A recent question about the precise meaning of “difference” led me to some past discussions of the word.
This came from Amia in August:
Hi Dr Math,
I have this question :
Does the difference between two numbers mean to subtract the small number from the large number?
I answered:
Yes, that is correct. We subtract the smaller number from the larger number to obtain a positive difference.
Or, as I would express it, the (absolute) difference between x and y is
|x – y|,
the absolute value of the difference in either order.
For example, the difference between 2 and 5 is \(5-2=3\), which could also be expressed as \(|2-5|=|-3|=3\). A difference in this sense is always positive, which can be ensured either by always taking the larger number first, or by using an absolute value.
But it depends on context; sometimes the phrase might be intended to mean either |x – y|, x – y, or y – x.
In some contexts, it is common to talk about the signed difference between x and y; sometimes this is written as “the difference of x and y”, taking the numbers in the order given:
x – y
I see this particularly in problems in elementary algebra, where the absolute value would be too complicated, and they want to be able to express x – y in few words.
We’ll see more about this later.
This can also be true in basic arithmetic. I see, for example, that Wikipedia says
https://en.wikipedia.org/wiki/Subtraction
“In a sense, subtraction is the inverse of addition. That is, c = a − b if and only if c + b = a. In words: the difference of two numbers [a and b] is the number [c] that gives the first one [a] when added to the second one [b].”
So they are using “the difference of a and b” to describe \(a-b\).
In the sense of absolute difference, we would say that the difference is the number that gives the larger when added to the smaller.
Similarly, MathWorld says,
The difference of two numbers \(n_1\) and \(n_2\) is \(n_1-n_2\), where the minus sign denotes subtraction.
Here, the context is that we call the result of subtraction the “difference”, as we call the result of addition the “sum”. So we are starting with any subtraction, rather than with two numbers and their “difference”.
In contrast, MathWorld says,
The absolute difference of two numbers \(n_1\) and \(n_2\) is \(|n_1-n_2|\), where the minus sign denotes subtraction and \(|x|\) denotes the absolute value.
I finished by quoting two pages from Ask Dr. Math:
Here are a couple answers I have given to this question, the first before I started teaching, and the second after I’d seen how elementary textbooks treat it:
Those two pages are what follows.
Difference in a word problem
From 2003, a year before I started teaching in community college:
The Difference of x and y... What does "the difference of x and y" mean? I pretty sure it means x - y. However, I have a problem with a word problem. It is: "the difference of a number and its square is 42" (actually, it wasn't 42, it was 100 something, but that shouldn't affect anything). It's not difficult or confusing, but I do have a disagreement. A friend says that the equation is x^2 - x = 42, so the number is 7 or -6. I disagree. I think the equation should be x - x^2 = 42, which would be no solution because the equation can be rewritten as x^2 - x = -42, which isn't likely to end as a real number solution. Who's correct?
I answered, taking the problem to mean the absolute difference:
Hi, Matt. I myself would say "the difference between ..." not "the difference of ..." But that doesn't affect the meaning. The important thing is that a difference is always positive, regardless of which number is larger; the order in which the numbers are given need not be larger to smaller. I would translate the phrase as |x-y|, the absolute value; if I knew which was larger, I could just use x-y or y-x.
In the problem, we might initially suppose that x is an integer greater than 1 (just because that is common in problems), in which case \(x^2>x\); but that wouldn’t necessarily be a valid assumption.
In your example, the equation would be |x - x^2| = 42 To solve this, we need two cases. First, we suppose that x - x^2 >= 0, and solve x - x^2 = 42 x^2 - x + 42 = 0 which has no (real) solution, since the discriminant (-167) is negative. Then we suppose that x - x^2 < 0, and solve x^2 - x = 42 x^2 - x + 42 = 0 (x - 7)(x + 6) = 0 x = 7 or x = -6 Now we have to check that in fact x^2 > x for these cases; not surprisingly, it is. So your friend's solution is correct, though not quite thoroughly supported.
Rather than use the absolute value, one might simply assume (consciously or not!) that the answer is expected to be a positive integer, which would imply that \(x^2>x\), so that \(x^2\) is the larger number, and the (positive) difference is \(x^2-x\). And the solutions are what I would expect in an elementary algebra class. Yet, as we’ll see, the wording is not what one would expect to see there.
On the other hand, if we allowed complex-number solutions, we could solve the first equation and get
$$x=\frac{1\pm\sqrt{(-1)^2-4(1)(42)}}{2(1)}=\frac{1\pm\sqrt{-167}}{2}=\frac{1\pm i\sqrt{167}}{2}.$$ Then
$$x^2=\left(\frac{1\pm i\sqrt{167}}{2}\right)^2=\frac{-83\pm i\sqrt{167}}{2},$$
and the difference in each case is
$$x-x^2=\frac{1\pm i\sqrt{167}}{2}-\frac{-83\pm i\sqrt{167}}{2}=42$$
So these are alternative solutions, but they are clearly not what one expects.
To check the answers against the original question (which is the ultimate basis for calling an answer correct), ask yourself, "Is the difference between 7 and 49 equal to 42? How about the difference between -6 and 36?" I think you'll say that they are.
It is unclear what the author expected the student to do here, but it seems likely that they expect the assumption that \(x^2>x\). (This is, in fact, true for all real numbers except where \(0<x<1\); and even then, the greatest possible difference is \(\frac{1}{4}\).)
Teachers take it differently
Three years later, a reader of the above answer asked a further question, which was appended to the page:
Recently I helped my eighth grade son with his algebra. He was to write out and simplify the following word problem: -54 decreased by the difference between -37 and 15 I wondered about the sign of the difference part of the equation and didn't know if it should be -54 - (-37 - 15) or -54 - (15 - -37) I found your posting here and advised him to solve it as: -54 - |-37 - 15| His teacher insists that it should be -54 - (-37 - 15) and that "difference between" should be translated to math as straight subtraction. All other web sites I have found seem to support her position as this seems to be the common practice. Can you comment on this issue?
This is a different sort of question, because here we have specific numbers rather than variables; it likely comes from an introductory algebra class before solving problems involving such expressions with variables. But the same question exists: Are we to take the positive difference, or the first minus the second, as the teacher says?
If we take the absolute value, as in the previous question, then the answer will be $$-54-|-37-15|=-54-|-52|=-54-52=-106,$$ whereas if we take the first-minus-second interpretation, we get $$-54-(-37-15)=-54-(-52)=-2.$$
I answered again, this time with more knowledge of current textbooks:
Hi, Mark. I would really call it ambiguous; that happens often in English. In this case, however, I think part of the ambiguity has been introduced by educators who want to make an easy problem out of a tricky one by pretending that English is under their control. In any real situation I can think of, if you were to ask someone for the difference between, say, 3 and 5, the answer would be 2 -- not 3 - 5 = -2! We tend to think of differences as positive numbers, and thus the proper rendition of that expression algebraically would be |3-5| (or |5-3|). This is the same idea as the "distance" between two numbers on the number line, which is always positive.
Alternatively, as we’ve seen, we can skip the absolute value and just take “larger minus smaller”, writing $$-54-(15-(-37))=-54-(52)=-106.$$
In textbooks, as you've observed, it seems common in "word problems" to make a different convention, that "the difference between a and b" means a - b. My guess is that they do that because an absolute value equation is beyond the students they are usually presenting this to, and they want to promote the illusion that everything you can write in words can be translated to algebra by a simple operation.
Specifically, the first-minus-second interpretation, which seems inappropriate when you know the numbers, is introduced in algebra books in preparation for word problems where you will not know ahead of time which number is larger, and they want to keep the algebra simple for students. Such a word problem might look like
A number, decreased by the difference between twice the number and 15, is 6. What is the number?
Here, if we take the number as x and use the teacher’s approach, the equation is $$x-(2x-15)=6$$ and the solution of \(-x+15=6\) is \(x=9\). Indeed, the difference between twice 9 and 15 is \(18-15=3\), and 9 decreased by 3 is 6.
If we did the subtraction in the other direction, the equation would be $$x-(15-2x)=6$$ and the solution to \(3x-15=6\) is \(x=7\). In this case, the difference between twice 7 and 15 is \(|14-15|=1\), and 7 decreased by 1 is 6.
Both solutions are valid if we take the absolute difference and write the equation as $$x-|2x-15|=6$$
But now consider this problem:
A number, decreased by the difference between twice the number and 15, is 9. What is the number?
Now the first version of the equation is $$x-(2x-15)=9$$ and the solution of \(-x+15=9\) is \(x=6\). Yes, the difference between twice 6 and 15 is \(12-15=-3\) by the first-minus-second definition, and 6 “decreased by \(-3\)” (that is, increased by 3) is 9, but that feels wrong. The difference as we normally think of it is 3, and \(6-3=3\), not 9.
Doing the subtraction the other way, the equation is $$x-(15-2x)=9$$ and the solution to \(3x-15=9\) is \(x=8\). In this case, the difference between twice 8 and 15 is \(|16-15|=1\), and 8 decreased by 1 is 7, not 9. Again, to make it work we have to use the negative difference, \(8-\left(-1\right)=9\).
So if we use the textbook meaning of “difference”, nothing feels right. All we can hope is that textbooks don’t give problems like this, where the (signed) difference ends up being negative!
What they’re doing is just keeping things simple for beginners. In better problems, it could be made clear that one number is greater than another, so you can know which to subtract from the other. In the best problems, you would use the absolute value or two cases.
But there’s a little more to it:
This convention does make some sense: when we do a subtraction, such as 3 - 5, we call the answer the "difference". So from the perspective of a textbook author, who has mostly been writing math problems rather than real-world problems, this is a natural way to interpret the question. The problem is just that you have to have that context in mind, rather than how the language is really used outside the class; and in some application problems, it's hard to tell which context should be in view! I've seen some texts use "difference of" rather than "difference between", which in my mind carries a little less of the real-world sense. (Oddly, in the page you refer to, the question WAS written that way, yet appears to have had the opposite order in view!)
There are other cases where we have to read English in a special way that textbook authors are used to, which I call “Mathlish”. This includes the use of inclusive definitions, and hyper-logical use of words like “not” or “or” or “if” or “any” or “all”. In general, we tend to mean exactly what we say, even when idiomatic English does not.
So in the classroom context, and especially if the text has explicitly stated what they mean by difference, you just have to go along with it. But if I were grading a problem and a student chose to use an absolute value, I would not mark it wrong -- I would just point out that, in order to make subsequent problems in the text doable, it would be prudent to adopt their convention for the time being.
It is not uncommon for a student to have insight or knowledge beyond what is being taught (such as absolute values), and make a problem harder than it is intended to be!
Our archivist added a link to the following separately archived question, which actually came a month later:
Twisting the meaning
Interpreting the Difference Between Two Numbers Not using absolute values, if a question reads "What is the difference between 15 and 12" or "What is the difference between 12 and 15," would your response be +3 and -3 respectively? And would the expressions be X = 15 - 12 (for the first example) and X = 12 - 15 (for the second example) with the idea that whatever number is stated first in the written expression is the first number in the equation?
Mary assumes (and presumably questions) the “first-minus-second” interpretation. Do we have to take the question as using a signed difference?
I answered:
Hi, Mary. Elementary algebra books tend to twist the English language a bit here, to make things easier for the students. In real life, the difference between a and b is |a-b| (or |b-a|, which is the same); differences are always positive. But that would lead to ugly equations that students would struggle to solve; so they pretend that when we say "the difference of a and b" we mean just a-b--in that order, even though if b were greater than a, the "difference" would be negative.
So the algebra books assume a signed-difference meaning, evidently for convenience. But not only for that reason:
(In fact, there are some cases in math where we _would_ allow a negative difference; the one that comes to mind is when we take "successive differences" in a sequence to see how it is made, and consider the difference from one term to the next to be the second minus the first, regardless of which is greater. But we wouldn't quite use the same phrase as in the textbooks!)
Note that this example is not first-minus-second!
Anyway, regardless of the usual rules in real life, for the sake of problems in this course, your "thoughts" represent what they want you to do.
Then I linked to the first answer above:
Here's a similar question from our archives that you might find interesting as it discusses the use of absolute value in these situations:
The Difference of x and y...
http://mathforum.org/library/drmath/view/63137.html
Mary replied:
Thank you for your complete and usable answer to my question! It was most helpful for our discussion and understanding.
How to figure out the intended order
It happens that, the day after Amia’s question, we got an example of a word problem using the word “difference”, which can demonstrate how we can handle it:
The difference between two positive integers is 50 and their ratio is 2:3. Form the equation by taking the two numbers as X and Y.
It’s far better if you tell us what you have tried, or ask questions, rather than just state a problem, implying you just want us to solve it for you. But here, one issue was clear. I answered:
Hi, Sadat.
There are several ways you could approach this problem; since they tell you to use two variables, they presumably expect you to write two equations, one representing each fact:
The difference between two positive integers is 50.
Their ratio is 2:3.
You haven’t told us where you are having trouble, but I know that my own first concern is the order of the variables in each equation. Do we subtract x from y, or y from x? Do we take the ratio x:y or y:x?
Interestingly, here both the difference and the ratio depend on order, and the two are somewhat in conflict, as we’ll see.
So the first thing I do is to define the two variables carefully:
x = the smaller of the two positive integers
y = the larger of the two positive integers
See what you can do with that start, and show us how far you get. (You aren’t told to solve the system of equations, but doing so will let you check that everything makes sense. If your solutions are not positive integers, for example, then either you wrote the wrong equations, or the problem itself is invalid.)
Unfortunately, Sadat never read the reply; possibly it went to his spam folder. But let’s carry out the work:
First, if we took the first-minus-second interpretation of “difference between x and y“, and the natural first-to-second interpretation of “ratio of x and y“, rather than focus on “larger and smaller”, then our equations would be $$x-y=50\\x:y=2:3$$ which becomes $$x-y=50\\\frac{x}{y}=\frac{2}{3}$$ We can rewrite the second equation as \(3x=2y\), and then replace \(x\) with \(y+50\) to get the equation $$3(y+50)=2y.$$ This gives us \(y=-150\), which is clearly not a valid solution. That’s the point of my suggestion to check the answer as a way to check the interpretation.
Instead, we can use the larger-minus-smaller interpretation of the difference, and recognize that the ratio must be of the smaller to the larger. This gives us the equations $$y-x=50\\\frac{x}{y}=\frac{2}{3}$$ We can again rewrite the second equation as \(3x=2y\), and then replace \(y\) with \(x+50\) to get the equation $$3x=2(x+50).$$ Solving, we get \(x=100\). That’s better! Now \(y=x+50=100+50=150\); and indeed the difference is \(150-100=50\), while the ratio is \(100:150=2:3\) as required.