High School Ideal Base for a Number System

[jbriggs444]

No, integer dollar amounts. It's a simple riddle and when you solve it you see the connection which is about counting in bases.

So you are going for a notion of radix economy based on the sum of the digits in the set of strings that encode values from 0 to 1000. This comes from a metric of goodness that is the maximum (over all values) of the minimum (over all conforming envelope selections) number of envelopes that produce the value. This under the constraint that all values must be encodable with at least one selection of envelopes. This amounts to a place value system where the place values are the denominations of the envelopes and the allowable digits are zero through the multiplicity of each denomination.

It seems clear that with this notion of radix economy that binary wins and you want place values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 489 or some minor variant.

Of course, the standard notion of radix economy uses a different metric of goodness and results in a different optimal base.

 

[bob012345]

So you are going for a notion of radix economy based on the sum of the digits in the set of strings that encode values from 0 to 1000. This comes from a metric of goodness that is the maximum (over all values) of the minimum (over all conforming envelope selections) number of envelopes that produce the value. This under the constraint that all values must be encodable with at least one selection of envelopes. This amounts to a place value system where the place values are the denominations of the envelopes and the allowable digits are zero through the multiplicity of each denomination.

It seems clear that with this notion of radix economy that binary wins and you want place values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 489 or some minor variant.

Of course, the standard notion of radix economy uses a different metric of goodness and results in a different optimal base.

Correct, you have ten envelopes and can make any sum to 1000 whereas a strictly decimal system would have 28 envelopes (9 hundreds, 9 tens, 10 ones). But if I give you a random number can you dish out the right envelopes in say about 2 or 3 seconds? What's the easiest system with the minimum envelopes?

 

[jbriggs444]

Correct, you have ten envelopes and can make any sum to 1000 whereas a strictly decimal system would have 28 envelopes (9 hundreds, 9 tens, 10 ones). But if I give you a random number can you dish out the right envelopes in say about 2 or 3 seconds? What's the easiest system with the minimum envelopes?

Optimizing for two criteria is not, in general, feasible. You are, perhaps, asking for the easiest system consistent with having the minimum number of envelopes total. In the case at hand, that's 10 envelopes. I do not see a quick (2 or 3 second) algorithm for a human teller given a decimal input in the range from 0 to 1000 to select an appropriate collection from an array of ten envelopes.

With 12 envelopes and a 3 (or 4 in the edge case) digit decimal request, it's easy, of course.

 

[bob012345]

Optimizing for two criteria is not, in general, feasible. You are, perhaps, asking for the easiest system consistent with having the minimum number of envelopes total. In the case at hand, that's 10 envelopes. I do not see a quick (2 or 3 second) algorithm for a human teller given a decimal input in the range from 0 to 1000 to select an appropriate collection from an array of ten envelopes.

With 12 envelopes and a 3 (or 4 in the edge case) digit decimal request, it's easy, of course.

Exactly. One possibility might be 1,2,3,4, 10,20,30,30 and 100,200,300,300. I think three seconds may be possible.

 

[Mark Harder]

Maybe it's obvious to all, but just consider polynomials in x given by p(x) = a₀ x⁰ + a₁ x¹ + a₂ x² + ... ,
and expressions for numbers (Integrals, at least) in bases. They can be expressed as polynomials. In base B:
p(B) = a₀ B⁰ + a₁ B¹ + a₂ B² + ...
For example take 123₁₀: 123(B=10) = 3₁₀ 10⁰ + 2₁₀ 10¹ + 1 10²

I don't know if the congruence between numerical systems and polynomials is good for anything, but it's interesting to think that when we write numbers in any base we desire, we're dealing with polynomials in that base with coefficients drawn from the base.

 

[SammyS]

Of course in base seven we write seven as 10 .

 

[jbriggs444]

I don't know if the congruence between numerical systems and polynomials is good for anything, but it's interesting to think that when we write numbers in any base we desire, we're dealing with polynomials in that base with coefficients drawn from the base.

Yes, for any [finite] digit string there is a corresponding polynomial using those digits as its coefficients. For a particular digit-string, one can find its associated value in base b by evaluating the corresponding polynomial function p at b.

The idea of "polynomials in that base" does not ring true. More typically one would talk of polynomial functions over a particular dummy variable (e.g. "a polynomial in x"). Or of formal polynomials over a particular field or ring (e.g. "a polynomial over the reals").

 

[Ralph Dratman]

Base 60 is one of the oldest used systems. (3300 B.C.)
https://en.wikipedia.org/wiki/Sexagesimal
A few decades ago as Assembler was still often in use 8 and 16 were useful to know.

But base 60 has fifty new "digits" beyond 0-9 for the K-12 math committee to invent, along with 1725 new multiplication facts that every child has to learn. I think in that case very few people would ever pass 4th grade. Certainly I never would have.

 

[Ralph Dratman]

I'm aware of that, thanks

It's just one of those odd things I forget, I guess. I'd edit my posts to make them accurate, but I no longer can edit them.

Why can you not edit your posts?

 

[Ralph Dratman]

Hm, base-8 is an interesting choice! Why do you choose that one? I think that base-12 would be the most reasonable number system to use. The transition from the decimal system to dozenal would be really easy. Base-12 is also convenient because 12 has a lot of factors--1, 2, 3, 4,and 6. Quick arithmetic while you're at the store or trying to divide something in your head during daily life would be simpler, I think.

I quite like binary, but the thought of living in a world that runs on the impractical base-1 is just ugly . . .

I think you hit it with twelve. Shall we use A and B for the extra digits? I like this already. About the only sad part is that we have 21 new multiplication facts to learn. I fear that would have broken me in 4th grade. I can remember crying over the multiplication table even in the old ten system we used in those days.

 

[Mark44]

Why can you not edit your posts?

There's a time limit on being able to edit one's posts. After that time limit has passed, you can't edit them any more.

 

[ProfuselyQuarky]

There's a time limit on being able to edit one's posts. After that time limit has passed, you can't edit them any more.

Which, I completely understand yet also hate sometimes.

 

[Mark44]

There's a time limit on being able to edit one's posts. After that time limit has passed, you can't edit them any more.

Which, I completely understand yet also hate sometimes.

Our reasoning is that we had many members post a homework question, and then after they had received a response, they would go back and delete their question. That would leave the rest of the thread not making much sense. Some members would post a question that they were supposed to do without outside help, and were therefore cheating. The time limit cuts down some of these kinds of problems.

 

[fresh_42]

Which, I completely understand yet also hate sometimes.

Me, too. Especially if I recognize I've made an ugly spelling or grammar mistake ... Nevertheless, I think per saldo it makes sense.
(I've been asking myself what would happen on FB if you state something, gain a lot of likes, and then turn the statement into the complete opposite? Diabolic, I know )

 

[ProfuselyQuarky]

(I've been asking myself what would happen on FB if you state something, gain a lot of likes, and then turn the statement into the complete opposite? Diabolic, I know )

Yes, I thought the same thing (), but of course, I would never have the audacity to do it. I make a lot of ugly spelling mistakes and inconsequential errors and I secretly feel that those typos make me lose my cred for that specific thread (even though that's probably not true ...).

 

[Mark44]

Yes, I thought the same thing (), but of course, I would never have the audacity to do it. I make a lot of ugly spelling mistakes and inconsequential errors and I secretly feel that those typos make me lose my cred for that specific thread (even though that's probably not true ...).

If it's something like typos that you would like to fix, you could either report the thread, asking a mentor to make the changes, or you could PM a mentor directly for the same purpose. Stuff like that we wouldn't mind changing.