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I had an argument with a friend of mine and I'd be glad if someone could clarify things a little bit.

So, let's say we have an integer, eight or seventeen, for example, doesn't matter. It has all the properties of an integer. In particular, it can be even or odd, i.e. has a property of parity.

From another point of view, integers are a subset of rational numbers, so integers 8 and 17 can be written as ratios 8/1 and 17/1, and also be written as rational 8.0 and 17.0.

The question is:

Do integer numbers keep their properties when expressed as an element of any of their supersets? E.g. if 8 is even, is it possible to say that rational 8.0 is also even as well as real 8.0?

If not, then why? Numbers 8, 8.0, 8/1 all express the same entity, so does notation influence the properties of an object?

Arthur
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Vsevolod Timchenko
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6 Answers6

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Let's say we define a number $n$ to be even if $n \in E$, where: $$E=\{ n \in \Bbb{Z} \ \mid \ (\exists t \in \Bbb{Z})(n=t+t)\}$$

Now, $8 \in E$ because $8=4+4$. However, only integers can be even because we defined $E$ as a subset of integers in our set-builder notation, so this definition will likely only be useful if we are thinking about integers.

However, if we look at $\frac 8 1 \in \Bbb{Q}$ and think in terms of the rational numbers, it does not change the fact that $\frac 8 1=8 \in E$. Furthermore, if we look at $8.0 \in \Bbb{R}$ and think in terms of the real numbers, it still does not change the fact that $8.0=8 \in E$. Therefore, yes, $8$ is still even when thinking about the rational or real numbers.

The question is, however, why does it matter? Even numbers is a property that only applies to integers, so when we are thinking in terms of rational numbers or real numbers, the property of evenness simply does not come up that much. It's just not very useful to talk about "even rational numbers" or "even real numbers" since they are both the same thing as just "even integers." Looking at $8$ as $\frac 8 1$ or $8.0$ does not change the fact that $8$ is even, but that property is probably not relevant when we are thinking about $8$ in these contexts.

Noble Mushtak
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  • "Why does it matter?" - I was trying to convey to my friend the idea of 0.(9) being identically equal to 1 and, ergo, 3.(9) being an even number. Despite best efforts this is something people are really reluctant to accept. A ridiculous non-existent 0.(9) + 0.(0)1 = 1 thing has somehow taken roots in common consciousness. – Vsevolod Timchenko Jun 11 '16 at 12:25
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    Hm...This is very odd. I'm not sure if this will help your friend, but it's easier to explain if you write $0.(9)$ as an infinite sum such as $\sum_{i=0}^\infty \frac{9}{10^i}$ and then you can use the infinite geometric sum formula to show that it really does equal $1$. You could also try to show them the Vi Hart video on this. – Noble Mushtak Jun 11 '16 at 12:28
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    Furthermore, you might want to explain hyperreals to them and how this notion of $0.(0)1$ is known as $\epsilon$ in the hypereals, but does not in decimal numbers because all decimal numbers are real numbers, so the notion is useless here. – Noble Mushtak Jun 11 '16 at 12:31
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    Thanks for suggestions, but I'm really afraid hyperreals may complicate things even further, not help with explanation, as I'm not very good at them myself. – Vsevolod Timchenko Jun 11 '16 at 13:01
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    @NobleMushtak I think hyperreals are a poor choice as a way to make sense of things like 0.(0)1. For one thing, I doubt that $\varepsilon$ is consistently used to denote a particular number in a hyperreal field (putting aside the fact that there are different hyperreal fields without CH), and for another, there's not really a way to tie a construction of hyperreals to an extension of a decimal-like notation. If you give up base 10 for base 2, you could argue that the sign expansions of surreals are a good interpretation, as I began alluding to at http://math.stackexchange.com/a/483582/26369 – Mark S. Jun 11 '16 at 17:52
  • @MarkS. I have never heard of Gonshor's sign expansion, but after reading this explanation, I agree that this is a better explanation. However, would this mean that $\epsilon=+---...$? I just want to make sure I am understanding you correctly. Thanks! – Noble Mushtak Jun 11 '16 at 18:11
  • @NobleMushtak That is the usual precise meaning of $\varepsilon$ in the surreals, yes. If you have further questions about the surreals that you can't answer by googling around, I suggest posting them on MSE. – Mark S. Jun 11 '16 at 18:32
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An even number is one that is $2$ times an integer. An odd number is an integer that is not even.

$8$ is an even number no matter how you write it -- as you yourself point out, the various notations are simply ways to denote the same mathematical object.

However: Writing down mathematical expressions is a way to communicate ideas to other humans. And how you write things will influence which properties of numbers will be on the reader's mind.

So if you have a variable letter and declared that it's supposed to be some number, chosen in a way that doesn't seem to force it to be an integer, then it will be confusing to speak about whether it is even, because the context has not primed the reader to think about evenness and oddness as properties that are relevant in the context. In that case it is good communication not to say "if $x$ is even, then ...", but instead something like "if $x$ is an even integer, then ...". This reassures the reader that you're aware that you're doing something slightly unusual, and means that he doesn't have to worry about whether he missed something that would make it natural to think about the relation between $x$ and the integers.

hmakholm left over Monica
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Oddly enough, Friedrich Engels the co-author, with Karl Marx, of the "Communist Manifesto", in a book on the Philosophy of science refers to 7 being "odd in base 10 but even in base 5". That is, of course, untrue. It is true that, in base 5, 7 is written as 12. However, "last digit even" is not the same as "number even" in an odd base. A number, n, is even if and only if n pebbles can be divided into two sets each having the same number of pebbles- that has nothing to do with the numeration system.

user247327
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I won't repeat the points made in other answers.

I'll make the point that sometimes the context in which a particular form of a number appears matters as to its interpretation/meaning in such a way as to make certain properties irrelevant.

For example, it's unusual to write $8.0$ unless the decimal serves a purpose, say to indicate accuracy of a measurement. So we take this to mean the actual quantity being measured is in the interval $[7.95,8.05).$ In this case, we aren't even referring to a single number, so evenness is irrelevant.

One would never write $\frac81$ unless it were useful to do so. For example, the odds of each side for a $9$ sided die. In this case the concept of evenness isn't applicable; odds ratios don't add meaningfully (at least not in all cases).

So the properties of the number itself don't change (i.e. $8=8.0=\frac81$ as pure numbers, representatives of the property of "eight-ness") as noted in other answers. However what that number is being used for may make certain properties irrelevant or inapplicable.

jdods
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  • Valid point. Though I might be wrong, but indicating accuracy with a number of naughts after period is more common in the applied part of mathematics: applied maths itself, physics, chemistry, engineering etc. When talking about pure mathematics and more specifically the case at hand (i.e. I use 8.0 to emphasize 8 being a real number) we're talking about the number 8, exactly it, not a measurement with approximate value of eight, but eight in all it eight-ness. – Vsevolod Timchenko Jun 11 '16 at 13:09
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    Yes, certainly as equivalent to a "pure number" in and of itself, it doesn't matter how you write it. But the underlying structure can still be important in pure math. E.g. $\sum a_n= s\in\mathbb R$ both sides are equivalent as numbers, but there is a certain structural difference. If you just wanted the fixed number $s$, you would never write $\sum a_n$. But certainly it wouldn't be wrong per se. I've never seen decimals being used to emphasize real number-ness, but it certainly isn't wrong to do so. As noted, the concept of even-ness is arguably irrelevant for real numbers. – jdods Jun 11 '16 at 13:20
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The way you call an object does not influence with his property. It is important to understand this. In mathematics when you speak about an abstract object you must necessarily name it. Its impossible to speak about something without giving it a name.

Marco Lecci
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  • I don't understand your point. How does it apply to the example, in which there is no issue about what "$8$" is (other the fact that it can be seen as an element of two sets in which the interesting properties are different)? –  Jun 11 '16 at 12:07
  • @G.Sassatelli If "%" is the number 8, then we can say that "%" has the same property of the number 8. – Marco Lecci Jun 11 '16 at 12:11
  • Quoting the OP: "The question is: 1) Do integer numbers keep their properties when expressed as an element of any of their supersets? E.g. if 8 is even, is it possible to say that rational 8.0 is also even as well as real 8.0? 2) If not, then why? Numbers 8, 8.0, 8/1 all express the same entity, so does notation influence the properties of an object?" The point, in my opinion, is that you never say that a certain rational number is odd. –  Jun 11 '16 at 12:19
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Contrary to other respondents, I say that, in a sense, it DOES matter.

Consider the statements: "8 is even, 5 is not'. Working in the integers, this means that 8 can be expressed as an integer multiple of 2 (i.e 4), whereas 5 cannot be so expressed. Therefore 'is even' is a meaningful and useful property - over the integers - because some have the property and some don't.

Now switch to the rationals. Then 8.0 is a multiple of 2.0 (i.e 4.0), but now 5.0 is ALSO a multiple of 2.0 (i.e 2.5). In fact, EVERY rational is a multiple of 2.0. So over the rationals the property 'is even', so defined, does not distinguish any numbers from any other numbers, hence the property is meaningless and useless.

Lest you think that this somehow still leaves '8.0 is even' intact, I will now show '8.0 is odd'. Over integers 'is odd' can be defined as 'can be expressed as 2 * X + 1'. Moving that definition up to the rationals, then 8.0 = 3.5 * 2.0 + 1.0, hence also '8.0 is odd'.

HOWEVER: I will add that this is more a matter of different numeric contexts, not different notations, except insofar as choice of notation implies a particular numeric context (which is not a very robust assumption, actually).

PMar
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  • While your point is valid, I am not sure notation and meaning are so strongly connected. –  Jun 11 '16 at 17:39
  • The whole point of an even number is being a multiple n = 2*k where k IS INTEGER. Saying that "5.0 is ALSO a multiple of 2.0 (i.e 2.5)" and "the property 'is even', so defined, does not distinguish any numbers from any other numbers" is just wrong, because if you change definitions in such an arbitrary way, you can prove that moon is made of green cheese and you yourself are the Pope. – Vsevolod Timchenko Jun 14 '16 at 16:49