# Why don’t American school textbooks recognize negative numbers as whole numbers?

Looking up for definition for whole numbers on Google yields a result which mentions:

The whole numbers are also called the positive integers (or the nonnegative integers, if zero is included).

1. According to American middle and high school textbooks, the set of whole numbers includes all positive integers and 0, and not anything else. In this context negative numbers cannot be whole numbers. I have no clue as to the origin of this poor usage.

2. Professional mathematicians (researchers) tend to use the term whole numbers somewhat more informally as a synonym for integers, with “whole” meaning without fractional part. Certainly −1, −2, … have no fractional part so they are whole. This is consistent with German terminology where the formal name for integers is “ganze Zahlen”, which literally means whole numbers, as well as French terminology “nombre entier” (often simply “entier”), with entier being a cognate to the English “entire” in the sense of whole. The best way to avoid ambiguity is to be explicit about which integers you are referring to by applying the appropriate adjective to “integer”. “Integer” by itself includes positive, 0, and negative; positive integers means 1 and up; non-negative integers means 0 and up; negative integers means −1 and down; non-positive integers means 0 and down. Do not use the terms natural numbers and whole numbers.

However, be careful with the concepts of positive and negative (“positif” and “négatif”, respectively), as the French think of 0 as both positive and negative, whereas English, German, and many other languages regard 0 as neither positive nor negative.

Does anyone know why American textbooks decided to go this way?

I ask because I’m using American textbooks to study maths and now I’m more curious about incorrect things in them. Yet, this is the first thing I found.

• I'm accustomed to using "integer" to include positive, negative and zero, while "natural number" includes only the non-negative integers. I rarely use "whole number" in my professional work, but when talking to children, I'd use it to mean "positive integer". As far as I can tell, most other professional mathematicians agree with my usage, except that some would not include 0 among the natural numbers. Oct 4 '20 at 22:33
• I agree with @AndreasBlass, and at least partly disagree with the Quora answer. This recommendation, "Do not use the terms natural numbers and whole numbers" seems like bad advice. Not being able to recognize $\mathbb N$ would be pretty bad. Oct 5 '20 at 0:33
• Welcome to Math Educators SE. I grew up in Ontario, Canada, where I was taught (note this is about 50 years ago, so it may be different now) similar to what's taught in the U.S., i.e., the natural numbers are the positive integers and whole numbers are the non-negative integers, i.e., 0 and the natural numbers. Nonetheless, I have only relatively rarely encountered (even in my school system), and I don't use it myself (I use non-negative integers instead), the term "whole numbers". As for why the Canadian & American school system defines whole numbers as they do, I don't have any idea. Oct 5 '20 at 1:26
• @Daniel R. Collins: Regarding the contract work I mentioned here, one of the things I've been editing out of math items (several graduate school admissions tests, not all given in the U.S.) is the use of "natural number" and "whole number", replacing with "positive integer" or "nonnegative integer" or "integers greater than 0" etc., because the meaning of natural number varies (continued) Oct 5 '20 at 7:53
• and whole number is rarely used as a precise designation outside of school mathematics. But in everyday language (when it's not absolutely essential to know whether, for instance, $0$ is included or excluded) and in situations in a class where other context is present or the meaning is understood, there is, of course, much less concern with saying natural number or whole number. Oct 5 '20 at 7:58

"The whole numbers" is not a term that professional mathematicians use to describe a certain set of numbers. The term is used in elementary education when fractions are introduced, so that one can distinguish between numbers that have a fractional part and numbers that don't. In the US, this happens in grades 3 or 4. As far as I can tell from the Common Core standards, negative numbers aren't introduced until around grade 6. So at the time when the term is introduced to kids, there is no question to address as to whether -1 is a "whole number," because those kids don't know that there is any number called -1.

• In German (and probably other languages too) mathematicians do say "Ganze Zahlen" which literally translates to "whole numbers" to denote integers. Oct 5 '20 at 10:01
• Sorry, but my 4 year old daughter uses -1 in her everyday life,as we have floor -1 in our house. It's sitting right there, right below level 0. There is nothing extraordinary in concept of negative numbers for children nowadays. Oct 5 '20 at 16:59
• @AskarKalykov As did my children, but they ran into 2nd-grade teachers (USA) who growled right back " you can use negative numbers at home, but not in my classroom". I'm still sad. Oct 6 '20 at 16:08
• @AskarKalykov: Whether your daughter knows something before school teaches it to her is irrelevant when discussing curriculum design decisions. This answer doesn't claim that no child could comprehend negative numbers before grade 6. This answer states that the school curriculum doesn't teach negative numbers before grade 6, and that it wouldn't make sense for a grade 3-4 curriculum to depend on something that up until then hasn't been taught to those children yet. It's possible they know about them, but the curriculum cannot assume/guarantee that they do. Oct 7 '20 at 10:41
• Early 30s as well (before Common Core), and I do remember when we were officially taught: Fractions at my school were 6th grade (elementary school), negative numbers 7th grade (middle school). But I too did understand negatives well before that. Oct 7 '20 at 15:47

I’m more curious about incorrect things in them. Yet, this is the first thing I found.

As Dave L Renfro noted in a comment:

and whole number is rarely used as a precise designation outside of school mathematics

there is no agreed-upon rigorous definition of the term, and in fact it's largely viewed by professional mathematicians as a matter of philosophy of mathematics and of one's particular school/ideology whether you consider "whole numbers" to include negatives or even zero. Mathematicians writing rigorously and with a concern for not raising annoying pedantic rehashing of this topic will tend to write things out explicitly as "positive whole numbers", "non-negative whole numbers", etc. (albeit typically using the more rigorous term integers, which is always understood to include negative numbers and zero). But more importantly, they'll take care to define the terms they use in the context they're using them in.

To me, the important takeaway for mathematical education is the idea that definitions vary by contexts and cultures, and that the underlying mathematical idea is independent of how a particular writer or teacher chooses to present it. Teachers who fail to acknowledge this and who insist (especially to young children) that certain definitions are "wrong" are doing a huge disservice to their students' appreciation of mathematics - it undermines the universality entirely and teaches them that this is just another subject where the teacher pompously thinks they're always right.

I don't think that "textbooks" decided this, usage did. The term "integer" covers positive and negative, so it would be redundant for whole numbers to refer to that category. And there is an argument to be made for the term linguistically: a negative number is sort of the opposite of having a whole thing.

But ultimately, there's not much to be said for "why" questions when it comes to words. You imply that this is an "incorrect" thing, suggesting that you have a Platonic view of words, that there is some objectively "correct" word. But words are just conventions. It is a convention that negative numbers are not whole numbers, therefore negative numbers are not whole numbers.

• Suggesting you cannot have a negative "whole" seems to go against the semantic meaning of the word. I have 3 apples, I give 1 away, now I have 1 less apple than I had before (-1 whole apple)
– DBS
Oct 5 '20 at 9:14
• "integer" is Latin for "whole number" ;-). Perhaps surprisingly, two words can refer to the same thing in natural languages. English specifically has a host of Romantic synonyms for Celtic/Germanic ones. You are eating cow when you are eating beef. Oct 5 '20 at 9:46
• @Peter-ReinstateMonica Yeah, but not an integer cow, just a fraction of one. ;) Oct 5 '20 at 14:30
• @DBS you had 3, then you had 2. You never had -1. Oct 6 '20 at 23:15
• @hobbs Not from a physical perspective (obviously negative "things" don't exist), but from a mathematical description of the events, the minus is the act of losing the apple. But as the answers are pointing out, the term "whole" seems to be used differently per country, so I guess there's no point arguing about it, haha.
– DBS
Oct 7 '20 at 8:54

You seem to describe "whole numbers" in this American usage as describing $$\mathbb {N}$$, the set of natural numbers, whereas you expected it to describe $$\mathbb{Z}$$, the set of integers. As others have pointed out there is nothing "incorrect" about them, it's a language difference. Although it is worth knowing those language differences since especially in maths (or "math" is it is known in either the US or the UK can't remember or be bothered to check) it can really make trouble!

Your quoted Quora answer brings French into it, for a French speaker it might be all the more confusing that "whole numbers" and "integers" both clearly derive from the word "entier", which means "whole" and can be used for either set ("entiers relatifs" would be $$\mathbb{Z}$$ and "entiers naturels" $$\mathbb {N}$$). If that is your situation I can understand your confusion all the more, but it is still merely a question of language convention, not being correct or not. It might also suggest a reason they started using "whole numbers" to describe $$\mathbb{N}$$: since they used "integer" to describe $$\mathbb{Z}$$, that freed up "whole number" to describe a different set.

You might be interested in an English dictionary of mathematics, that might clarify a lot of those nuances for you. There are some differences between American and UK English that are positively awful, like there are two geometric features that have exactly inverted names between the two. I cannot remember the exact example, I thought it was in the parallelogram family but I might be confusing with the trapezoid/trapezium difference, which is also a good example of inconsistent mathematical words between US English and others but doesn't seem to be a total inversion at least.

• The phrase "natural numbers" is five syllables. "Whole numbers" and "Integers" are both three. In abstract algebra, an element of a ring can be raised to a zero power (yielding the multiplicative identity) or a positive integer power, but not a negative integer power. Having a concise term to describe the allowable range of powers is useful, and "whole numbers" is more concise than "natural numbers", "non-negative integers", or other alternatives. Oct 6 '20 at 18:47
• Trapezoid/Trapezium it is, yes - mathwithbaddrawings.com/2015/05/20/… Opposite meanings in the US/UK.
– Joe
Oct 7 '20 at 19:33

In elementary school in a Dutch-speaking country, we were taught the concepts of $$\mathbb {N}$$, the set of "natuurlijke getallen" ("natural numbers") and $$\mathbb {Z}$$, the set of "gehele getallen" ("whole numbers"). At no point was there ever a hint that these terms were not unequivocal.

I remember $$\mathbb {Z}$$ and $$\mathbb {N}$$ then being used as examples to teach Venn diagrams, at which point it was emphasized again that the difference between the two sets was the negative numbers.

• I was never taught those concepts in elementary school in the Netherlands. I'm pretty sure I didn't even hear the term "natural numbers" until university. We were taught about "numbers", which at first just included non-zero integers, then gradually expanded to include negative numbers and fractions. Is that also what you mean, perhaps? Oct 7 '20 at 5:55
• @Buurman Why would I write one thing if I mean another?
– user14737
Oct 9 '20 at 16:44

I don't remember using the term "whole numbers" in High School. It's not a rigorously-defined mathematical term, but the idea of "whole [counting] numbers" as opposed to "fractions" is quite useful when you're young and the concepts are new.

Older students would be learning the proper mathematical terms (natural numbers, integers, rationals, etc) and so expanding the definition of "whole numbers" to mean "integers" doesn't really buy you anything with most older students. In fact it would be lose-lose: you'd be changing the meaning of a word they've known for many years to mean something else, and you'd be encouraging them to use that word instead of the proper term that they will encounter if they pursue mathematics.

I think a key dividing line is practical/physical math and more theoretical math. I can physically hold 3 glasses, or 2/3 of a pie. Having -3.14159 pies is more abstract/theoretical. At some point -- probably in High School -- students will come to the place that they will either continue into more abstract topics (algebra, calculus, etc) which will require more precise definitions, or more practical math (what used to be called "home economics": shopping, cooking, buying, etc) which to a large extent need only "whole numbers" and "fractions".

The natural numbers are the counting numbers 1,2,3,.... Add zero and then negative numbers and you have the integers. Then come the rationals, reals, complex, quaternions, and octonions. Then you have run out of numbers. There is NO whole number in those.

So the answer is because there are nothing called 'whole' numbers in the math I studied at the uni. Thus your textbook could define something new called whole numbers. And apparently they defined it in a way you do not like.

• Interesting. Today I learned what quaternions and octonions are. Oct 7 '20 at 10:22