Is "ten" independent of the chosen number base?

[jbriggs444]

I think it seems like a good idea to label the number bases in text form, since this is unambiguous.

I disagree with this. The KISS principle applies. ("Keep it simple, stupid"). We have an agreed upon form for expressing numbers in radix notation. It should be used.

If one takes pains to go outside the standard notation in the name of avoiding ambiguity, the result is to distract the reader from the essence of the communication. An important property of mathematical writing is that it is terse. One can express a concept simply and briefly.

Rather than writing $12_{\text{ten}}$, write $12_{10}$. Or better yet, just write $12$.

The more baggage that is tacked onto the writing, the more of the intended meaning is hidden. The reader is left thinking "why is the writer doing this -- is there some important meaning hidden in these baroque syntactical choices?"

Edit: In practice, there is little to argue about. The number of times I have needed to write down a numeric literal in standard place value notation using a radix other than 2, 3, 8, 10 or 16 can be counted on the fingers of one hand with five fingers left over.

 

[Dale]

The thing is, how can one write down ‘B’ without implicitly using another base?

To write a number in any base N you need N symbols. Those N symbols represent the integers from 0 to N-1 and N is represented by the combination of symbols with the symbol for zero in the first digit and the symbol for one in the second digit. By convention we use the symbols 0, 1, ..., 9, A, B, ..., but there is no need for that, it is purely conventional. Also by convention for any base other than base ten we use a subscript to denote the base in base 10 numbers. Again, it is convention and is not necessary if understood in context. To make the point, I will use base 16 numbers denoted by a (), b (*), c (**), d (***), e (****), f (**** *), g (**** **), h (**** ***), i (**** ****), j (**** **** *), k (**** **** **), l (**** **** ***), m (**** **** ****), n (**** **** **** *), o (**** **** **** **), p (**** **** **** ***), ba (**** **** **** ****)

So your $B3_{16}$ is my $ld$ which can be written purely in my notation as $ld = l \times ba^b + d \times ba^a$. So there is never any need to use base 10 digits, and any such use is purely a matter of convenience and convention.

 

[etotheipi]

Awesome, thanks a bunch for everyone's help. I think I have a better understanding now. Namely,

• The base of a number is specified in decimal by convention (note I'm careful not to write "10" here , I'd rather not have to deal with so-called regresum ad infinitum...). If the base is base-10, then we do not write the subscript.
• There exists a distinction between numbers and numerals (their representations). $10$, $14_6$, $A_{16}$ all refer to the number ten.
• $\text{abc}_{b}$ can be expressed as a power series, $\sum \alpha_{i}b^{i}$. This is usually done in base-10, but it does not have to be so. We could have $2B_{16} = 3 \times {{21}_{6}}^{1} + 4 \times {{21}_{6}}^{0}$; of course, this is needless complexity, but goes to show that the base $b$ - like all algebraic variables - exist independently of bases!
• The base $b$ could theoretically be negative, non-integer, complex etc. (i.e. we might have base $\phi$...)

It's just a peculiar thing since everyone gets an intuitive feel for numbers and arithmetic through school, but this intuition is unhelpful to understand bases, since it's easy to be inclined to resort to "well, this is obviously such and such" reasoning. Do let me know if I've gone mad.

 

[sysprog]

LaPlace tended to say things along the lines of 'from here, it is easily seen that . . .' when he didn't want to do the proof . . .

 

[jbriggs444]

The base of a number is specified in decimal by convention

Numbers do not have bases. Numerals (i.e. representations) sometimes do.

A course in computer languages, parsing and syntax nails some of these things down in painful detail and adds some useful terminology.

 

[etotheipi]

Numbers do not have bases. Numerals (i.e. representations) sometimes do.

A course in computer languages, parsing and syntax nails some of these things down in painful detail and adds some useful terminology.

Whoops, please do excuse my sloppiness...

 

[etotheipi]

Though it does beg the question of what a number actually is, rigorously. If "ten", "10", "$*7@1" are all numerals. I think like some have mentioned, it's for the best that I stop speculating and just get a hold of some actual study resources, I found some of Tao's notes which look like they cover some fun stuff:

https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

Anyway thanks for putting up with me for this long (!)

 

[jbriggs444]

Though it does beg the question of what a number actually is, rigorously.

At a simple level, a number is the abstract concept that all of those representations are trying to get at. It is, for instance, the abstract notion of four-ness shared by four rocks in your hand, four sheep in your field or four sides on a square.

Typically we have an operational notion that is good enough to count sheep and verify that the buyer has paid us for all of them. We typically leave it at that.

One way of approaching the foundations is to define numbers in terms of their properties. For instance, the Peano axioms. From memory (and there are a lot of flavors of these).

1. There is an undefined entity called "zero" that we say is a "number".
2. For every number A there is a next number that we refer to as the "successor" of A. We refer to it as S(A).
3. For every number A that is different from zero, there is exactly one number B such that A=S(B).
4. There is no number A such that 0 = S(A).
5. [Induction] -- If you have a set of numbers that contains zero and also contains the successor of every number in the set then that set contains all of the "numbers".

The key takeaway from this is that we've not defined what these "numbers" are. But we've described how they behave. That turns out to be good enough to do a massive amount of arithmetic and a ton of proofs.

You can spend a couple of semesters going from here to a description of the real numbers.

 

[etotheipi]

The key takeaway from this is that we've not defined what these "numbers" are. But we've described how they behave. That turns out to be good enough to do a massive amount of arithmetic and a ton of proofs.

Freaky... the more I learn the less I know...

Makes you appreciate the fact that people like Mr Russell (https://en.wikipedia.org/wiki/Principia_Mathematica) had to spend hundreds of pages to prove what people learn at around 3 years old, since nothing is ever "obvious" if you look at it in enough detail!

 

[PeroK]

Though it does beg the question of what a number actually is, rigorously.

The intuitive idea of whole numbers is as follows. You have a flock of sheep say. Every day you have to go out and bring all your sheep into a field for the night. How do you ensure you have all your sheep?

One approach is: as each sheep exits the field in the morning you add a small stone to a pile. Then, as each sheep enters the field at the end of the day you take a stone out of the pile. If there are stones left in your pile, then you know there are still sheep to be tracked down.

Note that this is a basic form of counting your sheep.

You do the same for everything: cows, chickens, pigs, whatever. For everything you need to keep track of you have a set of "counters" of some sort. But, you need a different set of counters for everything you need to "count"

You can improve on this by having a general set of things called numbers. You memorise the pattern of the sequence of symbols or words and each time a sheep leaves the field you move to the next number. Instead of having a set of stones, you write down (or remember) the number: 35 or XXXV or 100011 or thirty-five.

Now you can dispense with all the different sets of counters and use your general set of numbers to count everything.

In this intuitive context, the whole numbers form a defined, ordered sequence of symbols or words. This ties into the Peano axioms as "ten" is then the successor of the successor ... of the first symbol or word.

 

[zoki85]

Makes you appreciate the fact that people like Mr Russell (https://en.wikipedia.org/wiki/Principia_Mathematica) had to spend hundreds of pages to prove what people learn at around 3 years old, since nothing is ever "obvious" if you look at it in enough detail!

Makes me wonder how badly Russell banged his head against wall when he heard of Gödel's theorems

 

[pbuk]

Awesome, thanks a bunch for everyone's help. I think I have a better understanding now.

Yes I think you do, but one of the beautiful aspects of mathematics is that you never have to stop learning!

• The base of a number is specified in decimal by convention (note I'm careful not to write "10" here , I'd rather not have to deal with so-called regresum ad infinitum...). If the base is base-10, then we do not write the subscript.

A mathematician would write "numbers are expressed in radix 10 by convention" and know that every other mathematician would understand exactly what she meant.

• There exists a distinction between numbers and numerals (their representations). $10$, $14_6$, $A_{16}$ all refer to the number ten.

I would rather write

• There exists a distinction between numbers and and the symbols we use to represent them. $10$, $14_6$, $A_{16}$ and $\text{ten}$ all refer to the same number.

• $\text{abc}_{b}$ can be expressed as a power series, $\sum \alpha_{i}b^{i}$. This is usually done in base-10, but it does not have to be so. We could have $2B_{16} = 3 \times {{21}_{6}}^{1} + 4 \times {{21}_{6}}^{0}$; of course, this is needless complexity, but goes to show that the base $b$ - like all algebraic variables - exist independently of bases!

Not sure where you are going with this: $a_2 a_1 a_0_{b}$ implies the partial sum $\sum_{i=0}^{2} \alpha_i b^i$, not any other partial sum $\sum_{i=0}^{k} \beta_i c^i$ that happens to have the same value.

• The base $b$ could theoretically be negative, non-integer, complex etc. (i.e. we might have base $\phi$...)

Yes but with non-integer bases we are into a whole other game and we are no longer in a domain which includes Peano arithmetic. Although non-integer bases seem interesting and exotic at first, there are far more interesting things to discover in the "mainstream" analysis syllabus so I recommend you don't go down that track.

It's just a peculiar thing since everyone gets an intuitive feel for numbers and arithmetic through school, but this intuition is unhelpful to understand bases, since it's easy to be inclined to resort to "well, this is obviously such and such" reasoning. Do let me know if I've gone mad.

To learn mathematics you have to put your intuitive feel for everything you think you understand to one side and start again with an empty mind. Don't worry about this, once you have grasped the modern concept of an integer, and a real number, and a line, and a set, and an infinite set... they will be just as intuitive as those old notions and much more powerful and consistent.

I found some of Tao's notes which look like they cover some fun stuff:
https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

That might work but it would be better to hear Terry Tao deliver that course in person - it may be on YouTube. Failing that, it would be better to work from his book ISBN 978-981-10-1789-6 covering the same material. But whatever you do - move on from this fascination with number bases, it is not useful.

 

[charminglystrange]

Freaky... the more I learn the less I know...

Makes you appreciate the fact that people like Mr Russell (https://en.wikipedia.org/wiki/Principia_Mathematica) had to spend hundreds of pages to prove what people learn at around 3 years old, since nothing is ever "obvious" if you look at it in enough detail!

Don't forget the classic Four Colour Theorem, which took 139 pages, and some patient time in front of a computer to prove!

 

[Mark44]

One approach is: as each sheep exits the field in the morning you add a small stone to a pile.

And, in Latin, these stones are calculi (singular calculus), which makes the connection between what a dentist cleans from your teeth and the mathematical study of derivatives, integrals, and so on.

 

[PeroK]

And, in Latin, these stones are calculi (singular calculus), which makes the connection between what a dentist cleans from your teeth and the mathematical study of derivatives, integrals, and so on.

Calculus, calculum, calculi, calculo, calculi, calculos, calculorum, calculis.

I think that's right.

 

[DEvens]

There are 10 kinds of people in the world. Those who understand binary, and those who do not.

 

[etotheipi]

There are 10 kinds of people in the world. Those who understand binary, and those who do not.

And those who didn't expect this to be a ternary joke...

 

[Mark44]

There are 10 kinds of people in the world. Those who understand binary, and those who do not.

And those who didn't expect this to be a ternary joke...

Which it isn't...

 

[etotheipi]

Which it isn't...

It works better if both posts are read together

 

[jbriggs444]

Which it isn't...

Perhaps 10 interns should take turns interning the terns mentioned in this joke.

 

[Mark44]

It works better if both posts are read together

I quoted both posts. Your statement about ternary doesn't make any sense.

 

[DEvens]

I quoted both posts. Your statement about ternary doesn't make any sense.

In binary, 10 is 2 in base-10. In ternary, 10 is 3 in base-10. So the full statement would be.

There are 10 types of people. Those who understand binary, and those who don't.

[Pause for the pained groan "yeah, binary 10 is 2" response.]

And those who didn't expect this to be a joke about ternary...

[Pause for people to go "Ternary? Base 3? Oh, ternary 10 is 3.
So there must be a third type of person.]

 

[Mark44]

@DEvens, yeah, I got it.

Anyway, I think we've beaten the topic to death, so I'm closing the thread. If anyone has anything of substance to add, send me a PM and I'll reopen the thread.