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In a college class I was asked this question on a quiz in regards to sets:

All integers are fractions. T/F.

I answered False because if an integer is written in fraction notation it is then classified as a rational number. The teacher said the answer was True and gave me the link http://www.purplemath.com/modules/numtypes.htm. As a teacher of mathematics in the K-12 system I have always taught that integers were all whole numbers above and below zero, and including zero. All of the resources I have used agree to my definition. Please clarify this for me.

What is the truth, or are she and I just mincing words?

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    $\begingroup$ You've given your definition of integers. What is your definition of fractions? $\endgroup$ – Chris Eagle Mar 2 '13 at 21:54
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    $\begingroup$ Humans are animals, math questions are questions, birthday cake is cake, chairs are furniture, integers are rational numbers. See en.wikipedia.org/wiki/Subset $\endgroup$ – anon Mar 2 '13 at 21:56
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    $\begingroup$ I sympathize. In ordinary English, no one would call $5$ a fraction. A fraction is a "broken number," and $5$ ain't broken, it is whole. Unfortunately, the words that were being used in the question, though they looked like ordinary English words, did not all have their ordinary English meanings. $\endgroup$ – André Nicolas Mar 2 '13 at 22:22
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    $\begingroup$ Note that if the statement had been "All integers are rational numbers," then True would have been the correct response. $\endgroup$ – André Nicolas Mar 2 '13 at 22:53
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    $\begingroup$ Chris Eagle: Good question. A more relevant one for the purposes of the quiz is: What is the teacher's definition of "fractions"? I think we can guess that for the teacher, the words "fraction" and "rational number" are interchangeable, which is not true for all of the Math.SE community, but would be good for the OP to understand when taking future tests or quizzes. $\endgroup$ – Todd Wilcox Mar 3 '13 at 5:06

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Integers are fractions, because a number is itself no matter how you write it.

A relevant section from Lockhart's A Mathematician's Lament:

In place of a natural problem context in which students can make decisions about what they want their words to mean, and what notions they wish to codify, they are instead subjected to an endless sequence of unmotivated and a priori “definitions.” The curriculum is obsessed with jargon and nomenclature, seemingly for no other purpose than to provide teachers with something to test the students on. No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re equal for crying out loud. They are the same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade?

Here's a relevant comic from SMBC:

Header: A+ Math Student / Woman #1: "No! Pi is IRRATIONAL, meaning it can't be expressed as a relationship between two numbers. Johann Lambert proved this in 1761." / Header: Future Mathematician / Woman #2: "pi/1"

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I want to point out a different perspective. This is a typical issue where non-rigorous use of language leads to confusion. We often treat objects that are equivalent (under a possibly unspoken equivalence relation) as if they are equal.

Integers and rational numbers are not fractions, in the strictest sense of the word "are". For example, the fractions $1/1$, $4/4$, and $8/8$ are all different fractions, but they all represent the same integer. (Actually, I should say "the fractions represented by the expressions $1/1$, $4/4$, and $8/8$.") None of the fractions $1/1$, $4/4$, and $8/8$ literally is the integer 1. If they were all the same as the integer $1$, they would all be the same as each other - but they are not, because they are different fractions. What is true is that they have the same value as each other.

Similarly, the fractions $1/2$ and $2/4$ are not both the same as the rational number $1/2$, because they are not the same as each other, because they have different numerators. Who would claim that a rational number has a numerator?

When we write $$ 1 = 4/4 = 8/8 $$ the "equals" there only means that fractions represent the same number (they have the same value), not that they are the same fraction. In other words, the $=$ we write is actually an equivalence relation on a set of fractions - the equivalence relation being "has the same value". We have no symbol that we typically use to denote "actual" equality of fractions. This point is often ignored completely in lower-level texts.

The first place that people start to consider these things in detail is in abstract algebra. In that context, we run into many interesting distinctions that would not be visible in elementary math. For example, under the definitions in most algebra books, $\mathbb{Z}$ is an extension of the semigroup $\mathbb{N}$ to a group; $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$; and $\mathbb{R}$ is the Dedekind completion of $\mathbb{Q}$. It follows from these definitions that the natural number 1 is not the same as the integer 1, both are different than the rational number 1, and all three of those are different from the real number 1.

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    $\begingroup$ +1 because I also agree with this perspective :) Depending on the mathematical situation, we might want to distinguish (or not distinguish) things that can be identified. Depending on the educational situation, we also might be expected to, even if we don't want to. $\endgroup$ – Zev Chonoles Mar 2 '13 at 22:27
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    $\begingroup$ I explain the difference between a fraction of the form, 2/4 and 1/2 to my probability students as follows. Suppose that we have two populations, one population with four people, and two women, and a second population with two people and one woman. Then the proportion of women in the first population is 2/4 and the proportion of women in the second is 1/2. It turns out that the two proportions are equivalent but not identical. The lesson is when we take equivalence classes we lose information, namely the size of the population. $\endgroup$ – Baby Dragon Mar 2 '13 at 22:35
  • $\begingroup$ @Carl Mummert . In my comment to an answer I ended up with "It is equal because there is a mathematical convention that makes it equal.", which I think it is incorrect in light of your answer. So they are not equal as mathematical entities. Are we left to say they are equal as a language convention? That is when we say Z is a subset of Q or z=z/1 for z integer, we make a language convention (shortcut, compromise)? Thanks $\endgroup$ – Theta33 Mar 2 '13 at 23:14
  • $\begingroup$ Your last sentence, to have any meaning, requires a definition of "the same" and "different than". It seems you may have in mind "different sets" but even this need not be true, because some authors do explicit set surgery to keep subobject inclusions set-theoretical. $\endgroup$ – Math Gems Mar 2 '13 at 23:39
  • $\begingroup$ @Theta30: I would say that they are equivalent under a particular equivalence relation. The language convention is that we often say "equal" when we only mean "equivalent". That happens all the time in mathematics, and it lets us focus on things that matter while ignoring distinctions that are irrelevant in context. But it only works as long as we don't ask whether things are "really" equal - when we start asking that kind of question we have to make the equivalence relation explicit to eliminate the confusion. $\endgroup$ – Carl Mummert Mar 2 '13 at 23:52
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Every integer $x \in \mathbb Z$ can be expressed as the fraction $x \over 1$

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  • $\begingroup$ I wouldn't say that $x$ can be thought of as $\frac{x}{1}$. $x$ is equal to $\frac{1}{x}$. You might say that $x$ can be expressed as $\frac{x}{1}$... $\endgroup$ – Thomas Mar 2 '13 at 22:05
  • $\begingroup$ @Thomas: But it is not equal in the sense of identity. For example $4/2$ has numerator $4$ and $8/4$ has numerator $8$. If these were the same fractions, we would have $4 = 8$, because the operation "take the numerator of a fraction" is a function. Strictly speaking, what is true is that the value of $x/1$ is exactly $x$, under the map that assigns a nondegenerate fraction of reals to its value. $\endgroup$ – Carl Mummert Mar 2 '13 at 22:24
  • $\begingroup$ @CarlMummert: You know more about strict logic than me. I was trying to say that $x$ and $\frac{1}{x}$ are equal. It is not as if one can be thought of as the other one. I am not sure I understand what you are saying. I would say that "take the numerator of a given fraction" is a function from the set of fractions ($F$) to the set of integers $\mathbb{Z}$. However as elements in $F$ we for example don't have equality of elements $4 / 2$ with $8 / 2$. I think that in the OP's question we are working with the rational numbers where $4 / 2$ and $8/4$ is the same element. $\endgroup$ – Thomas Mar 2 '13 at 23:30
  • $\begingroup$ @CarlMummert: Maybe my confusion is that I read the "fraction" as "rational number". Would you read "fraction" in the original question as meaning just the set of formal expressions $a/b$ where we haven't made the identification that $a/b = c/d$ iff $ad = bc$? $\endgroup$ – Thomas Mar 2 '13 at 23:37
  • $\begingroup$ @Thomas: yes, that exactly what I think of. There is an evaluation function $v$ that takes a fraction $a/b$ with a nonzero denominator and produces a value $v(a/b)$. When I write $1/2 = 2/4$ in my calculus class, what I actually mean is $v(1/2) = v(1/4)$. $\endgroup$ – Carl Mummert Mar 2 '13 at 23:59
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I think you have a case to take to the teacher on this based on the teacher's use of the word "fraction" instead of "rational number". All integers are rational numbers also, everyone agrees (or should agree) on that. That is because all integers can be expressed as a ratio of integers. So here's the case:

Is $3$ a binary number?

I'd be surprised if too many people say "yes" to that question. Note that even though $3$ can be expressed as a binary number ($11$), we don't call it a "binary number" unless it is expressed as one.

Similarly, even though every integer can be expressed as a fraction (of integers, even), even mathematicians are likely to agree (for consistency's sake at least) that an integer is not itself a fraction even if it is always a rational number.

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    $\begingroup$ As I think about this more I realize what is at issue is whether we are talking about what the number is versus how it is written. Some may say that an integer is always a fraction because it can be written that way at any time. In that case it would seem to me the word "fraction" is used to characterize the number itself, not the way it is represented. On the other hand, the term "binary number" seems to only be used to denote the type of representation of a number, not any of the number's characteristics. $\endgroup$ – Todd Wilcox Mar 3 '13 at 5:24
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The important question is what is a fraction. If it is any number that can be expressed as $\frac ab$ with $a,b$ integers, then any integer $c$ can be expressed as $\frac c1$. If the fractions exclude the integers, integers are not fractions.

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  • $\begingroup$ I think it's more than that. One can ask why c is the same as c/1? One might answer because we see them as embedded in the quotient ring Q, that is c is the same as equivalence class of (c,1) . Or one can maybe say because "ordinary" division is preserved, that is c divided by 1 is c. In other words, the answer because "it is a subset" or because "c=c/1" is not sufficient. It is equal because there is a mathematical convention that makes it equal. $\endgroup$ – Theta33 Mar 2 '13 at 22:28
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$$ \begin{align} -7 &= \frac{-7}{1} \\ 0 &= \frac{0}{1}\\ 1 &= \frac{1}{1} \\ 4 &=\frac{4}{1}\\ \end{align} $$ Note here the $=$. So for example $-7$ is indeed an integer and as a number (element in the set of integers) it is the same as $\frac{-7}{1}$. It is not about thinking about numbers in different ways.

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  • $\begingroup$ As I point out my answer, the $=$ here is an equivalence relation, rather than literal identity. $\endgroup$ – Carl Mummert Mar 2 '13 at 22:22
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I'll assume by "fraction" you mean "rational number". If you want to be extremely pedantic, the answer is, in some sense, no. We don't really have $\mathbb{Z} \subset \mathbb{Q}$, but rather, we identify the rational numbers of the form $a/1$ with the integers. In cases like this, however, we always say that $\mathbb{Z}$ is a subset of $\mathbb{Q}$, even though it isn't strictly true. It would be pointless to so pedantic to say that integers are not rational numbers.

We have the same thing going on when we ask whether $\mathbb{Q}$ sits in $\mathbb{R}$. But, sometimes when doing math, I find it useful to switch between thinking of things as subsets, and thinking of things embedding into another thing, even when something isn't actually a subset.

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Integers are a subset of rational numbers. This doesn't make them not-rational numbers.

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  • $\begingroup$ In many definitions the integers are isomorphic to a subset of the rational numbers, but they are not literally a subset of the rational numbers. This is the case, for example, if $\mathbb{Q}$ is defined to be the field of fractions of $\mathbb{Z}$. $\endgroup$ – Carl Mummert Mar 2 '13 at 22:25
  • $\begingroup$ Of course the integers are not isomorphic, as a ring, to the rational numbers. $\endgroup$ – Carl Mummert Mar 2 '13 at 22:29
  • $\begingroup$ @CarlMummert Well, when we build up $\Bbb Q$ from $\Bbb Z$, we embed $\Bbb Z$ into $\Bbb Q$ isomorphically and forget about the "original" $\Bbb Z$. Under that idea, I would agree that every integer is a rational number. $\endgroup$ – Pedro Tamaroff Mar 2 '13 at 22:37
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The set of rational numbers can be partitioned into the set of ratios and the set of integers. Five clearly falls into the second partition. So the question is, do you define "fraction" as "ratio" (the first partition only) or as "rational".

I would tend to agree with you and define fractions as being non-integral rational numbers, or ratios.

If an integer is a fraction, that legitimizes nonsense like: 5 is 3, plus a fractional part of 2.

Also, note that we never say that numbers are divisible by 1. Prime numbers simply have no divisor, and two relatively prime integers have no common divisor, and that is that. The phrase "other than one" is added so that school children don't get excited. "Fractions" with a denominator of 1, do not count, in a certain sense.

Oh, and by the way, I slipped today and fractured my leg. Luckily, it was fractured in zero places so it is not actually broken. Each bone manifests itself in one whole, healthy fragment!

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Any integer is a rational number being that the definition of a rational number is any number that can be expressed as the ratio of two integers. It is enough to have the property without having to explicitly express it as such. As far as the semantics go I would disagree and say that any integer is not a fraction. Any integer can be written as a fraction but until you write it in that format its an integer. Just like you do not say 10/5 is an integer but call it a fraction.

Also, I see a lot of people mentioning that all fractions are rational numbers which is not true. sqrt(2)/2, while a fraction, is not rational

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  • $\begingroup$ Out of curiosity, what do you feel that your answer adds to this post that (for instance) Zev's, Carl's, and Todd's do not? $\endgroup$ – Cameron Buie Sep 16 '15 at 4:33

protected by Cameron Buie Sep 16 '15 at 4:33

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