Wikipedia:Reference desk/Archives/Mathematics/2023 December 16

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December 16

Prime number determiner

If a number can be broken up into two strings of digits, and one string is divisible by the other, does that mean the number itself cannot be prime?

For example, I’ll have the letter string abcd stand in for a number (the letters stand for place value, not multiplying variables together). This can be broken up three different ways:

1. As “a” and bcd.

2. As ab and cd.

3. As abc and d.

If bcd is divisible by “a” in #1 or if abc is divisible by d in #3 (provided that the lone digit in both situations is not the number 1), or if either ab or cd are divisible by each other in #2, does that mean that the original number abcd cannot be prime? Also, what would be a more easily understandable way of wording this question? Primal Groudon (talk) 14:46, 17 December 2023 (UTC)

Indeed, the original number cannot be prime.

Let's say that the original string of digits of some number $M$ is $a_{m}a_{m-1}...a_{0}$ (you'll see why this numbering is useful in a second.) Also, let's first assume that $a_{m}a_{m-1}...a_{i}$ divides $a_{i-1}a_{i-2}...a_{0}$.

Our original number $a_{m}a_{m-1}...a_{0}$ is just $a_{m}a_{m-1}...a_{i}0...0+a_{i-1}a_{i-2}...a_{0}$. You can notice that $a_{m}a_{m-1}...a_{i}0...0$ is $a_{m}a_{m-1}...a_{i}*10^{i}$. So if $a_{m}a_{m-1}...a_{i}$ divides $a_{i-1}a_{i-2}...a_{0}$, then since it also divides $a_{m}a_{m-1}...a_{i}*10^{i}$, it must also divide the sum $a_{m}a_{m-1}...a_{i}0...0+a_{i-1}a_{i-2}...a_{0}$, which is indeed $a_{m}a_{m-1}...a_{0}$, the original number $M$.

Assuming that you didn't just let $a_{m}a_{m-1}...a_{i}$ be the string $1$, this means that you have a factor $a_{m}a_{m-1}...a_{i}$ greater than 1, making $M$ not prime.

If you consider the other way around, that $a_{i-1}a_{i-2}...a_{0}$ divides $a_{m}a_{m-1}...a_{i}$, then it still works. $a_{i-1}a_{i-2}...a_{0}$ then divides $a_{m}a_{m-1}...a_{i}*10^{i}$, and as such it divides $a_{m}a_{m-1}...a_{i}0...0+a_{i-1}a_{i-2}...a_{0}$, which as mentioned before, is the original number $M$. GalacticShoe (talk) 15:17, 17 December 2023 (UTC)

More in general, if the two parts are not coprime – that is, have a non-trivial common divisor – their combination is not prime. For example, 793 and 3523 are both divisible by 13:

793 = 61 × 13 and 3523 = 271 × 13

Then

7933523 = 610271 × 13 and 3523793 × 271061 × 13.

 --Lambiam 15:37, 17 December 2023 (UTC)

Fun fact: sometimes a number can be split up in several ways that have common divisors, for example the number 1463869:

146 = 2 × 73 and 3869 = 53 × 73

So

1463869 = 20053 × 73.

But we can also break it up into:

1463 = 133 × 11 and 869 = 79 × 11

So

1463869 = 133079 × 11.

 --Lambiam 16:10, 17 December 2023 (UTC)

Overlooked proof

(Copied from talk:Division by zero)

Let imagine a graph (1/x) is divided into 2 halves where the negative half is x<0 and the positive half is x>0. The primary proof against 1/0 being a defined value is that the two halves directly contradict one another (negative half shows -∞ but positive half shows ∞) so it must be an undefined value right? No, so lets change our way of thinking. We will be adding a new rule which states: "Any division that might return a negative result shall be written in a way in which a negative number will never be dividing another or divided against.". Since this only applies to HOW we write down or calculate the equations, this does not change anything ((wrong) 2÷-5 = -0.4 --> (correct) -(2÷5) = -0.4). By applying the rule, we would never actually divide by the negatives which completely removes the negative half. This will leave only the positive half with nothing contradicting it, which could only mean x/0 = ∞.

I don't study math like a mathematician but this is easy. I need yall to say im wrong rn holy moly. Also the rules are made ambiguous, think of the best case scenario of my explanation as the primary one. SussusMongus (talk) 04:52, 16 December 2023 (UTC)

It's not "wrong" per se. You are welcome to define your own new number system that behaves however you want. It's just that most mathematicians and other technical professionals have settled on a few conventional choices of number system which were found to be the most convenient in the broadest class of existing uses, and your proposed system is not one of them. (Your number system is quite limiting in that you have to constantly check that you aren't dividing by negative numbers, which is even more burdensome than checking to not divide by zero, and there are many kinds of algebraic manipulation that have the potential to introduce illegal divisions which you will have to either disallow or be very careful about. Many of the most convenient and useful tools of algebra are going to be made much more troublesome, if they aren't entirely broken.)

I'd recommend you take this kind of question to Wikipedia:Reference desk/Mathematics, since it's not really about improving this article per se. –jacobolus (t) 05:05, 16 December 2023 (UTC)

I am going to reply since i already posted this and the rule actually NEVER specified about when you check, you just need to know that before you actually divide it. If it's a variable then just look if it have a sign. Afterall, the variable itself might be negative but the value might NOT be negative (or vise-versa) and the rule never specified any of that. This reply is the primary example of why I left the phrase "The rules are made ambiguous" there (i knew my initial post isnt going to be accurate at conveying what i literally meant, especially the little details). SussusMongus (talk) 06:28, 19 December 2023 (UTC)

Which are the rules that are ambiguous? The statement that $1/0$ is undefined is not a rule but an observation: the observation that a question is being asked that has no answer.

    Let me explain. In mathematical notation, $a/b$ is convenient shorthand for, "the unique number $x$ such that $a=x\times b.$" Usually there is an answer to the question, "Which is the unique number $x$ such that $a=x\times b$?" But if $b=0$ this question fails to have an answer.

    One can always ask questions that have no answers. "The successor of Puyi as Emperor of China" is, by definition of "successor", the answer to the question "Who succeeded Puyi as Emperor of China?". Puyi was the last Emperor of China; he had no successor. So this question has no answer and the expression "the successor of Puyi as Emperor of China" has no definable meaning. You may try to formulate a rule that one is only allowed to use the phrase of the form "the successor of X as Emperor of China" if one has checked that X was an Emperor of China who had a successor, but is this rule helpful?  --Lambiam 08:22, 19 December 2023 (UTC)

That a/b and a = x/b stuff could just be made invalid for cases where b is 0. It's a shorthand, not an important number like π. The Emperor of China stuff is a good metaphor to explain why things like 1/0 just cannot exist even when I force it, but I'm not forcing it. This is a quote from RDBury: "So it's not so much that you can prove 1/0 doesn't exist at all...". It is possible.

Also the rule i'm talking is "Any division that might return a negative result shall be written in a way in which a negative number will never be dividing another or divided against." anything wrong with that statement is covered in my second reply (above)

Also, i know you did not really mean it but: https://www.quora.com/Who-would-be-the-Chinese-emperor-today

To: Lambiam SussusMongus (talk) 10:32, 19 December 2023 (UTC)

The primary proof against 1/0 being a defined value is that the two halves directly contradict one another — no, not really. There are more fundamental problems that show up long before you try to graph the function $1/x$. If you have 10 cookies and try to share them among 0 people, how many cookies does each person get? XOR'easter (talk) 08:09, 19 December 2023 (UTC)

@Jacobolus: A system where 1/0 is defined already exists; it's called the projective line. It's useful in geometry but as a number system it leaves a lot to be desired. Instead of just one operation not being fully defined, none of them are: for addition and substraction ∞±∞ is undefined, for multiplication 0⋅∞ is undefined, for division ∞/∞ is undefined and 0/0 is still undefined. Cancellation laws don't hold since 0+∞=1+∞, 1⋅∞=2⋅∞. Also, it's harder to define < and >; is ∞ positive or negative? Basically it removes one exceptional case but introduces many more, so a net loss in terms of simplicity and ease of use. It's hard to interpret in terms of the real world as well; if I have $∞ is that good or bad? It depends on whether ∞ is positive or negative, but ∞ doesn't have a sign in this system. So it's not so much that you can prove 1/0 doesn't exist at all, but you can prove it doesn't exist as a real number and real numbers are just better for everyone's sanity than trying to do arithmetic on the projective line. --RDBury (talk) 09:12, 19 December 2023 (UTC)

To: RDBury

Ah, I see, but I will keep trying to do it anyway and you can't stop me. 😈 SussusMongus (talk) 10:10, 19 December 2023 (UTC)

10 cookies to 0 people actually means 10/1 in context since YOU are the one giving the cookies. But if you throw away the cookie anyway, that would be subtraction, so that does not work. Also I meant like the primary reason wikipedia said it's not defined which i will assume is the very first proof shown. My theory is that if my theory is right then the other paradoxical stuff should be solved.

To: XOR easter

SussusMongus (talk) 10:07, 19 December 2023 (UTC)

If expressions of the form $a/b$ are ruled to be erroneous whenever $a<0$ or $b<0$, this new rule invalidates a major swath, very likely even the vast majority, of the mathematical achievements in the field of analysis since Fermat. Do the advantages of introducing the definition $x/0=\infty$ outweigh this loss? For one thing, it is not clear that this is an advantage in the first place. Most of analysis takes place on the real or complex numbers, and $\infty$ is neither a real number nor a complex number. One can work in the extended real numbers or the projectively extended real line, but in these systems some properties of the "vanilla" real numbers are not generally valid. For example, $(a+c)-(b+c)=a-b$ is not a theorem in the extended real numbers. Most achievements just do not carry over.  --Lambiam 22:25, 19 December 2023 (UTC)

Well from my estimated 2 minutes of hard work, dedication, sweat, and blood trying to find a = x/b and (a+c)-(b+c), I have found nothing about it. Please link a source so i will have a source to look at thanks. Also my goal is to cause destruction and chaos within mathematics so we don't need to learn math at school anymore (reason that i made up like 2 seconds ago), I am well aware of the mind twisting monster I am trying to raise. (BTW the entire reason why I thought of this proof is due to PURE boredom) SussusMongus (talk) 00:27, 20 December 2023 (UTC)

You want to get rid of the restraint on division by $0$ by introducing $\infty$. If $\infty$ is admitted as a first-class citizen among the numbers, without introducing new restraints, you need the identity $a+\infty =\infty .$ If the familiar identities $a-0=a$, $a-a=0$ and $(a+c)-(b+c)=a-b$ are allowed to remain without restraint, we can now prove that any number, say $x,$ is equal to $0$:

$x$ $=x-0$ $=(x+\infty )-(0+\infty )$ $=\infty -\infty$ $=0.$

 --Lambiam 09:36, 20 December 2023 (UTC)

i just realised this, if we assume division by 0 is possible then 0 divided by 0 would be valid (and according to the definition of 0, it will be 0), so: b=0

, a/0=∞ OR 0 (if a=0), now let say: a=0, 0/0=0, 0=0/0. x=0. But this only proves 0/0 is 0. Nothing divided into infinity is nothing. This sort of make sense (a little bit) (kinda) (maybe not). Does this satisfy a/b a=x/b? SussusMongus (talk) 10:34, 20 December 2023 (UTC)