Do different notations imply different properties of a number?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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).