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Since there is no way to divide 1 by any natural numbers other than 1 itself, how can 1 be divided into pairs until either 0 or 1 units are left, letting us call it even or odd?

Perhaps the definition of odd numbers is irrational and wrong, at least for one.

Why is one considered an odd number?


Additional Details:

I think an odd number is a number that when taking that number of units and dividing the units into pairs, you have either get an unpaired unit or not when you are finished. This definition does not include the number one. All natural numbers are testable except for one. A number must be able to be tested in this way before being called an odd number. 1 cannot be tested in this way, and therefore it cannot be called an odd number. The un-testability of the number of one makes it impossible to judge whether it is odd or even.

closed as off-topic by Mauro ALLEGRANZA, Tim B II, Dennis, Jordan S, MichaelK Apr 4 '18 at 17:41

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "While this question may be related to philosophy or occur in a philosophical context, the question itself doesn't seem to be about philosophy, and is therefore not a good fit for our site." – Mauro ALLEGRANZA, Tim B II, Dennis, Jordan S, MichaelK
If this question can be reworded to fit the rules in the help center, please edit the question.

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    The wording of this question is almost exactly the same as this one ._. – Challenger5 Mar 29 '18 at 3:26
  • Huh, what a strange coincidence. Personally, regarding that question, I feel the comment section has some wisdom that would better serve as an accepted answer. In any case the asker can choose what they want. I just hope there are as useful answers to my own distinct question here as there are in there. – World Outsider Mar 29 '18 at 3:40
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    I'm not convinced it's a coincidence. The paragraphs are laid out the same way, you've used similar formatting, header, and titles, and the line "Perhaps the definition of X is irrational and wrong, at least for Y." is exactly the same. The two questions were posted within two days of each other. – Challenger5 Mar 29 '18 at 4:00
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    Because it is not even. – Mauro ALLEGRANZA Mar 29 '18 at 6:09
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    The question has nothing to do with "phil of math". – Mauro ALLEGRANZA Mar 29 '18 at 6:09
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Your test can be applied to 1. The result of dividing up 1 into pairs and observing whether there is an unpaired unit or not results in 0 pairs and 1 unpaired unit, making it odd.

The only reason why 1 would not be odd by your definition is because you explicitly define it as such, with no rationale given other than the desire to make it not be odd. It took you extra sentences in order to exclude it, which weren't even part that described the test itself.

As with the other mathematical question linked, the real answer is one of lingusitics and the nature of mathematicians. Both the sets {1, 3, 5, 7, 9, ...} and {3, 5, 7, 9, ...} are indeed sets that can be defined. One of them could be given the title "odd numbers." As it turns out, mathematicians prefer the simple definition which does not arbitrarily exclude 1 instead of the definition which does arbitrarily exclude 1. You are welcome to care more about the latter set, but the former set is the one which mathematicians as a whole have given a single-word name to.

You also might be interested in the concept of the parity of zero. Parity is the concept behind the definitions of odd and even. The system of parity states that zero is even. Of all the numbers which might get arbitrary exclusions, zero is much more common, and not even zero gets excluded here.

  • Normally, I believe, odd and even numbers are defined in terms of multiplication and addition, not division. Under that definition (2 * n is even, 2 * n + 1 is odd, for n being an integer), 1 must be odd under that definition. – Clearer Mar 30 '18 at 17:59
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It is not unusual in mathematical definitions for the first member of a set to be defined differently, or seemingly more arbitrarily than the subsequent members of the set. This is particularly true in cases where the first member plays a key role in the general definition.

Mathematics in general, however, has a odd metaphysical status. It simultaneously seems like something we create, and something we discover. Many mathematical rules and procedures seem arbitrary from one point of view, yet written in stone from another. There is a seeming definiteness and purity to mathematics that can be deceptive.

What this means in practice is that there are times when we assign a number to a set --1 to the set of odd numbers, for example, or 2 to the set of prime numbers --less because it fits a general definition, but because it "works," because it is useful, or because it creates less problems and exceptions when defined that way. This may be frustrating to those who think this heuristic falls short of mathematical perfection and precision. Every good idealist should know, however, that reality always inevitably falls short of the ideal.

  • Nice answer Chris. In mathematics this question is trivial but in metaphysics the precise status of the first few numbers becomes critical. (This is suggested by what you say about the first members of sets). For this reason I see the question as more metaphysical than mathematical. – PeterJ Mar 29 '18 at 17:49
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An integer x is considered an even number when dividing x by 2 leaves rest = 0, it is considered an odd number number when x dividing by 2 leaves rest = 1.

Now:

  • 3 / 2 = 1 rest 1: 3 is odd
  • 2 / 2 = 1 rest 0: 2 is even
  • 1 / 2 = 0 rest 1: 1 is odd
  • 0 / 2 = 0 rest 0: 0 is even
  • Using remainder with multiplication is confusing. Can you change to 3 / 2 = 1 remainder 1, etc? – amI Mar 29 '18 at 22:31
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Note that the natural numbers contain the positive integers as well as 0.

Of the four operations, addition, subtraction, multiplication and division, only two of these operations are closed in the natural numbers, addition and multiplication. That is, given two natural numbers, n and m, we can add them, n + m, and multiply them, n * m, and the results will be natural numbers.

This doesn’t always work for subtraction and division as you observed. So, these operations are not closed. That does not mean that there is "no way to divide 1 by any natural numbers other than 1 itself". There are workarounds using multiplication with a remainder.

For example, we can say that if n = s * t where n, s and t are natural numbers, then both s and t “divide” n. Also, if n = q * m + r where n, q, m, and r are natural numbers and r > 0 then neither q nor m "divide" n because their product leaves a positive remainder. This gives us a way to work with division even though the division operation is not closed in the natural numbers.

Furthermore, we can restrict the remainder to be less than m and greater than or equal to 0 because there is an order on the natural numbers. This simplifies finding natural numbers, q and r, and makes the pair, q and r, unique for any given n and m.

Given this notion of division, we can say that a natural number, n, is even if 2 divides that number, that is, the natural number is even if its remainder upon division by 2 is 0. If not then n is odd.

In the case of the natural number 1, we see that 1 = 0 * 2 + 1. Since the remainder is 1, the natural number 1 is odd.

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