Physics education research, electronic materials, and the musings of Christopher Moore, Ph.D.
Physics education research, electronic materials, and the musings of Christopher Moore, Ph.D.
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I am submitting this question here because I don't know where else to put it.
My question concerns the use of number systems other than base 10. Specifically, how to express a non-integer in a base other than ten.
Example: How does one express the number 4.5 in base 8? From an abundance of ignorance, I reason that in base 10, 4.5 means 4 plus 5/10 . Therefore shouldn't 4.5 base 8 mean 4 plus 5/8?
And, shouldn't 4.5 base 10 be equal to 4.4 base 8? Or, is some other notation used for non-integers in bases other than ten?
And, what about base 7? What would the equivalent of 4.5 base 10 in base 7 be? 4. 3.5??? but 3.5 also in base seven must equal 3. 3.5 and, uh.... hmm.
Further: how would a scientist from another planet (the 'people' there have just four digits on either 'hand') on which base 8 was used express the number pi ? It would still be the ratio of the circumference and the diameter of a circle (assuming earth-like geometry) wouldn't it? Would pi base 8 be a non-ending 'decimal' number? 3 point what?
Also, how is the number pi expressed in binary? Or, for that matter, how is any non-integer expressed in binary? Surely 'whole' numbers aren't restricted to base 10, are they?
As I say, I am nearly totally ignorant of this and would appreciate any insights.
Jack Murd
Thanks
Jack.
Last edited by JackMurd (2006-08-06 20:16:35)
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See if this article (<—click on this link) helps.
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Firstly, pi can never have a finite expansion, because it is irrational. After all, if a number goes to N digits after the point in base b, then it is a multiple of
Secondly, when representing non-integers in other bases, the system is as follows:
Take 0.625 (base 10).
For conversion to binary: 0.625 is 5/8.
Since 5 is binary 101, 5/8 is 1/2 + 0/4 + 1/8
This comes to 0.101
Since 0.625 is a multiple of (1/8) = (1/(2^3)), there are 3 binary places.
Now 5.2 to base 3.
5.2>9*0 - 0
5.2>3*1 - 1
2.2>1*2 - 2
.
0.2>(1/3)*0 - 0
0.2>(1/9)*1 - 1
0.0[8rec]>(1/27)*2 - 2
0.0[148rec]>(1/81)*1 - 1
So the ternary expansion begins 12.0121...
(EDIT: Incidentally, having checked using a program I wrote some months ago, it appears the expansion is 12.[0121rec]. But anyway)
Since 5.2 is not a multiple of any 3^-N, it has an infinite ternary expansion.
In the same way, 1/3 is not a multiple of any 10^-N, so it has an infinite decimal expansion - 0.[3rec].
I hope this makes sense (knowing me, it probably won't) and I hope it is of help.
Last edited by Phase Theory (2006-09-11 17:34:12)
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