3$, which is indeed an irreducible fraction. You didn't say why it is irreducible, but it is easy: since $3$ is prime and $3nmid2^n5^n$, $3$ and $2^n5^n$ are coprime and therefore, yes, the fraction is irreducible.
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
1
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,*is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such ascdotandtimes, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.
– JMoravitz
Jul 30 at 16:16
As for the pros and cons of usingcdotversustimes, again both are quite common in my experience.timesruns the risk of being confused for a variable $x$ depending on the handwriting or font, whilecdotruns the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefertimeswhen it is purely arithmetic such as $2times 5$ while I tend to prefercdotwhen it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.
– JMoravitz
Jul 30 at 16:23
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
It is almost correct, but you should not write $2.5$ when what you mean is $2times5$.
And, yes, $frac(2times5)^n3=frac2^n5^n3$, which is indeed an irreducible fraction. You didn't say why it is irreducible, but it is easy: since $3$ is prime and $3nmid2^n5^n$, $3$ and $2^n5^n$ are coprime and therefore, yes, the fraction is irreducible.
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
It is almost correct, but you should not write $2.5$ when what you mean is $2times5$.
And, yes, $frac(2times5)^n3=frac2^n5^n3$, which is indeed an irreducible fraction. You didn't say why it is irreducible, but it is easy: since $3$ is prime and $3nmid2^n5^n$, $3$ and $2^n5^n$ are coprime and therefore, yes, the fraction is irreducible.
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
answered Jul 30 at 15:48
José Carlos Santos
112k1696172
112k1696172
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
1
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,*is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such ascdotandtimes, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.
– JMoravitz
Jul 30 at 16:16
As for the pros and cons of usingcdotversustimes, again both are quite common in my experience.timesruns the risk of being confused for a variable $x$ depending on the handwriting or font, whilecdotruns the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefertimeswhen it is purely arithmetic such as $2times 5$ while I tend to prefercdotwhen it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.
– JMoravitz
Jul 30 at 16:23
add a comment |Â
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
1
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,*is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such ascdotandtimes, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.
– JMoravitz
Jul 30 at 16:16
As for the pros and cons of usingcdotversustimes, again both are quite common in my experience.timesruns the risk of being confused for a variable $x$ depending on the handwriting or font, whilecdotruns the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefertimeswhen it is purely arithmetic such as $2times 5$ while I tend to prefercdotwhen it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.
– JMoravitz
Jul 30 at 16:23
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
A . is a perfectly common multiplication symbol several places in the world. For instance, it appears on occasion on the brown paper of Numberphile. One should still probably be careful with using it, though. Particularly when writing to an international audience like here in this site. I personally think $times$ looks almost as bad, since I was raised to use $cdot$, and that's just the way it is.
– Arthur
Jul 30 at 15:57
1
1
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
I know that, but in this instance it is ambiguous, and therefore it should be avoided. It's like using the notation $|cdot|$ for the determinant. No problem for $ntimes n$ matrics with $n>1$, but would you really write that $|-1|=-1$? It is correct, nonetheless…
– José Carlos Santos
Jul 30 at 16:00
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,
* is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such as cdot and times, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.– JMoravitz
Jul 30 at 16:16
On the topic of 2.5 to represent the multiplication of 2 times 5, it is incredibly ambiguous whether you mean 2 and 5 tenths or if you mean 2 times 5, with most people leaning to the first interpretation. If you are stuck using only the symbols directly available on a keyboard,
* is more commonly used to notate multiplication. We do have access to MathJax and $LaTeX$ here, giving us access to better alternatives such as cdot and times, both being common symbols used for multiplication in literature. TLDR: 2.5, bad, 2*5 acceptable, $2cdot 5$ and $2times 5$ good.– JMoravitz
Jul 30 at 16:16
As for the pros and cons of using
cdot versus times, again both are quite common in my experience. times runs the risk of being confused for a variable $x$ depending on the handwriting or font, while cdot runs the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefer times when it is purely arithmetic such as $2times 5$ while I tend to prefer cdot when it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.– JMoravitz
Jul 30 at 16:23
As for the pros and cons of using
cdot versus times, again both are quite common in my experience. times runs the risk of being confused for a variable $x$ depending on the handwriting or font, while cdot runs the risk of being confused for a decimal point. They key for handwriting these is to center them vertically. For that reason, I tend to prefer times when it is purely arithmetic such as $2times 5$ while I tend to prefer cdot when it is purely algebraic such as $xcdot y$, of course never switching notations mid-problem.– JMoravitz
Jul 30 at 16:23
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I'd say $0.33333...$ is a perfectly fine decimal representation of $1/3$.
– Henning Makholm
Jul 30 at 15:45
Yes it looks Ok. But you should maybe write "cannot be finitely represented by decimal number system" or something.
– mathreadler
Jul 30 at 15:45
It is fine, though you can immediately write $1/3=b/10^n$ where $n$ is the number of decimals.
– Yves Daoust
Jul 30 at 15:47
1
Yeah, I meant finite decimal representation, And 0.333... isn't a finite representation.
– Dvir Peretz
Jul 30 at 15:48