Would Maths 'Work' in a Different Base?
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So there I was, wallowing in the bath when it suddenly occurs to me 'would maths work the same if we used a different base.' What I mean is, lets say instead of the 10 fingers that we have and on which our decimal counting system is based we had say 14 fingers so we would be counting in base 14 as the norm. Would things like Pythagoras theorem, the Fibonacci sequence, Pi, prime numbers, calculus, square roots, sins, cosines and tangents work? Not being a mathematician, I can't get my head round whether these would work or not. Intuition tells me it would. Would this work in base 14? http://news.bbc.co.uk/1/hi/sci/tech/2911945.stm[^] Does Riemann's hypothesis work in any base? What do the great and good think?
I am the Breeg, goo goo g'joob Aici zace un om despre care nu sestie prea mult
Benjamin Breeg wrote:
Not being a mathematician, I can't get my head round ...
You don't have to be a mathematician to understand that a number is an abstract quantity, and does not need digits to exist and have a meaning. It is just most humans seem to need digits to visualize numbers, and most of them do that in decimal. BTW: do you really think you can prove your computer performs its calculations in binary arithmetic? :)
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Benjamin Breeg wrote:
Not being a mathematician, I can't get my head round ...
You don't have to be a mathematician to understand that a number is an abstract quantity, and does not need digits to exist and have a meaning. It is just most humans seem to need digits to visualize numbers, and most of them do that in decimal. BTW: do you really think you can prove your computer performs its calculations in binary arithmetic? :)
Luc Pattyn [My Articles] Nil Volentibus Arduum
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CP Vanity has been updated to V2.3Luc Pattyn wrote:
do you really think you can prove your computer performs its calculations in binary arithmetic?
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So there I was, wallowing in the bath when it suddenly occurs to me 'would maths work the same if we used a different base.' What I mean is, lets say instead of the 10 fingers that we have and on which our decimal counting system is based we had say 14 fingers so we would be counting in base 14 as the norm. Would things like Pythagoras theorem, the Fibonacci sequence, Pi, prime numbers, calculus, square roots, sins, cosines and tangents work? Not being a mathematician, I can't get my head round whether these would work or not. Intuition tells me it would. Would this work in base 14? http://news.bbc.co.uk/1/hi/sci/tech/2911945.stm[^] Does Riemann's hypothesis work in any base? What do the great and good think?
I am the Breeg, goo goo g'joob Aici zace un om despre care nu sestie prea mult
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So there I was, wallowing in the bath when it suddenly occurs to me 'would maths work the same if we used a different base.' What I mean is, lets say instead of the 10 fingers that we have and on which our decimal counting system is based we had say 14 fingers so we would be counting in base 14 as the norm. Would things like Pythagoras theorem, the Fibonacci sequence, Pi, prime numbers, calculus, square roots, sins, cosines and tangents work? Not being a mathematician, I can't get my head round whether these would work or not. Intuition tells me it would. Would this work in base 14? http://news.bbc.co.uk/1/hi/sci/tech/2911945.stm[^] Does Riemann's hypothesis work in any base? What do the great and good think?
I am the Breeg, goo goo g'joob Aici zace un om despre care nu sestie prea mult
Yes and no. With sequences of whole numbers, base does not matter. However, some properties of a sequence, such as the "shallow diagonal" property of the Fibonacci sequence, take on different characteristics in different bases. Geometric values, such as pi, phi and e, will be eqivalent regardless of base: given any circle, the ratio of its circumference and its diameter will equal pi regardless of what base you are using. The most notable problem when using different bases is the representation of a fractional part. For example, the ratio of 1/10 in base-10 evaluates to a rational number, 0.1. The eqivalent in base-2, 1/1010, evaluates to an irrational number, 0.000110011.... There is a whole branch of mathematics that looks at translation between bases, although I cannot remember what that field is.
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Yes and no. With sequences of whole numbers, base does not matter. However, some properties of a sequence, such as the "shallow diagonal" property of the Fibonacci sequence, take on different characteristics in different bases. Geometric values, such as pi, phi and e, will be eqivalent regardless of base: given any circle, the ratio of its circumference and its diameter will equal pi regardless of what base you are using. The most notable problem when using different bases is the representation of a fractional part. For example, the ratio of 1/10 in base-10 evaluates to a rational number, 0.1. The eqivalent in base-2, 1/1010, evaluates to an irrational number, 0.000110011.... There is a whole branch of mathematics that looks at translation between bases, although I cannot remember what that field is.
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Bassam Abdul-Baki wrote:
A hexadecimal representation of pi was discovered a while back, but no decimal representation is possible
Huh? Then what is 3.14159...
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14 in base-10 is the same as E in base-14 (though perhaps then we'd call it base-E). The relations are all the same, the numbers would just be presented differently. Consider, for example, that the approximate value of pie is just as easy to represent on your computer in binary (base-2) as it is on paper using base-10. Dividing by 2 in any number system is just cutting in half.
AspDotNetDev wrote:
Dividing by 2 in any number system is just cutting in half.
In Binary, it's cutting 10 in 0.1
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Bassam Abdul-Baki wrote:
A hexadecimal representation of pi was discovered a while back, but no decimal representation is possible
Huh? Then what is 3.14159...
An approximation. In decimal, a few billion digits have been calculated. In hexadecimal, a representation was found.
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An approximation. In decimal, a few billion digits have been calculated. In hexadecimal, a representation was found.
Link or it didn't happen!
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Link or it didn't happen!
You have to wait until he makes a Wikipedia page for it... may take a second or two... :laugh:
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You have to wait until he makes a Wikipedia page for it... may take a second or two... :laugh:
If he really wants to make it convincing, he should ask for help from a pro, such as DD.
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You probably mean 16/7. It is too bad there is no Nobel prize for Maths. :((
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An approximation. In decimal, a few billion digits have been calculated. In hexadecimal, a representation was found.
If you are talking about this, it just allows you to calculate the n-th digit of pi in a variety of number bases, but the number of digits is still infinite. Well, unless your number system is in increments of pi (0π, 1π, 2π...). :rolleyes:
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So there I was, wallowing in the bath when it suddenly occurs to me 'would maths work the same if we used a different base.' What I mean is, lets say instead of the 10 fingers that we have and on which our decimal counting system is based we had say 14 fingers so we would be counting in base 14 as the norm. Would things like Pythagoras theorem, the Fibonacci sequence, Pi, prime numbers, calculus, square roots, sins, cosines and tangents work? Not being a mathematician, I can't get my head round whether these would work or not. Intuition tells me it would. Would this work in base 14? http://news.bbc.co.uk/1/hi/sci/tech/2911945.stm[^] Does Riemann's hypothesis work in any base? What do the great and good think?
I am the Breeg, goo goo g'joob Aici zace un om despre care nu sestie prea mult
It would have evolved that way - they didn't have borrowed characters form alphabets. And its not about conversion from based 10 to base x, everything would have been developed according to base x, so there is a probability that they would have figured out something similar but can't be said exactly....
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Yes and no. With sequences of whole numbers, base does not matter. However, some properties of a sequence, such as the "shallow diagonal" property of the Fibonacci sequence, take on different characteristics in different bases. Geometric values, such as pi, phi and e, will be eqivalent regardless of base: given any circle, the ratio of its circumference and its diameter will equal pi regardless of what base you are using. The most notable problem when using different bases is the representation of a fractional part. For example, the ratio of 1/10 in base-10 evaluates to a rational number, 0.1. The eqivalent in base-2, 1/1010, evaluates to an irrational number, 0.000110011.... There is a whole branch of mathematics that looks at translation between bases, although I cannot remember what that field is.
Gregory.Gadow wrote:
The most notable problem when using different bases is the representation of a fractional part. For example, the ratio of 1/10 in base-10 evaluates to a rational number, 0.1. The eqivalent in base-2, 1/1010, evaluates to an irrational number, 0.000110011....
Actually, the base of a number has nothing to do with whether it's rational or irrational. Your example of 1/10 in base-2 is still rational as a rational number is a number that can be represented by: P/Q Where P and Q are integers. In base-10 your example is 1/10. In base-2 your example is 1/1010. Both are the ratio of two integers and thus are rational. Don't confuse a decimal fraction that is infinitely long as irrational. 1/3 is a rational number yet its decimal representation is 0.3333333... If the decimal fraction is finite length the number is rational. If the decimal fraction is infinite length but repeating the number is rational. If the decimal fraction is infinite length and non-repeating the number is irrational.
-Sean ---- Fire Nuts
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You probably mean 16/7. It is too bad there is no Nobel prize for Maths. :((
Luc Pattyn [My Articles] Nil Volentibus Arduum
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He wanted a hex formula! :)
Luc Pattyn [My Articles] Nil Volentibus Arduum
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Gregory.Gadow wrote:
The most notable problem when using different bases is the representation of a fractional part. For example, the ratio of 1/10 in base-10 evaluates to a rational number, 0.1. The eqivalent in base-2, 1/1010, evaluates to an irrational number, 0.000110011....
Actually, the base of a number has nothing to do with whether it's rational or irrational. Your example of 1/10 in base-2 is still rational as a rational number is a number that can be represented by: P/Q Where P and Q are integers. In base-10 your example is 1/10. In base-2 your example is 1/1010. Both are the ratio of two integers and thus are rational. Don't confuse a decimal fraction that is infinitely long as irrational. 1/3 is a rational number yet its decimal representation is 0.3333333... If the decimal fraction is finite length the number is rational. If the decimal fraction is infinite length but repeating the number is rational. If the decimal fraction is infinite length and non-repeating the number is irrational.
-Sean ---- Fire Nuts
D'oh! That's embarassing. Yes, I meant to say "finite" for base-10 and "infinite" for base-2. As a former math major, I should have known better.
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So there I was, wallowing in the bath when it suddenly occurs to me 'would maths work the same if we used a different base.' What I mean is, lets say instead of the 10 fingers that we have and on which our decimal counting system is based we had say 14 fingers so we would be counting in base 14 as the norm. Would things like Pythagoras theorem, the Fibonacci sequence, Pi, prime numbers, calculus, square roots, sins, cosines and tangents work? Not being a mathematician, I can't get my head round whether these would work or not. Intuition tells me it would. Would this work in base 14? http://news.bbc.co.uk/1/hi/sci/tech/2911945.stm[^] Does Riemann's hypothesis work in any base? What do the great and good think?
I am the Breeg, goo goo g'joob Aici zace un om despre care nu sestie prea mult
I remember having read that a lot of common problems and complication of decimal system would be non-existent if we used duodecimal system.... Clickety!
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If you are talking about this, it just allows you to calculate the n-th digit of pi in a variety of number bases, but the number of digits is still infinite. Well, unless your number system is in increments of pi (0π, 1π, 2π...). :rolleyes:
That's the one. In base 10, you need to calculate all n digits to calculate the (n+1)th. With that equation, you calculate the nth in base 16 (or 2) without having to calculate the previous ones. Some things are unique to some bases.