There are a lot of misconceptions around negative numbers (see this post) and I sometimes wonder if this is down to confusion caused by the language we use when describing a number that is below zero. I know some maths teachers interchange between calling something negative or minus depending on the situation or the mood they are in. Outside of the maths classroom, the word “minus” is used frequently and almost always when describing temperature. Is it correct though?

I put a poll on Twitter to try and find out what other maths teachers think and the results were surprising for me. More than half of the teachers, it seems, have a different view to me..

Maths teachers: How do you say the number “-5”?

— Mo Ladak (@MathedUp) January 24, 2017

I know that I can be a bit pedantic but I argue that we should always use the word “negative” when we are describing a number below zero. For me it is a case of opposites. When talking about operations, the opposite of “add” is “subtract”. When talking about the operators themselves, then the opposite of “plus” is “minus”. So when talking about the property of a number, then the opposite of “positive” has to be “negative”.

So why does it matter? I find that there are many barriers to understanding the fundamentals of mathematics that stem from the language we use. An example is the fact that we say “half”, “third” and “quarter” rather that “twoth”, “threeth” and “fourth” when talking about fractions (all the other unit fractions seem to follow a more intuitive logic). I’m not sure if other languages have similar barriers so would love for anyone to let me know..

Anyway, when it comes to negative numbers, I have found that even some of the best mathematicians I’ve taught get themselves into a pickle with unhelpful phrases such as “two minuses make a plus” and struggle with the misconceptions this brings. It is much more useful, in my experience, to be able to separate the language and always use the words “positive” and “negative” to highlight a property that belongs to a number and add/plus or subtract/minus as an operation that we apply to a number. Being clear and consistent with the language removes this barrier and allows students to concentrate on getting to grips with the conceptual understanding.

Do you agree/disagree? Have you been convinced? Please comment below to let me know..

## jemmaths

January 26, 2017 at 6:33pmI like to interchange them, so that my students become familiar and happy with both words, simply because they will encounter both throughout their lives. Having said that, I do explain that I will use both, and why. I sometimes say things like “negative, or minus, 5” to build familiarity with them as synonyms. I also expose them to negatives written with and without brackets around, so that they don’t get thrown if they see them presented in the way they haven’t been taught.

## MathedUp

January 26, 2017 at 8:57pmI completely agree that they need to have an awareness of both. I guess my point is that I want them to have clarity in their thinking when calculating with them and I would prefer their internal voice to say “negative” rather than “minus”.. I like the fact that you expose them to multiple written forms – I don’t think I do that enough!

## jemmaths

January 26, 2017 at 9:47pmI can totally see the arguments for choosing one way and sticking with it for lots of areas of maths. This is one where I choose to go both ways!

## Chris Hawkins

January 26, 2017 at 11:47pmI would posit that ‘plus’ is used as a descriptive word, particularly in relation to temperature (and particularly in relation to low positive temperatures), possibly because ‘minus’ is used as a descriptive, and therefore ‘plus’ is the logical opposite. For example “last night it was minus two degrees, but today it rose to plus three degrees”.

## MathedUp

January 27, 2017 at 12:00amI had never thought of that before..

## Jenny B

January 27, 2017 at 8:19pmI’m trying hard with my negatives 🙂

## MathedUp

January 28, 2017 at 11:01pmI know you are!!

## phil

January 31, 2017 at 12:56pmI like the word take for subtract – quick to say and obvious inverse of add

## MathedUp

February 20, 2017 at 8:35pmI like it too!

## Nick Edwards

February 7, 2017 at 1:14amI am quite adamant about the use of ‘negative’ over ‘minus’ if describing a number less than zero. ‘Negative’ is an adjective – describing the property of the number. ‘Minus’ is a verb ‘to take something away’. So if I see the following in a lesson 4 – -3 I would always teach my students to read it as ‘4 minus negative 3′ (though to be honest, I actually very rarely use minus at all – I’d normally say 4 subtract negative 3’.

## MathedUp

February 20, 2017 at 8:37pmI agree. I’m not very consistent with the word I use for “subtract”..

## teachingbattleground

February 15, 2017 at 12:08pm“Negative five” is unambiguous. “minus five” could mean “negative five” or it could mean “subtract five”.

## MathedUp

February 20, 2017 at 8:44pmAbsolutely. I recently realised there is an exception in my language.. For example, when solving x^2 = 16

I would say the solution is “plus or minus four”..

## Matt Rosenberg

March 20, 2017 at 12:52pmI find it’s best to mix them up, not only for real life resilience but also because they are essentially the same thing. 3x*-2 = -6x “Three x multiplied by negative two equals negative 6x” but 3x(y – 2) = 3xy – 6x, here you need to do the same calculation when you expand the bracket but you are much more likely to read the second term as “minus six x” or “subtract six x” because of the first term.

## arkalot

April 2, 2017 at 2:10pmThis is all fine but there is a problem, which is why I interchange minus/negative all over the place

Consider: 20 – 10 + 4

So you say: twenty minus ten plus four

no problem you think

now use BIDMAS/BODMAS

the tendency is to do 10 + 4 = 14 then 20 – 14 = 6 but this is not the correct answer!

You need to do -10 + 4

So is it negative 10 or minus 10? (thats why I tell kids to work L – R for +/- only questions)

## MathedUp

April 10, 2017 at 8:06pmA better solution would be to not use BIDMAS 🙂

## arkalot

April 27, 2017 at 11:47amMaybe – but still may arise as part of a BIDMAS question 🙂

2 x 5 – 20/2 + 2 x 2

## Oliver Gray

May 15, 2017 at 3:08pmI think it’s a really important distinction to make, and try to emphasise it by writing the negative sign as a superscript proceding the digit, wheras I write the minus sign in the usual place. I think we as mathematicians ought to separate our objects from our processes, our nouns from our verbs.

## MathedUp

May 15, 2017 at 5:29pmAgree!

## Tom

January 5, 2018 at 3:05pmHow about the following. Would you consider there is a mathematical difference between the following two statements?

(The x represents multiplication)

-3 x 5

And

0 – 3 x 5

The first is “negative three times five”.

The second is “zero minus three times five”.

I reckon that for all intents and purposes, both statements are the same thing, and it doesn’t matter whether the “-“ sign is read as:

(1) A property of the negative 3, or

(2) An operator on the positive 3

Happy to hear an alternate opinion, but would also like to hear whether you think that for school students of age 12 to 13 (or so), why the above would not be a reasonable explanation.

I reckon the above could help students visualize repeated subtraction, when multiplying by a negative number, as in the following.

First, Repeated Addition:

3 x 5 =

0 + 3 x 5 =

Zero PLUS 3 lots of 5 =

0 + 5 + 5 + 5 = 15.

Repeated Subtraction:

-3 x 5 =

0 – 3 x 5 =

0 MINUS 3 lots of 5 =

0 – 5 – 5 – 5 = -15

## MathedUp

January 9, 2018 at 11:45pmHi Tom. Thanks for commenting. I understand where you are coming from but I have a counter-argument. How about if it was “five multiplied by negative three” ie.

5 x -3

If you write that in the way you suggested, this would lead to

5 x 0 – 3

We could argue that we now need brackets around the “0 – 3” however would this lead to further confusion? Would be interested in your thoughts..

## Mohammad

January 28, 2018 at 9:20pm-3 here “-” is a unary operator whereas in 4-3, “-” is a binary operator. In the former when you say minus three you refer to the process by which negative three is obtained but when you say negative three you refer to the object itself. There is no way to denote negative three without the minus operator, either in unary or binary usage. Although one might say the unary one could be seen as zero minus three hence being a binary. Nonetheless, “-” is an operator and -3 is not a monolithic object but rather a process through which the object of negative three can be applied in arithmetic and elsewhere. Assuming what I said is true, -3 can be read minus three or negative three, like reading 2^(0.5) as square (second) root of 2 (object) or two to (the power) half (process).

I should mention that I am not a mathematician and English is not my mothe tongue. I have a degree in Physics and Two masters in Engineering. So my comment is my lay understanding of the matter and may not live up to the standard rigour of the Maths folks!

## Tom

October 15, 2020 at 1:30pmHi MathedUp.

WIth 5 x -3, if thought of as “5 lots of negative 3”, this is tangible as it is.

With -3 x 5, if thought of as “negative 3 lots of 5”, this is hard to vizualize, as in what does “negative 3 lots of” mean in concrete terms?

The only explanation I can find online that appears reasonable as a concrete model, is to treat the negative sign of the first number as subtraction, ie:

(-3) x 5 = subtract 3 lots of 5.

If there was a way to model (-3) x 5 in a concrete way, and leave the negative sign as a property of the first number, I would love to hear it. I know we can use the commutative law, and change the order of the numbers, to 5 x (-3). This is also a proper explanation, but is not a model of (-3) x 5.

Thanks