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

Question

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).

I was suspicious about this answer and I decided to dig more into this. I found a Quora answer which mentions:

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.

Answer 1

"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.

Answer 2

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

There's absolutely nothing "incorrect" about this.

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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".

Answer 7

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.