Can irrational numbers be made more meaningful with a different counting system?

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Offline bizerl

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Probably more maths than science, but I was thinking specifically of pi and the fact that the decimal places just seem to keep going.

If we changed our number system from base-10 to something else like base-7 or base-19 for example, would it change the spectrum of irrational numbers and would we be able to find a better way to resolve something like pi?

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Offline RD

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I think you're you're looking for an irrational base, e.g "base pi", which is impractical ...

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Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating pi to 1 in "base pi," ... Even more unexpectedly, the representation of a given integer in an irrational base may be nonunique
http://mathworld.wolfram.com/Base.html
« Last Edit: 13/09/2012 03:14:50 by RD »

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Offline CliffordK

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Keep in mind that fractions of rational numbers and decimal repeats are all considered rational numbers.  So, ⅓ = 0.33333....  is a rational number.
If you chose base 12, then ⅓ no longer gives a repeat decimal, but rather is 0.4 (still rational).

The problem with an irrational base is that many irrational numbers are not whole factors of other irrational numbers.

So, if one had base pi, then [tex]\sqrt{2}[/tex] is still irrational.
If one used base [tex]\sqrt{2}[/tex], then [tex]\sqrt{3}[/tex], as well as [tex]\sqrt[3]{2}[/tex] would still be irrational.

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Offline imatfaal

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Probably more maths than science, but I was thinking specifically of pi and the fact that the decimal places just seem to keep going.

If we changed our number system from base-10 to something else like base-7 or base-19 for example, would it change the spectrum of irrational numbers and would we be able to find a better way to resolve something like pi?

And to focus on on you last question (ie not to go down route of irrational bases) - no it cannot make a difference.  irrationals cannot be formed as a fraction of integers - our bases are just a short hand to allow us to manipulate, they do not change the essence of the number or the position on the number line.  Division - making a fraction - is how many times the top will go into the bottom, and you could do that sum with pebbles, or with sweets and children etc - ie with no bases at all.  And all of your answers would be rational - 1 mars bar for every two kids (1/2 bar each) , 3 kitkats between 4 children (3/4 each - ie 3 fingers); never would your answer be an irrational amount each (and never would the kids be happy)

Basically integers are integers in any base, and rationals/irrationals follow integers in their definition
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Offline evan_au

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There are a number of symbolic maths packages around, and they can represent numbers like "pi", "i" and "square root of 2" directly (and some of them produce precise answers like sin(pi)= 0, exactly).

http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems
« Last Edit: 16/09/2012 08:49:14 by evan_au »

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Offline Bored chemist

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If you chose a number base that made pi or root two rational then all the integers would be irrational.
It would take an infinite amount of time to write down exactly how many fingers you have.
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