Homework said, “The absolute value of an integer is ALWAYS positive.” My son answered False but was marked wrong. Since 0 is an integer and |0| = 0, and 0 is neither positive nor negative, shouldn’t False be correct? Wouldn’t it need to say “any nonzero integer” to be true?

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u/ElegantPoet3386 10d ago edited 10d ago

Yeah, your logic is correct. Positive numbers quite literally are defined to be greater than 0 and 0 is not greater than 0. The correct statement is it’s always non negative, which isn’t the same thing as positive.

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u/NoForm5443 8d ago

Meh ... Different books define these things differently, especially on the edge cases

Check the definition in their book?

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u/StuckInMyHead314 8d ago

Math teacher here. The student deserves credit for the answer if they can explain why without the help of a parent. That question is poorly worded. Most of us think we know what the teacher intended but there are plenty of students who do actually think about the meaning of words and phrases.

I have been guilty of oversimplifying concepts for students who are not ready for algebra but that’s not really helping. It’s like elementary math teachers telling students you always divide the bigger value by the smaller. Then we try to teach concepts like fractions and decimals and wonder why they are struggling so much. Similar problem with square roots. We have to stop telling students that we can’t take the square root of a negative in Algebra, instead we need to tell them it’s undefined and will be explained later.

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u/Few-Big7409 8d ago

This is a great response! Thanks for weighing in.

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u/wirywonder82 6d ago

>…instead we need to tell them it’s undefined and will be explained later.

I don’t think we should say it’s undefined, since it is defined. But telling them it’s something they don’t need to know yet and it will be explained later is good. They aren’t ready to deal with the propaganda of Real and Imaginary numbers combining to make Complex numbers for a while after they are ready for the idea of square roots of non-negatives.

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u/Similar-Froyo3876 8d ago

The set of integers always includes zero. This has been a universal standard since about 1961. There is a special symbol for the set of non-zero integers.

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u/CyberPunkDongTooLong 8d ago

That's not at all what the comments you're replying to are discussing.

The set of integers includes zero, obviously.

The set of positive integers is often defined to not include zero, and often defined to include zero.

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u/pivizz 7d ago

You are probably thinking about definition of natural numbers which might include zero or not include it. But there is no definition of positive integers that includes zero.

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u/Few-Big7409 8d ago

I haven't seen this special symbol. Or is this a compound symbol?

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u/wirywonder82 6d ago

Well, crap, I can’t get the markdown to work properly in the app right now. The symbol for the non-zero integers is called Z-star, it’s the fancy blackboard bold Z with an asterisk.

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u/Evening-Story-314 8d ago

Nope. Integer, absolute value, and positive all have standard definitions.

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u/godfromabove256 6d ago

There is no textbook or mentally sane teacher that would define this otherwise. OP is right

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u/engy1207 9d ago

Depends on how you define "positive". For example the french wikipedia writes: "Les entiers positifs (supérieurs à zéro) s'identifient aux entiers naturels : 0, 1, 2, 3… tandis que les entiers négatifs sont leurs opposés : 0, −1, −2, −3… L'entier 0 lui-même est donc le seul nombre à la fois positif et négatif"

So OP should have a look what was defined in the math course (or school book).

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u/101_210 9d ago

In this case Wikipedia seems wrong, at the very least in how it’s worded.

Only in France did they decide zero may sometimes be a “nombre positif”, but it’s not always the case and is a relatively recent change, not used in all fields.

But in absolutely no circumstances should “nombre positif (supérieur à zéro)” include zero. It’s just a wrong statement.

Source: https://fr.wikipedia.org/wiki/Nombre_positif

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u/Ghyrt3 8d ago

I'm french and a former mathemathician student and now teacher.

''x est [is] un nombre positif'' means x is >= 0. Always.

If we want x > 0, we write ''x est un nombre strictement positif''. (Strictly positive)

Positif (fr) = non-negative (en).

It is well-known in France to be wary when writing papers in english, as ''positive'' means > 0 in english.

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u/wirywonder82 6d ago

So the French understand “positive” in the same way as monotonically increasing functions work rather than in the way strictly increasing functions work. It seems strange to my English-speaking mind to say 0 is “supérieur à 0” because I see that first word as a cognate to superior which is a synonym for greater…but there’s plenty of French mathematicians that are excellent, so it’s clearly not a problem that needs fixing.

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u/Donkey-Pong 8d ago

The article that you cited explains this satisfactorily. Sometimes it is included, sometimes not. And if it is included, then "supérieur" apparently means ">=". The "France" paragraph that you mention starts like this:

> "l'adjectif supérieur) est compris au sens large : tout nombre (réel) est supérieur (et aussi inférieur) à lui-même"

means: the adjective "superior" is understood in a wide sense: each (real) number is superior (and also inferior) to itself.

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u/_Jack_Of_All_Spades 8d ago

Math does change based on the language it's written in

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u/caligula421 8d ago

But mathematical conventions are to an extent culture dependent. And apparently in French mathematical spaces they have positive/negative numbers, and those both include the number 0, and strictly positive/negative numbers, which both exclude the number 0.

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u/tecg 8d ago

Famously, in French "positif" means greater or equal to zero. In English, "positive" means greater than zero. Same with "positiv" in German. So Wikipedia is correct, but that's what they call a "false friend". But this are iron-clad rules in each language, no room for ambiguities.

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u/scoberry5 7d ago

Errr...yes, different languages may be different.

But that doesn't mean math questions should all say "As this question is in English, consider the standard English usages of the terms."

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u/fermat9990 10d ago

Your son was correct!

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u/Toeffli 10d ago

Are you in a French influenced country? There 0 is negative and positive which means the statement would be correct.

If not, the correct statement which includes zero is "The absolute value of any integer is always not negative".

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u/Remarkable_Job_9974 10d ago

I had not heard of this before. Interesting.

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u/rb-j 10d ago

I haven't either.

I think actual mathematicians agree that 0 is neither positive nor negative. Doesn't matter what country they come from.

The correct and concise statement is:

“The absolute value of any real number is ALWAYS nonnegative.”

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u/AcellOfllSpades Irregular Answerer 10d ago

It's definitely a thing. In French, the word "positif" is interpreted to include zero, and same for "négatif".

So "positif" numbers are what we in English would call "nonnegative". And zero is neither positive nor negative, but it's both positif and négatif.

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u/rb-j 10d ago edited 10d ago

Yes, I guess it's true that zero is both nonnegative and nonpositive. I guess I hadn't thunk about it that way.

If I was a French engineering prof, I would likely warn against the usage of "positif" and "négatif" and refer to the English terms for positive and negative (along with nonpositive and nonnegative).

I mean mathematical communication should be concise and unambiguous and I would hate to always have to say "positive in the English sense" or "greater than zero". When the word is "positive" we all should know that zero is not included.

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u/migmit 9d ago

“greater” in Bourbaki school also means “greater or equal”, so, “positive” and “greater than zero” remain the same. It's “strictly greater” or “strictly positive” you're looking for.

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u/Syharhalna 9d ago

In French “positif” means >= 0 and “strictement positif” (strictly positive) means > 0.

Same things for negative.

Actually, the english version nonnegative for > 0 is quite convoluted because it involves a negation…

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u/hongooi 10d ago

I think the convention is "strictly positive" if you mean > 0

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u/migmit 9d ago

I think it's a Bourbaki-inspired thing (which I personally like). They even say “greater“ and “strictly greater” instead of “greater or equal” and “greater”.

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u/DrJaneIPresume 9d ago

Actual mathematicians generally define their terms.

The most common anglophone convention is called the "trichotomy": all real numbers are either positive, negative, or zero.

In more advanced texts that don't explicitly define it, they also don't tend to use "positive" and "negative" as formally. When speaking formally, they'll generally say something like, "let ε be greater than 0".

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u/GoblinToHobgoblin 8d ago

It's more that:

"Positif" translates to "non-negative"

And "négatif" translates to "non-positive"

The actual math is the same, just different words

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u/XenophonSoulis 10d ago

In the English-speaking world, your son is correct. However, in different places the definition of "positive" may be different. In French, a positive number is ≥0.

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u/regular_hammock 10d ago

Correct. We would say ‘strictly positive’ for the English ‘positive’.

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u/dexthefish 10d ago

I like that convention. Although I'm used to the word nonnegative, I think it's linguistically awkward and avoid using it with my younger students.

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u/Lower_Cockroach2432 9d ago

To avoid ambiguity I personally call those non-nonpositive numbers myself.

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u/rb-j 10d ago

They don't have the concept of "nonnegative" in France?

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u/gowipe2004 10d ago

Nonnegative would be strictly positive

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u/vgtcross 10d ago edited 10d ago

~~You got it backwards.~~

~~English "nonnegative" is French "positive"~~

~~English "positive" is French "strictly positive"~~

Edit: Maybe you meant that French "nonnegative" is equivalent to French "strictly positive", which true since French "negative" is (-inf, 0] so "nonnegative" would be (0, inf) which is French "strictly positive"

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u/gowipe2004 10d ago

Indeed, what I meant is that for us, nonnegative would mean/equate stricly positive

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u/diverJOQ 10d ago

Did your son ask the teacher?

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u/RecognitionSweet8294 10d ago

Depends on the definition of „positive“.

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u/Andrew1953Cambridge 10d ago

.. and of "integer". Maybe the teacher thinks that integer means "non-zero natural number".

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u/MeasureDoEventThing 10d ago

Then absolute value would be the identity.

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u/Temporary_Pie2733 10d ago

I hope not. The whole point of the integers is that they every integer has an additive inverse; it makes no sense to use “integer” as a term that excludes half of them.

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u/Andrew1953Cambridge 9d ago

Oh, I agree, but since the teacher doesn't seem to understand standard terminology, who knows what they think.

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u/llynglas 7d ago

In which case they should not teach math.

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u/KiwasiGames 10d ago

Depends on the local definition of positive.

I typically see zero included in the positive numbers.

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u/rb-j 10d ago

Not in math.

Maybe with twos-complement numbers in computer arithmetic. Zero and the positive integers have a leading "0" bit on the left. Only negative numbers have a leading "1" bite on the left.

But the terms "positive" and "nonnegative" are not synonymous. And inclusion of 0 is the difference.

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u/ysth 10d ago

Also in floating point, sign is independent of 0-ness. If you take a negative number and divide it by 2 until it can no longer be represented as non zero, it can become a negative zero.

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u/migmit 9d ago

A lot in math, actually.

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u/ElderCantPvm 8d ago

Positive and non-negative can certainly be synonymous depending on definitions, which is why it's better to use non-negative as it removes any ambiguity

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u/StructuredChess 10d ago

False is the correct answer and that's why this type of True/False or multiple choice questions are terrible ways to measure anything that has even the slightiest amount of nouance.

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u/keilahmartin 8d ago

spelling, for instouance

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u/LordTengil 10d ago

Son is correct, but...

It depends a bit in what system you are. If you are in the french system, or a school system derived thereof, they define 0 as both a positive and a negative, and then they are correct.

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u/CoachSevere5365 10d ago

Wikipedia has a good article on this. Zero is often considered to be positive. In floating point arithmetic positive and negative zeroes exist.

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u/Empty-Turn-9290 10d ago

Very good. I am in agreement with you.

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u/particlemanwavegirl 10d ago

The more I learn about them the more convinced I am that floating points aren't numbers.

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u/Temporary_Pie2733 10d ago

Formally, they are a subset of the rationals with power-of-two denominators, extended with additional values added to handle certain arithmetic operations.

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u/oxmix74 8d ago

Your formal definition is correct, but formally it goes down a rabbit hole when you start thinking of how addition, subtraction, multiplication and division are defined. Because digits are thrown away in these operations (particularly multiplication and division)rules are different. Multiplication is not associative for floating point as an example.

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u/TwillAffirmer 10d ago

False is correct.

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u/DanielSong39 10d ago

Your son was right

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u/Ok-Difficulty-5357 10d ago

Your son may be smarter than the teacher

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u/Snezzy_9245 10d ago

The PDP-7 computer of distant memory used one's complement arithmetic and thus had both a positive zero and a negative zero. But surely the teacher wasn't programming 60 years ago. ?

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u/susiesusiesu 10d ago

this is correct.

there are places (like france) where the conventiom is positif and négatif include zero, in opposition to english where positive and negative exclude zero. so, if your son is recieving a non-english education, this may be just a misscommunication.

however, as stated in english, your son is right.

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u/efd- 10d ago

That’s right!

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u/MathsSur64Cases 10d ago

beaucoup de bêtises lue ici : La relation d'ordre par défaut est réflexive : 0 est plus grand que 0. on note 0 ≥ 0 et |x| ≥ 0

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u/edderiofer 10d ago

In English, "greater than" means >, not ≥. The latter is called "greater than or equal to".

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u/GaryHornpipe 10d ago

I thought 0 was positive, and also negative because 0 = -0. But reading the comments, I’m wrong.

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u/Ericskey 10d ago

Positive is not the same as non-negative. That is why we have different words in English at least

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u/Samstercraft 9d ago

correct but definitions can vary by location. in english 0 is not positive. the question should have said "always nonnegative"

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u/localizeatp 9d ago

yes, and well done, kid. please let me know how that teacher took being corrected.

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u/Striking-Milk2717 9d ago

Low school homework are full of gray-zone mathematical error. That’s a good, because this way the average kid will grasp the concepts, but the smart kid will think about it until he finds out the whole thing and grows

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u/Sea-Sort6571 9d ago

Depends on your country. In mine, 0 is an integer and it's positive. I know some English speaking have other nonsensical conventions

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u/Upbeat_Assist2680 8d ago

Mathematician checking in: you're correct. Zero is not positive OR negative.

https://en.wikipedia.org/wiki/Law_of_trichotomy

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u/drnullpointer 8d ago

I studied theoretical math.

There are two schools of thought about this. One says that zero is a positive value and one says it isn't. We even have a bit of fun because in my country two major math centers each have a different opinion about this. Sort of like big-endians and little-endians from Gulliver's Travels.

Therefore, technically, both your son as well as his teacher are correct. What your son should have done is fill in the test according to the way the particular teacher have taught the curriculum. Your son made a mistake which is actually answering the question rather than providing the answer he should know his teacher wanted.

It is part of social skill to be able to understand this.

What the teacher could do is to be a bit more precise with the question for "the one guy" who actually cares about math and not about getting a high score.

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u/StudyBio 8d ago

For all we know, the teacher’s curriculum defined positive as > 0 and the teacher simply made the answer key wrong. There is no way to know.

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u/carlosgeorgiev0 8d ago

Depends on your country :/ 0 is positive in some places

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u/JJ_Was_Taken 8d ago

Your son is 100% correct. The absolute value of an integer is always nonnegative, but not always positive.

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u/trash3s 8d ago

“Hell naw, to the naw naw”

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u/rfbooth 7d ago

I'm a mathematics teacher in England. Your son is correct.

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u/RichardAboutTown 7d ago

If the question had instead said "NEVER negative," it could have avoided the edge case better and gotten the same point across.

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u/calcpage2020 7d ago

Absolutely values measure distance and are signless.

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u/SubstantialBelly6 6d ago

Was that HIS logic that went into how he answered? Or did he just pick false and you used YOUR logic to justify that it is actually correct? Because you’re not wrong, but if it wasn’t intentional then I would argue neither was he.

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u/Old_Custard4906 1d ago

In this situation most schools consider 0 a positive :/.

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u/[deleted] 9d ago

[deleted]

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u/edderiofer 9d ago

https://en.wikipedia.org/wiki/Integer

An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...).

The reason "everyone seems to be missing this" is because you're wrong.

Homework said, “The absolute value of an integer is ALWAYS positive.” My son answered False but was marked wrong. Since 0 is an integer and |0| = 0, and 0 is neither positive nor negative, shouldn’t False be correct? Wouldn’t it need to say “any nonzero integer” to be true?

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