This is an observation in abstractmath that I think needs to be publicized more:

Two symbols used in the study of integers are notorious for their confusing similarity.

The expression “$m/n$” is a term denoting the number obtained by dividing $m$ by $n$. Thus “$12/3$” denotes $4$ and “$12/5$” denotes the number $2.4$.

The expression “$m|n$” is the assertion that “$m$ divides $n$ with no remainder”. So for example “$3|12$”, read “$3$ divides $12$” or “$12$ is a multiple of $3$”, is a true statement and “$5|12$” is a false statement.

Notice that $m/n$ is an integer if and only if $n|m$. Not only is $m/n$ a number and $n|m$ a statement, but the statement “the first one is an integer if and only if the second one is true” is correct only after the $m$ and $n$ are switched!

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4 thoughts on “The most confusing notation in number theory”

I’d say the Legendre / Jacobi symbol notation is more confusing, since it’s exactly the same as the notation for fractions (granted, with parentheses).

The most confusing? I beg to differ. The lines are never in the same direction.

One symbol, which is also used in the study of integers, and is even more confusing because of its similarity to a notation for division, is the notation for the Legendre symbol $\left(\dfrac{a}b\right)$. That uses the horizontal line and parentheses even though those already have different meanings.

Then the vertical bar can also be used for “such that”. That can make some expressions hard to parse. For example $\{a|f(a)|n\}$

(Sorry if the notation doesn’t come out right — your blog should provide a “preview” facility and say what notation is available.)

Rosie: The part about the lines having different directions is a good point. My experience has been mostly teaching engineering students. They tend to overlook similar but not identical symbols. For example, they have to be told that in math, whether a letter is uppercase or lowercase is almost always significant.

The point about the vertical bar occurring both in division and in setbuilder notation is a good one, and I expect to point this out in the abstractmath article on sets.

Mike Stay: That is a good example, but there are many other examples where the same notation is used with two different meanings. For example the notation “(a,b)” can mean the GCD of a and b, an ordered pair, or an open interval.

The problem with $m|n$ and $m/n$ is that (1) both of them involve division and (2) the placement of $n$ and $n$ are reversed.

I will admit that I have taught discrete math lots of times, where this notation comes up and the Legendre symbol does not. I have never taught number theory, introductory or otherwise. So my experience is distinctly biased.

I’d say the Legendre / Jacobi symbol notation is more confusing, since it’s exactly the same as the notation for fractions (granted, with parentheses).

The most confusing? I beg to differ. The lines are never in the same direction.

One symbol, which is also used in the study of integers, and is even more confusing because of its similarity to a notation for division, is the notation for the Legendre symbol $\left(\dfrac{a}b\right)$. That uses the horizontal line and parentheses even though those already have different meanings.

Then the vertical bar can also be used for “such that”. That can make some expressions hard to parse. For example $\{a|f(a)|n\}$

(Sorry if the notation doesn’t come out right — your blog should provide a “preview” facility and say what notation is available.)

Rosie: The part about the lines having different directions is a good point. My experience has been mostly teaching engineering students. They tend to overlook similar but not identical symbols. For example, they have to be told that in math, whether a letter is uppercase or lowercase is almost always significant.

The point about the vertical bar occurring both in division and in setbuilder notation is a good one, and I expect to point this out in the abstractmath article on sets.

Mike Stay: That is a good example, but there are many other examples where the same notation is used with two different meanings. For example the notation “(a,b)” can mean the GCD of a and b, an ordered pair, or an open interval.

The problem with $m|n$ and $m/n$ is that (1) both of them involve division and (2) the placement of $n$ and $n$ are reversed.

I will admit that I have taught discrete math lots of times, where this notation comes up and the Legendre symbol does not. I have never taught number theory, introductory or otherwise. So my experience is distinctly biased.