About unsolved math problems, and the numbering base

[Kyriakos]

A number ( ) of still unresolved issues, often famous problems, seem very tied to the base used, 10 numbers (0-9) and iterations. Examples include the four colour theorem, and (afaik proven a brief time ago) the last theorem of Fermat. Yet i am wondering if there isn't yet sufficient work generalising the use of base so as to present adequate insight on why our own 0-9 system produces these patterns when logically followed through.
As a (very amateur) math-enthusiast, i am often let down by not finding the info i personally deem as important in examining such issues. An example is with the plane used to present all numbers (including imaginary ones). I have read about the basic reason imaginary numbers were presented (to account for some solutions where there is the square root of a negative integer), but i couldn't find any treatises on how irrational (and by extension, complex) numbers fit into out mental sense of numbers, which usually is about ideas of proportion, comparison, position, progression, likeness and difference, distinctness and lack of it.

Maybe some of the uni educated mathematicians in the forum could help provide readings where hopefully such issues are contemplated? By the way, i mean mathematically contemplated, not philosophically (i can do that by myself, thanks, let's not make having a philo degree more than asymptotically approach zero in value :/ ).

 

[AdrienIer]

Why do you say that Fermat's last theorem and the four color theorem (which were both proved a few years ago) are linked to our decimal system ?

Edit : you also forget to mention a major property of numbers : rationals then algebric then real then complex. In field theory (and pushing into Gallois' theory) the algebric vs transcendental difference is the main one.

Generally, looking into groups, rings (like the rings of gaussian numbers which are really fun) and fields is the best way to look at numbers

 

[Kyriakos]

Math people here have brought up algebric numbers re rings, and related mathematicians (Galois and Abel, iirc) but i am only having an amateur's angle at this, and haven't looked those up by now
I am mostly familiar with the ancient greek math, due to philosophy being part of them and vice-versa, and then some other fields to a degree (probability and limits, mostly).
Re the four colour theorem, afaik it is not rigorously proven? Afaik it currently used computers running groups of cases? -i did observe a local lecture on this last year, but maybe i misunderstood).
I thought it was like providing proof that the different arrangements of A,B,C are (without repetition) ABC, ACB, BAC, BCA, CAB, CBA, thus 6, with the computers running the x numbers of cases.

 

[Zelig]

Computer-assisted proofs by exhaustion are absolutely rigorous. (See the Boolean Pythagorean Triples for a good recent one.)

Relatedly, computers don't care much about number base.

 

[AdrienIer]

I just checked for the four color theorem : the question is now whether the algorithm they used to prove it really proves it, and whether they programmed it well enough. Not the fact that a computer did it. You could argue that a proof without the use of computers would be more elegant, but the current one is still a proof.

Do you know anything about modular arithmetic ? If you want I could write a summary of what groups, rings and fields are, and then about how modular arithmetic works. Going into Galois's stuff would be overkill if you just want to understand number theory better.

 

[Kyriakos]

I just checked for the four color theorem : the question is now whether the algorithm they used to prove it really proves it, and whether they programmed it well enough. Not the fact that a computer did it. You could argue that a proof without the use of computers would be more elegant, but the current one is still a proof.

Do you know anything about modular arithmetic ? If you want I could write a summary of what groups, rings and fields are, and then about how modular arithmetic works. Going into Galois's stuff would be overkill if you just want to understand number theory better.

I know basic stuff about modulo, if that is what you meant. I do plan to read more on it, so if you have any online text which in your view is helpful, i'd be interested
I think modulo may be good to use to build a proof about whether 196 (or similar numbers) really never reach a palindrome regardless of how many iterations of that procedure (adding the number starting from the opposite side when there is no palidrome, eg 196 isn't a palindrome and next becomes 196+691; the so-called Lychrel numbers).

My own interest in math is primarily due to logic. We are built in such a manner that those progressions are intelligible for us. In reality they are our own projection, so they can help us understand ourselves. Ultimately i think that the human thinker is the collection of unknown axioms where the entire system is based (way past the axioms we have consciously set about math).

 

[AdrienIer]

Unfortunately my references in maths are in french... I think that wikipedia articles in mathematics are generally good, but I don't know how deep and detailed they are on these topics in particular.

Modulo are also great to understand the reason for some of our basic rules like n is divisible by 9 if and only if the sum of its numbers is divisible by 9. Or the lesser known rule for 11 (4851 is divisible by 11 because 4-8+5-1=0).

Mathematical logic (and first order logic) is fun as well, and is a good way to formalize human logic.

 

[Kyriakos]

^Hey, i do want to read about the 9 thing, cause it has interested me for a while ^^ I didn't know it was rigorously accounted for through the modulo 10 thing with adding numbers in that manner ^^

 

[AdrienIer]

Actually you never use modulo 10 to prove it. You use the value of 10^k mod 9. Let me explain.

You can add, substract and multiply modulos (if m = l mod k then m² = l² mod k, and additionally if p = q mod k then m*p = l*q mod k and m + p = l + q mod k). Because 9, 99, 999, 9999 (and so on) are multiples of 9 it means that 1, 10 (= 9 + 1), 100 (= 99 + 1), 1000 (and so on) are all equal to 1 modulo 9. So (to take an example of a number < 1000) if n = a*1 + b*10 + c*100 then, modulo 9, n = a*1 + b*(1+9) + c*(1+99) = a + b + c + 9*b + 99*c = a + b + c. Which means that for n to be a multiple of 9 you need a + b + c to be a multiple of 9.

For divisibility by 11 you do the same thing, except that 10 = -1 mod 11, then 100 = 99 + 1 = 1 mod 11, then 1000 = 1001 - 1 = -1 mod 11 etc... So the rule is if your number is written n = abcd then you need a - b + c - d to be a multiple of 11 for n to be a multiple of 11.

 

[Kyriakos]

Actually you never use modulo 10 to prove it. You use the value of 10^k mod 9. Let me explain.

You can add, substract and multiply modulos (if m = l mod k then m² = l² mod k, and additionally if p = q mod k then m*p = l*q mod k and m + p = l + q mod k). Because 9, 99, 999, 9999 (and so on) are multiples of 9 it means that 1, 10 (= 9 + 1), 100 (= 99 + 1), 1000 (and so on) are all equal to 1 modulo 9. So (to take an example of a number < 1000) if n = a*1 + b*10 + c*100 then, modulo 9, n = a*1 + b*(1+9) + c*(1+99) = a + b + c + 9*b + 99*c = a + b + c. Which means that for n to be a multiple of 9 you need a + b + c to be a multiple of 9.

For divisibility by 11 you do the same thing, except that 10 = -1 mod 11, then 100 = 99 + 1 = 1 mod 11, then 1000 = 1001 - 1 = -1 mod 11 etc... So the rule is if your number is written n = abcd then you need a - b + c - d to be a multiple of 11 for n to be a multiple of 11.

Actually you never use modulo 10 to prove it. You use the value of 10^k mod 9. Let me explain.

You can add, substract and multiply modulos (if m = l mod k then m² = l² mod k, and additionally if p = q mod k then m*p = l*q mod k and m + p = l + q mod k). Because 9, 99, 999, 9999 (and so on) are multiples of 9 it means that 1, 10 (= 9 + 1), 100 (= 99 + 1), 1000 (and so on) are all equal to 1 modulo 9. So (to take an example of a number < 1000) if n = a*1 + b*10 + c*100 then, modulo 9, n = a*1 + b*(1+9) + c*(1+99) = a + b + c + 9*b + 99*c = a + b + c. Which means that for n to be a multiple of 9 you need a + b + c to be a multiple of 9.

For divisibility by 11 you do the same thing, except that 10 = -1 mod 11, then 100 = 99 + 1 = 1 mod 11, then 1000 = 1001 - 1 = -1 mod 11 etc... So the rule is if your number is written n = abcd then you need a - b + c - d to be a multiple of 11 for n to be a multiple of 11.

Great ^_^ Then i will also have to look into identities about powers with modulo, does the proof for that require anything more advanced than simple limits? Cause i should at least be aware of why those identities are such, if i am to form any insight myself on using modulo at all I will look into it these days...

 

[AdrienIer]

You mean why this :

You can add, substract and multiply modulos (if m = l mod k then m² = l² mod k, and additionally if p = q mod k then m*p = l*q mod k and m + p = l + q mod k)

is true ? In arithmetics you usually don't use any limits (limits are useful in R not in Z). Here you just say that if m = l mod k then m = a*k + l (with a, k and l in Z). So m² = (a*k + l)² = a² * k² + l² + 2*a*k*l = k * (a² * k + 2*a*l) + l². If you look at it mod k you have m² = l².