This post was requested by frequent commenter, Terry Mills, famous for his divisibility test for 53. Simply put, Terry asks the question in the title:
Is a percentage a number?
To give Terry’s question some context, we’ll include a few exercises from the Year 7 Cambridge text. To be clear, we haven’t chosen Cambridge because it is a very bad textbook; that is by the by. Cambridge‘s is just a good example of pretty standard treatment. If and when the conversation gets going, commenters can introduce better sources.
To begin, here is a typical percentages exercise:
It is a very difficult question, of course, and almost impossible without an accompanying picture of lemonade being poured. Thankfully, Cambridge included such a picture, which we have reproduced above.
Here is another (picture-free) exercise:
Finally, here are a few straight number exercises:
What do people think? Are these exercises kosher? If not, why not? I have my own, correct, opinion, of course. But I don’t want to come across as the CEO of Assholes R Us or anything, so I’ll be quiet for now.
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UPDATE 05/08/26
God, we need a Tony Gardiner or an Ed Barbeau. Instead, we have a Peter Sullivan and a Greg Ashman.
Thank you to everyone who has commented so far, and to Terry, especially, for suggesting the question-post. I’m intent on waging wars with nitwits and Terry keeps insisting that instead I provide something helpful for others. Some people.
I think there is a lot left to discuss and there is probably good purpose for a dedicated post to try to set all this stuff straight. The fundamentals of mathematics are hard, and it requires a strong and intelligent and empathetic but firm mathematician, such as Ed or Tony, to write it all clearly, to provide the proper semantic and syntactic structure for curriculum authorities and textbook writers and maths ed lecturers to follow.
To begin the substance, I will say outright that I strongly agree with Ed in his comment, and seemingly some other commenters, that I do not believe a percentage is a number, or anything. I do not believe that 88% means 88/100, quite independent of and prior to how we might calculate or represent that percentage. In fact, I do not believe that anyone here really believes that a percentage is a number. (Give up, Terry: I can pick a fight with anyone, anywhere.)
Along with Ed, I do not like Questions 1, 12, and 13 above, and I think they are wrong. I understand Q1 is very standard, and has been for a long time. As Craig points out in his comment, even David and Gareth’s Enrichment text, which I regard as an absolutely wonderful text, does the percentage is a number thing (Vol 1, p 49, and subsequent exercises).
How to convince yourself that a percentage is not a number? I think Q13 above does it. I can understand that people have gotten used to Q1 and its ilk. People may feel Q1 is not quite right, but obviously a percentage is somehow a fraction thingy and so it is understandable to ignore any small sense of unease. Q12, then, feels quite a bit less right, but maybe still not enough: “an interest rate of 11/200” is not sounding kosher but we’ve bought into this with Q1 so, whatever. But then, with Q13, I think the game is up.
I cannot imagine many people are genuinely comfortable with picturing 8% and 9% on a number line. Yes, if you have bought into percentages as numbers via Q1 then you may feel you have no choice but to accept Q13. But I cannot imagine anyone being comfortable with that choice. Percentages as fractions – and really it is as ratios – may be ok, but percentages as full blown real numbers on a number line just feels wrong. However, we got here, it feels the wrong place to be.
So, what are percentages if not numbers? As tom suggests, and others have at least hinted at, percentages are operators: you cannot have 87%, you can only have 87% of something. Without the “of”, the percentage is meaningless.
Tony Gardiner discusses this in his amazing book on the English curriculum at the time. The line from the English curriculum is
interpret fractions and percentages as operators
This is Tony’s comment on that line (and his entire discussion around here is really, really worth reading):
1.6.3 The final comment [the excerpt above] is slightly awkward. It has become common in England to require pupils to treat “50%” as if it were a number equal to 
. This is not only false, but thoroughly confusing (and shows that textbook authors, editors, and examiners have themselves failed to distinguish between numbers and operators). The number ” 
” sits midway between 0 and 1. In contrast “50%” on its own has no more meaning than the “
” in 
: it is an operator, and gives rise to a quantity or value only when it is given a “whole” (or an “
“) to act upon. “50% of” is another way of writing ” 
of”, which is in turn another way of writing ” 
of”. But this is an operator, and is not the same as the number . In particular, the arithmetic of fractions only applies to numbers: there is no similar way (at this level) of making sense of “adding and dividing operators”.
There it is. What has has happened is a flattening, where the subtle meaning of percentages and, similarly, ratios, has been flattened so that all there is is number. One can understand this flattening, the conscious or unconscious motivation to simplify and the subsequent evolution of ideas. There are short-term advantages to the flattening. But, whatever the short-term advantages of flattened language, the mathematical subtleties are still there and are not going anywhere, and in the long run it is much preferable to have the language, properly presented, with which to teach these subtleties.
Finally, how did David and Gareth get it wrong? They didn’t listen to Marty. I argued hard with David against his percentage exercises. But, he wanted the flattening. I understood, but I think he clearly made the wrong decision. People should really learn that Marty is always correct. It would save us all a lot of time.
************************************************************
UPDATE (19/08/26)
Commenter WellWellWell has noted the alternative viewpoint of Hung-Hsi Wu. In his exceptionally thorough and careful book Understanding Numbers in Elementary School Mathematics, Wu states forthrightly (p 323),
[A percent] is a number, end of discussion.
Notwithstanding Wu’s discussion-ending pronouncement, there is some subtlety in what Wu writes, and I quote him at length, below. But let’s first note the situation: Wu and Gardiner are/were two of the hardest working and most thoughtful mathematicians in the maths ed world, and they fundamentally disagree. One must greatly respect their authority, if not appeal to it. Which means we are stuck between a rock and another rock. Such is life. I guess we’ll all have to think more.
Now some more detail of Wu’s discussion of percent (Chapter 20). Wu begins with a bald definition:
A percent is by definition a complex fraction whose denominator is 100.
Here, a complex fraction is a quotient of fractions – that is, something of the form 
, with the numerator and denominator both fractions (possibly integers). Wu defines and discusses complex fractions in the previous chapter.
Wu then notes the familiar language, “N percent”, and notation, N%. Wu then emphasises the requirement of such a definition, in order to avoid the barely comprehensible mush of the familiar teaching of percentages. Then, after carefully working through many examples on the standard problems of percentages, Wu returns to the definition of percent and discusses the “operator” viewpoint (p 322):
We have so far interpreted the notation of “
” as the denominator of a complex fraction or fraction, e.g., 
” means 
. In ordinary language, however, “percent” is often used as an adjective, as in “I am getting 5 percent interest from my savings bank”. There are educators and mathematicians alike who would argue that 
” is not a number but an “action” or an “operator” (whatever they are), and should therefore be treated differently from ordinary fractions or complex fractions. Let us put such a statement in perspective. Every number, be it whole number or fraction, is often used as an “action” or an “operator”, e.g., “three apples”, or “a quarter pound of beef”. To say that a percent is a number therefore in no way contradicts its common usage as an”operator”. This book promotes the primacy of the concept of a number, because the belief is that if students have a firm grasp of what a number is, can explain all the arithmetic operations with numbers, and can compute with them fluently, then how we use numbers (as a noun or an adjective) will hardly be an issue. In particular, there is no cause for singling out “percent” to make believe that it is anything different from a complex fraction. It is a number, end of discussion.
I have thoughts on this, but i’ll leave it for now, for others to ponder.
Choose one point on the number line.
How many numbers live at that point?
One. How does that help?
It is going to take more than a few words to explain.
Well, make sure respect this blog’s word limit of
.
Interesting questions, thanks, Marty.
I am old enough to have grown up with textbook questions like this, and have taught textbook questions like this.
This means I have been indoctrinated in the theory and language of percentages, and how to translate example of percentages into fraction-speak, and vice versa.
Thanks to my indoctrination (or is this a matter of being fluently mathematical-systems bilingual?) I don’t see any difficulty with these textbook practice exercises.
I classify them as “exercises”: mathematics questions that are like problems for beginners, but just practice tasks for students who have learned how to do these.
Of course, thinking of such questions in terms of language, we have synonyms: answers that mean the same thing.
For example, in the first Cambridge question number 6, the answer to 6a could be 75/100 — or written in the two-line format, 75-over-100. And there are many synonyms, the “mathematically simplest” being 3/4, or the word-expression “three-quarters”.
Now IF you see 6a differently, or have more to say about the goodness or badness of these “translate between percentage-speak and fraction-speak questions and situations”, I will be interested to hear more.
Otherwise, my indoctrinated retired mathematics teacher brain doesn’t see a big issue.
Pedagogically, part of the larger point is also training students to grasp words such as “equivalent to”, and “simplify”, because being able to re-express an equation-in-x as a solution, “x = whatever”, or understanding that “simplify” has a conventional mathematical meaning that varies according to the context — these are big-picture, more nebulous skills about mathematical words that are, in mathematics-land, different in meaning and use from the way the words are used in everyday-land.
Faced with the question: “If x-squared plus 4x plus 4 = 0, find x”; the student who draws a ring around the first “x” and writes the label “here it is!”, (or draws a circle around any or all of the x’s , and adds the label “here they are!”), has missed the maths-speak point, the CONVENTION that is the intended target of the mathematics teacher.
Or I am missing the point.
Thanks, John. It’s not the difficulty of these exercises at issue (except for the lemonade one), it’s the meaning.
My 2 bob’s worth.
2% means 2 per hundred. So I suppose 2% of 300 means 2 out of each hundred, giving 6.
But if we come to 2% of 50 one could argue that there is no hundred so 2% is meaningless. But I feel that 2% means just a proportion, 2 per hundred is the same as 1 per 50. So this is still kosher, answer 1.
Now what about 2% of a half? Cut a half into one hundred bits and take 2 of them, answer 2 of 1/200, which is 1/100. So a percentage of a number is all above board.
Now that word “of”. We write 2/100 of
meaning cut 
into 100 bits and take 2 of them. So in this context 2% is interchangeable with the fraction 2/100. So a percentage is the same as a fraction? A fraction has come to mean a number on its own. But strangely, a percentage in common usage is always a percentage “of” something.
Except perhaps when we write 2% of 50%. That second percentage is not of something? well maybe we are thinking it is potentially of something??
My conclusion. There would be no harm in including percentages as numbers, but I will keep them as an operator only.
Thanks, Tom. *Would* you write something like 2% of 50%? Are you OK with Q12 and Q13?
In my world, I define the percentage symbol to represent 1/100. So 25% = 25 * 1/100 = 25/100 = 0.25.
– To express a percentage as a fraction or decimal is to replace % with 1/100 and evaluate.
– To express a fraction or decimal as a percentage is to manipulate the number such that 1/100 = % appears.
I thus get (unreasonably?) annoyed when textbooks specify percentage conversions as times/divide by 100.
0.25 times 100 = 25, which by my definition of % is NOT equal to 25%.
Rather, by taking % to equal 1/100, conversions can be thought of similarly to other changes in representation – we are simplying multiplying/dividing by 1 = 100%, to change how the number looks. (Similar to creating equivalent fractions by multiplying by 1 = n/n.)
Questions like (25% of 30) or (2% of 50%) are therefore simple multiplication problems (where “of” is taken to mean times)(cf. “out of” to mean divide). Questions asking to calculate a percentage (‘calculate the percentage of students taking specialist maths’) are identical to proportion questions (‘calculate the proportion of…’), with the stipulation that your answer should be expressed including the % symbol.
Thanks, Marcus. So, for you, X% *means* X/100, and that’s it? So, ignoring the methods by which one might calculate or convert percentages, you are ok with the questions above?
I don’t quite understand what is (perhaps reasonably) annoying you about the textbook conversion. Can you give an exercise, a description of how textbooks teach students to approach the exercise, and why you disapprove?
Yes, to me, X% *means* X/100, and that’s it. The questions seem fine to me.
Examples in textbooks that annoy me include (year 11) Heinemann Chemistry 1’s treatment of percentage composition and Cambridge’s (all years/streams) treatment of compound interest. The formulas they present include factors of 100 that (to me) obscure (or clutter) the underlying structure. I would much rather take 25% = 25/100 = 0.25 as three representations of the same number, and keep formulas ‘clean’. (Surely senior students do not need prompting nor assistance to shift between these representations…hmm)
In Heinemann Chemistry 1 worked example 7.4.1, the percentage by mass of carbon in carbon dioxide is calculated.
I would proceed:
%(carbon in CO2) = (mass of C in 1 mol CO2) / (mass of 1 mol CO2) = 12.0/44.0 = 0.273 (3 s.f.) = 27.3% (3 s.f.).
The textbook includes “x 100” in the formula, which means (to the textbook) ‘the proportion’ =/= ‘the percentage’. To me, they are the same, with a different preferred representation (decimal/percentage). Here, “x 100%” would appease me.
P.S. Reviewing this example, I guess I also regularly use % as a function, where %(x) = x / (some implicit whole) or similar.
Thanks, Marcus (I presume).
I can see why question 6 could be confusing for a child, especially given that odd picture, which looks like it was lifted from a marketing poster. Part 6d, in particular, feels awkward.
Questions 12 and 13 are also poorly designed. I think I understand what the authors were trying to ask, but they could have chosen a much clearer and more appropriate context.
Thanks, Dr. M. In what way do you think Q12 and Q13 are poorly designed?
5 1/2 = 11/2 I am not sure what to simplify here. Question 13 is a kind of ‘so what’ exercise. A child is going to tell you a fraction between 8/100 and 9/100, but they would lose the denominator of 100, so again, not sure how that is a ‘percentage educational’ question. Question 1 is also more of an exercise on how to simplify fractions.
Let’s say that 10% = 0.1. Does “Increase 18 by 10%” mean the same thing as “Increase 18 by 0.1”?
And I note that one of the first things that Enrichment Mathematics says in chapter 3 is “A percentage is a number” and that 37%, 37/100, 0.37 and 37:100 are all representations of the same number. Is a number its representation?
Nice example. Do you agree with Enrichment Mathematics?
Also, I’m updating the post and noting your comment. Quickly, a number is the thing in our minds. Definitely, a number is not the same as some representation of that number.
Increase 18 by 10% is a short version of “increase 18 by 10% (of it, i.e. of 18)” or “increase 18 by 0.1 (of it, i.e. of 18”.
I think that is a convention.
Like “derive y= 3x+1” is “derive y=3x+1 with respect to x”
But questions / problems could be designed/worded better.
Why not: inflate 18 by 10% (of it), increase 18 1.1-fold, …?
“Derive”. Ugh!
This is a very interesting and delicate issue. I think in the early education of children , it is important to cement the idea of ratio separately from the idea of a fraction as a number. This can be done naturally because there are lots of examples of sharing proportionally that can be used, as well as the use of fractions in measurements (as for example, on a ruler or measuring jug).
A ratio is simply a pair of numbers a:b, the antecedent and the consequent . A percentage indicates a ratio whose consequent is 100; and i would prefer that percentages not be confounded with numbers; I really do not like 12, 1 and 13. Such questions are not a modern novelty; if you look at old Grade 8 exams of the early 20th century, you will see similar stuff.
To ask for 60% or 300 is to ask for the antecedent of a ratio whose consequent is 300 that is proportional to 6:10, although no one in his right mind would talk this way in front of kids, but technically that is what is going on and you have what used to be called a “rule of three” problem. Good older textbooks dealt with ratio and proportion separately and kept the proper distinction between a vulgar fraction when used as a ratio and as a number. To be sure, in working out percentage problems, you are doing arithmetic and at that point dealing with numbers directly.
Fractions have a dual role as a ratio and as a percentage, and like a lot of things in language are hard to delineate formally but kids learn to negotiate through the context. When you say “1/4 of”, it is using fraction as a ratio, when you say “1/4 times”, then the context is some situation in which you are using 1/4 as a real number and multiplying it by another number. Thus problems 6 and 2 are ok for me.
From the teacher’s point of view, it is necessary to be fastidious in your use of language and the kids will pick it up (the modern term is “modelling”). I think the process is similar to how young children learn to speak grammatically long before they get any schooling in grammar.
Thank you very much, Ed, and everyone who has commented. This has been very interesting, and I think there is a lot more to say and to fight about. (In a collegial way, of course, not in an “All Bow to the Queen of Haileybury and the King of Clarendon” way.)
Ed’s is not quite along my lines of thinking, which suggests to me that my lines of thinking are not quite right, but it is close. I will update the post soon.
Thanks again to everyone. I’ve added a long update to the post.
Somehow I was thinking about this on my bike riding to school. It’s not categorically watertight, but I feel like numbers are something that you can “put in front” of things, units, pronumerals, and it makes sense. So 3.5 apples, 3.5 metres, 3.5x. All good. It doesn’t “work” with percentages – all of 3.5% apples, 3.5% metres, 3.5%x make me feel quite ill.
I’m happy with % as an operator – a clean distinction.
Yes, questions 1, 12, and 13 all make me nervous – and if I see questions like this again, I’ll hesitate before setting them.
Thank you very much, F E B. I think your 3.5% apples and 3.5% meters are really great examples, illustrating how percentage as number is wrong. Even people who are comfortable with 3.5% on a number line, and I really cannot imagine there are many such people, are gonna struggle with 3.5% apples and 3.5% meters.
Thanks – very interesting and the sort of thing that really should be in teacher training for maths!
Yes, but there’s all that N-E-T pedagogy to get through, and which is obviously way, way more important.
I wonder if someone could elaborate on the operator perspective/provide an illustration how this perspective can insightful? I’m not sure what perspective my teachers held, but certainly I have driven myself down the ‘percentage is a number’ road, and I would like to see what I am missing.
On my own ‘percentage as number’ perspective, I agree with what Banacek Spaces said that “Increase 18 by 10% is a short version of “increase 18 by 10% (of it, i.e. of 18)” “. Indeed, percentages (or numbers that are expressed as percentages) frequently arise when calculating proportions. In these calculations, if the numerator and denominator are expressed in the same unit, the proportion (=?= the percentage) is ‘unit less’. Yet, at the same time, since it is calculated via division, it is also a rate.
e.g. %(bad apples) = (number of bad apples) / (total number of apples) has units apples PER apple
Interest rate = (interest paid) / (principal) has units dollars PER dollar. **Just discovered that latex works here**
Rather that quoting an awkward specific unit (e.g. the interest rate is 0.05 dollars per dollar) or unit less (the interest rate is 0.05) the convention is to quote ‘per 100’ (the interest rate is 5%).
Thanks, Marcus. “Operator” is just a two dollar word for a function.
You have the notion of functions, I presume. (I’m not meaning to sound condescending: I simply have zero clue of your background.) So, the square root function, for example, takes a number and spits out the square roots. Chuck in 9 and you get back 3. Chuck in 5 and you get back √5.
The point is that the √ thing is has no independent existence as a number thing to manipulate. In the same way, 83% is not a thing to manipulate. It only (properly) exists as a function, in the form of “87% of”. Chuck in x and you get out 87% of x, but the 87% has no independent existence.
Thanks Marty. I have in the past completed MAST3**** and MAST9**** code subjects. My question is what insight is to be gained from adopting a ‘percentage as operator’ mindset?
Maybe I can jump in here Marcus – as a maths teacher, it really helps to have an excellent conceptual understanding of the maths I teach and how it fits into the bigger picture. This is especially important for kids who struggle with maths, particularly for the ways we use language with so many different meanings. The most concrete example I can think of the 3 meanings of the negative sign I read about in learning about algebra manipulatives recently: ‘-‘ as opposite eg -(-2) = 2, denoting a negative number, and the operation of subtraction. Every little bit helps inform the teaching, particularly in something as critical and badly understood by the general population as %.
Thanks, Marcus. I agree with what JJ has written, but I think yours is a very good question and there is more to be said. Others may wish to jump in but in any case I’ll reply when I catch my breath (maybe tomorrow).
Thanks again, Marcus.
Certainly we *could* flatten percentage to simply mean a number, in the manner of Q1 above, and we are all aware that this flattening is now very standard. Such flattening is not uncommon and not necessarily bad, at least with proper timing. In secondary school, for example, division gets flattened into fractions: we quickly cease caring about the process of
and go straight to the result of that process, 
. Another example are prisms and pyramids, which, for mathematicians, are simply cylinders and cones: at the mathematician level, the similarities to the circular things are more important than the differences.
I think there are two separate answers for why the flattening here is a bad thing, why it is preferable to keep percentage solely as an operator.
The first reason is that most of us, most of the time, think of percentage as an operator, and that’s the way it is. Banacek’s example of “increase 18 by 10%” is a very good one. It’s preferable, if manageable, to not have the “of 18” only implicit, even if including the “of 18” can be very awkward reading, and even if we naturally read the implicit “of 18”. But, given we use such phrasing, it is very not good to have “increase 18 by 1/10” and “increase 18 by 10%” mean fundamentally different things. Consistency may be a bit of a hobgoblin, but it’s still worth some attention and effort.
Then, as I and others have noted, once you try to percentage as real number, as a thing on a number line, it clearly feels wrong. Fuzz E Bear’s examples are excellent.
So, the first argument to treat percentage as an operator is simply to be consistent with what we do. But I think there may be a second, more positive reason. I haven’t thought hard about this, but I think there’s something there.
I think there is a real problem with teaching multiplication, and a double think, or even triple think, with what it means. In the beginning, and despite what some loud mathematicians may claim, multiplication is repeated addition:
means 
. But then the question is, what does 
mean? Or, 
? In the beginning, and very non-obviously, this fraction multiplication means “fraction of”: 
means “one third of 
“.
I do not know how this multiplication/of understanding fits in with the introduction and the teaching of percentages. But I cannot imagine not introducing percentages other than as an “of” operator, and implicitly as fraction multiplication. We eventually, and pretty quickly, have to give up the idea of fractions as fractions “of” something, and they become numbers in their own right, living on the number line. But it is certainly worthwhile keeping in mind the intuition, and the definition, of fraction multiplication as an “of” thing.
As I said, I haven’t really thought this one out. I have an earlier post, on
, which resulted in some very interesting discussion.
If 25% is a number, what is the value of
or 
?
My calculator makes sense of these things, but I don’t like them.
I think the answers are obviously 0.25, 1/2 and ≈ 0.004.
“ ‘of’ means ‘times’. “ This was taught to me in upper primary school.
I attempted to try out the ‘percentage as operator’ perspective last night.
As has been mentioned it is unlikely anyone would say “2 of 300” or “0.1 of 300”. But people do say “half of 300”. I wonder(ed), if in the construction “50% of 300”, “50%” is not a number, rather “% of” is an operator applied to 300, is the same true of “1/2” in “1/2 of 300”? (I think you have suggested this in your reply above, Marty.) I came to this:
Let ⋅ denote implicit multiplication.
Consider the phrase “m/n of Y”, read “m n-ths of Y”. Is this to be interpreted as
m⋅(1/n of Y)
where (1/n of ) is an operator defined
1/n of Y = Y/n,
returning one n-th of its input/its input divided by n.
If % of Y = Y/100, then
50% of 300 = 50⋅(% of 300) = 150
since % of applied to 300 returns 3.
2/5 of 300 = 2⋅(1/5 of 300) = 120
since 1/5 of applied to 300 returns 60.
Where I am at so far:
Define the (% of) operator:
% of Y = Y/100
So
50% of 300 = 50⋅(% of 300) = 50⋅3 = 150
Define a percentage, P, as referred to in expressions
“Calculate the percentage of X in Y.”
“What percentage of Y is X?”
In these expressions, the percentage, P, is a number such that
X = P⋅(% of Y)
In words, a percentage is the number of (outputs of % of applied to Y) that equals X. Equivalently, X is equal to P (outputs of % of applied to Y).
Rearranging for P, we have
X = P⋅(% of Y)
X = P⋅Y/100
⇒ P = X/Y × 100
There it is! “times 100”! At least I can justify it now.
But, can’t “of” just mean “times” instead? Pedagogically, ‘percentage as operator’ seems like it would be harder to teach to year 7s than ‘percentage as number’. I still can’t see what the payoff is?
Thanks, Marcus. I have two thoughts in response.
1) It is not that “of means times”, it’s that “times means of”.
We know what half of a pie is and then, carefully, half of 3 or half of 4. THEN, we turn half of 3 into
. So, the clear meaning of “of” is handballed to the previously mysterious meaning of fraction multiplication.
2) It is only fraction multiplication that effortlessly is interpreted as “of”. The preposition “of” is intrinsically about possession, or about being part of something, belonging to. You sort of get this with integers, with a sentence like “I’ll have two of those weasels, please”, but that’s selecting two from a group. If you were simply counting (i.e. multiplying) weasels, you’d say “two weasels”, not “two of weasels”.
Thanks to a journey down a related rabbit hole some years ago, here is one way to think about it so you can sleep at night.
Always think “fraction of
“.
of 
.
of 
will have a value, e.g. 
of 
.
For example
For cases when X is a number,
Now suppose I am wondering what number lives on the number line at a point between the integers.
” when 
.
of 
. I can do it, and it makes sense to do it.
Consider the cases of “fraction of
Now think about putting a point on the number line in the interval between 0 and 1 that represents
Now what would make sense in the interval between 1 and 2?
For simplicity, now let’s drop the “of
” business when 
and represent 
of 
as just 
. And so on.
Voila, we have fractions (
=1) on a number line, and so I can think about fractions (
=1) as numbers!
1 is such a powerful little loner.
It has been a while since I thought about this, but I believe this way of thinking causes zero issues going forwards.
Now to the question of whether or not a percentage a number.
Seems to me I could argue either way.
It is always going to come down to “the” definition.
But following the above way of thinking leads me to the affirmative.
However, I have never felt any sensible need to put a percentage on a number line.
And there could be a very good reason not to, to avoid confusion about how percentages are generally used and spoken about.
(Thanks to my friends who have helped me to think about this along the way.)
Thanks, Anthony. I agree with all that except the dismount. I think it is really ignoring a lot of misfit and non-use to hold to percentages being numbers.
Dismount. 😂
I do not think we are in disagreement regarding putting stuff in front of typical students that potentially confuses them regarding the best/most useful way to think about a percentage.
Per is in its name, after all.
My dismount tries to make the point that one could argue, mathematically correctly, as we did for fractions, that a percentage is a number.
While discussing this with a cleverer-than-me friend he made the point that 1.5 million apples sounds aOK. How about 1.5 twenty apples? It might sound “weird” and be unhelpful, but it is not mathematically wrong.
That’s a silly example. 1.5 million is standard terminology for a number, and 1.5 twenty is not.
As a teacher, I should frame problems such as “Find 20% of $150” rather than “Write 20% as a fraction.”
I thought a percentage, like a fraction, was a part or a proportion of the number.
[and decimals are also numerical parts].
Hi, Adelaide. I’m not sure I understand how you’re thinking of percentages/fractions/decimals.
Fractions obviously begin as fractions of something, but eventually become numbers in and of themselves, as placed on the number line, and with which we can do arithmetic. That is a notoriously difficult conceptual leap. And, it is made more difficult because we don’t give up the idea of fractions as fractions of things, and we don’t give up the idea of a/b being a parts out of b. We end up having to double-think.
Decimals don’t carry the same “of” baggage/uses, and i don’t think it is totally obvious why not. Partly, I think it is because the “a parts out of b” message is not so explicit in the notation. And partly I think it is because decimals are naturally ordered into larger and smaller, so that they are more readily treated as real numbers: they are more easily pictured on the number line, and they are more claerly useful for measuring.
Percentages are a bit of a weird thing of themselves, and hence the blog post and discussion. I am strongly of the opinion that percentages are not numbers: you can only take a percentage of something else. But other people, who are of course wrong, disagree.
I’m (again) late to the party, but I agree (with 100% “of” your argument) that an “of” is necessary for a percentage to have meaning. Percentage is an operator and needs something on which to operate for it to have meaning. Ignoring the mathematics that is expected to be undertaken by the hapless and bemused student, Q6 is a trainwreck of literary nonsense.
To add some further context, Marty, I am now secondary maths teacher, and ?guilty? of peddling a ‘percentage as number’ conception of percentages to students.
I don’t remember when (sometime in upper primary or year 7) I realised that fractions were ‘just numbers’, and I could position them on a number line, and I knew how to perform the usual operations on them, and suddenly they (and percentages, in the same way) made a lot of sense. I knew that I could choose what representation I wanted to use in my calculations, and noted that percentages were never a useful representations for calculations, and should only ever be specified as a final answer.
Language (English) is fickle; we have many different words with the same or similar definitions, and many words that are the same but with completely different definitions. I liked, and still like, mathematics for its definitions – nothing means anything unless we give it a definition, and we get to decide the definitions!
And so my experience learning mathematics was that ‘in the beginning’ you are given ‘numbers’, and you learn how to play with them, until at some point in secondary school you are introduced to new objects (e.g. equations*, graphs, functions, …) and learn how to play with them and so on.
In mathematics, we get to create the (mathematical) ‘world’ in which we wish to exist, and so my question has been why would you, especially at the beginning, want to make that world more complex by including different types of objects? Mathematics made sense to me once I realised that fractions (and percentages) could be conceived as ‘just numbers’. You said that we “eventually, and pretty quickly, have to give up the idea of fractions as fractions “of” something” – is it necessary (and if so, why) to start there?**
I suspect that the answer has everything to do with how humans move from the concrete to the abstract. (And I do not feel particularly well-prepared by my secondary teaching degree, in this regard.)
*statements of equality, that we get to manipulate
**And, is it necessary for students to remain there, into secondary school? I do like my ‘percentage as number’ conception of the world very much.
Thank you, Marcus. I agree with 95% OF that, and am sympathetic with 100% OF that.
I don’t think you should feel !guilty! for using percentage as a number. It is very common and there is an argument for it. As I wrote, David’s book does it, and David did it even after I argued strongly about it. I think you, and David, are wrong, but not “You’re out of your mind wrong”. It’s not on a par with using PowerPoint or something.
Why begin with fractions others than as numbers? Because fractions are *hard*. In the beginning, it is way hard enough to get across the idea of fractions as fractions of something, and that is a really really important idea. You gotta start with “half a pie” or whatever. But, you cannot stay there.
This is true with most mathematical concepts, and I think you simply have to accept that different meanings are appropriate and digestible for different levels. The example I used above is multiplication as repeated addition. I don’t think there is any other sane approach to introducing multiplication. But students eventually, and pretty quickly, need to adjust, to learn that multiplication is its own thing, not simply a sub-case of addition.
On the ambiguity and muddiness of language, I mostly agree with you, I think we have to learn to live with it. There are certainly scenarios where we could do better, and that bad language or notation is simply due to tradition rather than best-we-can-do sense. The expression “equivalent fractions”, for example, is diabolical, as is the notation
for the inverse of a function. But sometimes it seems best to muddle along.
The word “fraction”, for example, is a mess, confusing the type of number with its representation. If fraction means a number then it cannot make sense to talk about the numerator of a fraction. But, we want to do that. The alternative is to use both “fractional representation”, for the symbols, and “fraction”, for the number represented by those symbols. That is doable, and I do it on occasion. But more often such an approach seems too pure and forced.
This doesn’t really address the issue of how to test percentages (although I’m not sure there is a perfect method anyway), but it seems to me that (here in the UK at least) there’s a bit of a ‘thing’ about the whole Fractions-Decimals-Percentage (FDP) mantra and drumming all manner of possible conversions into the minds of impressionable young children (I was going to write ‘obsession’ in place of ‘thing’ but I think that’s a bit too strong, although I couldn’t settle on a precise term).
Thinking about fractions for a minute, I’m sure life was easier for earlier generations – before the widespread adoption of metrication and decimalisation (and digital timepieces) it was common to think in terms of quarter of hours, eighths of an inch, half of a penny and so on. So as a child in the 70s I would happily trot up to the local sweetshop with my 10p piece, buy some sweets for 6 1/2 p and come back with my 3 1/2 p in physical coins. No problems with accepting what a ‘half’ meant there. Later on I could still buy 1/2 p value stamps (or some larger value like 2 1/2p) to stick on an envelope. Tradesmen and engineers had no need of more accuracy than 1/16 inch or similar. So generally there were more opportunities to work with fractions, and almost no need to worry about decimals.
Here’s something I literally just discovered today (unless I knew it 50 years ago and never used it) – while thinking about the above I took out an old plastic 30 cm / 12 inch ruler and looked at the markings. The metric scale is divided into mm as you would expect. But on the imperial side, the section from 0 – 6 inches is divided into sixteenths of an inch (indicated by a legend of “16 THS”) but the section from 6 – 12 inches is divided into tenths of an inch (indicated by a legend of “10 THS”) ! I thought the whole thing would be consistent. Is this the same in other parts of the world?
Jeremy, your ruler is one of the stupidest things I’ve ever heard of. I’ve had a thousand metric-imperial combo rulers over the decades, and I’m pretty sure I’ve never had one where the imperial side had a split personality.
I don’t think there is a particular problem with testing percentages, just that the solution depends upon how you regard percentages. If percentages are numbers then it is reasonable to “obsess” over the conversion thing. If not, then the emphasis would be more on application questions, where you’re calculating the percentages of things (and preferably more interesting things than random lemonade bottles).
Your suggestion about the “easier” old days is very interesting. Off your point, but important, the arithmetic of imperial units and psd money was absolutely horrendous, not remotely easier. On the other hand, the imperial units in and of themselves often worked much better: feet and inches and yards, work much, much better in everyday life; and Celsius is the absolute stupidest unit ever imposed upon the general populace.
You are very importantly correct, however, that imperial units made the practice of fractions a natural daily occurrence, reenforcing their meaning, and in turn encouraging their strong practice in school. Decades ago, the real world was much more important for schooling, which is obvious but note this post). Ironically, the current real worldism in Australia is fetishistic and gratuitous and distracting. And, because the real world is now decimalised, this real worldism results in a massive over-emphasis on decimals and decimal arithmetic, with fractions a very poor second. This has been an unmitigated disaster.
There are many crimes to lay at the feet of mathematics education clowns, but one of the greatest crimes has been the marginalisation of the teaching of fractions. Thankfully, however, that is all changing. With the new emphasis on explicit teaching, it is undoubtedly the beginning on a new era.
Well, I agree that Q12 and Q13 are weird questions. No one would *really* want to do that kind of thing with a percentage. But I disagree that a percentage is not a number. I would say it definitely is a number, but that very strong convention is that it is a proportion of something.
I don’t think it *has* to be a proportion of something, because maths can be very abstract. I just think it almost always *is* a proportion of something – as a matter of convention rather than definition.
It reminds me a bit of how torque and energy seem to have the same physical units. Obviously they are different concepts, but both seems to be measured in Newton meters. Are the ‘radians’ embedded in torque somehow physical units too? Many so. So then does π itself somehow have some kind of mathematical units embedded in it that make it an angle? … well probably not, because π shows up in places that don’t seem related to angles.
Anyway, my point is just because something tends to be used in a particular way doesn’t mean that the usage needs to be part of its definition.
Thanks, Zac. I think you’re walking a bit of a tightrope, distinguishing a definition from a strong convention, but I take your point. Also, whatever the definition *was*, if enough teachers and texts garble it, then, for better or worse, the definition changes. Still, I’m pretty sure percentage did not begin as a number, and I see no advantage to making it a number.
I guess for me the advantage of saying “it’s a number” is that we can then go ahead and use all the rules and properties of numbers that we’re familiar with without having to worry about whether it is allowed. One thing that comes to mind is that probabilities are often expressed as percentages, and probabilities are also used in various calculations; sometimes multiplying them together, sometimes adding them, sometimes dividing one by another… I guess for me if it looks like a number and it quacks like a number… then I might as well call it a number.
On the flip side, there are lots of things that you can do with numbers that don’t really make sense when you’re talking about percentages – but I think that’s true with basically everything that has context. The context makes it so that some things make sense and some things don’t.
Actually, now that I think about it, I’ve got a friend who argues that it was a mistake for people to start thinking of irrational numbers as numbers. From their point of view things like sqrt(2) only ever arise when trying to ‘measure’ stuff; and so they argue that they are a ‘measure’ rather than a number. (I don’t think I can do their argument justice, because I don’t agree with them – but I do think it’s at least something to think about.)
Thanks, Zac. I don’t think your example is very quacky. A probability percentage is naturally thought of as the fraction of 1/certainly, particularly in the discrete setting, where a probability is essentially a fraction: (favourable outcomes) ÷ (total outcomes), or whatever.
Probability: a function mapping events(outcomes) of a random experiment onto…?
(N.B. I tutored a student who had a mark taken for fractional (correct) answer to probability question, crossed by the teacher and replaced with a decimal equivalent).
It would be wise to jump on Mark Katz (Kac?) ‘s postulate that when teaching mathematics to young students we tell the truth but not necessary the whole truth – here: % and numbers.
So, too much philosophysing here?
Thanks, Banacek. I like Kac’s principle (if it is his) but only as a principle.
My personal and entirely vibes-based assessment is that 6 and 2 are fine, while 12, 1, and 13 are not, on the basis that something like ‘50%’ only means something in relation to another quantity and has no independent numerical meaning. I haven’t thought through carefully if this is the best approach.
Thanks, bell. Vibes are not to be underestimated. They can be the indication of the more solid sense that is being obscured by modern sloppy practice. I think that is exactly the situation here.
Thanks, Marty, for your update. In my earlier comment, I neglected to mention the perception of a percentage as an operator, which in my opinion, is another perfectly valid way of looking at it. This just illustrates how mathematical ideas can assume different guises depending on the context.
Some children have always had difficulty with different aspects of arithmetic, but in my view, until the last few decades, fractions were no harder other things in that respect. Jeremy in the UK says something that chimes with my own experiences as a pupil. Fractions (as numbers) occurred in a lot of situations involving measurements and money that a child would be exposed to in everyday life, so even in kindergarten, kids could handle simple fractions. At the level of schooling, to be sure, children are exposed to proportionality (in sharing, for example). But the relationship between the two concepts developed over time in context as the problems given to the children became more complex.
I think that what has buggered up children’s understanding of fractions is that they really do not know what the hell they are talking about, and the looseness of lot of modern teaching and text writing does not help at all.
Thanks, Ed. I’m not sure I agree, that fractions were once no harder than other things. If fractions were too hard for the Greeks to think of, there has to be something intrinsically difficult about them. But we agree on Jeremy’s point. I am sure, however, that the new emphasis on decimals and the introduction of calculators have been pedagogical own goals, which have made things much, much worse than they needed to be.
“It is a number, end of discussion.”
– Hung-Hsi Wu
Well, one ignores Wu at one’s peril (although I enjoy peril). Do you have a source?
Quote is from page 323 from his ‘Understanding numbers in elementary school mathematics’
Starts chapter 20 ‘Percent’ on page 319 with ‘The main point is that a percent is a number.’
Also remarks on percent as an “action” or an “operator” on page 322.
Thank you. I’ll look.
What about Eddy? What does he say? So important, with his leverage.
Eddy Woo? Jesus. Eddie Barbeau, yes.
Was intrigued by 0 to the power of 0 analysis…
Indeed.
Thanks again, WWW. I’ve updated the post with Wu’s paragraph discussing the matter. I have my own thoughts but thought it better to give other people a go first.
Wu’s book is, of course, generally very interesting. I’d read other stuff by Wu, obviously in the same vein, but I hadn’t looked at the book previously.
I can see Wu’s point. It is a different form of a number like decimals. However, you do have to write a percentage as a complex fraction or decimal to do any arithmetic with it. So x% does seem seperate or adjacent to numbers, since you are not going to write x% + 3 for instance. Whereas 0.3 + 3 or 1/3 + 3 is fine. But 1/3 + 0.3 does require converting to a single form. Having said that, x% + y% makes sense, I guess, but it would be confusing when considering what multiple percentage increases mean.
Thanks, Potii. i pretty much agree. There is some simplicity to stating that a percentage is a number, and simplicity is always to be encouraged. But the actual usage of percentages as numbers, if at all, is restricted. Your x% + 3 is yet another excellent example of usage that Wu would presumably permit but is self-evidently very off-key.
I don’t think your x% + y% is much of a counterexample. One can add operators just as one can add functions.
I think of it as:
Fractions and decimals can be both numbers and proportions. So fraction and decimal arithmetic should be part of a “Number” unit.
Percentages are ONLY a proportion, not a number. So there should be a seperate “FDP” unit, a proportion unit, that deals with fractions, decimals and percentages OF AMOUNTS.
If percentages could be stand-alone numbers, then couldn’t the answer to “increase 30 by 10%” be 30.1? (I have seen this misconception in real life)
Thanks, Dan. I think you mean ratios, rather than proportions. (“Proportion” has a different meaning, which is getting lost.)
But your suggestion to teach percentages separately from rational arithmetic is spot on, and it used to be standard. I checked out a number of texts from the 60s and 70s, and they all taught percentage as a quite separate topic, often far away from the arithmetic and often in conjunction with ratios. The texts had *some* suggestion of percentage as fraction, akin to Q1 above. But such exercises were very limited, and I spotted nothing like Q13.
I did mean proportion, meaning “part compared to a whole”. As opposed to ratio, which we use to mean “part compared to part” (which also describe a relationship between 2 amounts in “direct proportion”, slightly confusingly)… But understand this terminology might vary between countries.
An example of describing fractions, decimals and percentages as “proportions”: https://mmerevise.co.uk/gcse-maths-revision/fractions-decimals-and-percentages/
Thanks, Dan. That usage is a little peculiar to me, but sure. And I’d see percentage as creating a part rather than comparing a part. Btu I get where you’re coming from, and certainly old texts are treating and placing percentage in that kind of context, rather than as a number thing.