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I've always wanted to know what the name of the vertical bar in these examples was:

$f(x)=(x^2+1)\vert_{x = 4}$ (I know this means evaluate $x$ at $4$)

$\int_0^4 (x^2+1) \,dx = \left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$ (and I know this means that you would then evaluate at $x=0$ and $x=4$, then subtract $F(4)-F(0)$ if finding the net signed area)

I know it seems trivial, but it's something I can't really seem to find when I go googling and the question came up in my calc class last night and no one seemed to know.

Also, for bonus internets; What is the name of the horizontal bar in $\frac{x^3}{3}$? Is that called an obelus?

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    $\begingroup$ I don't know if it has a name, and if it does, I don't know how useful that information would be to you for talking about mathematics (since I am pretty sure most people would not recognize it). $\endgroup$ Commented Jul 20, 2011 at 14:58
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    $\begingroup$ "bonus internets" - ??? Are these the internets that Al Gore didn't invent ? $\endgroup$ Commented Jul 20, 2011 at 15:07
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    $\begingroup$ As Qiaochu Y. notes, knowing a Latinate name would not help in actual communication, since it is pronounced "evaluated at" in math. Maybe "pipe" in computer science, but that's unrelated. The double vertical line does have a Latinate name, "vel", which is just "or" in Latin. Horizontal lines for grouping (as in large fractions, but also in radicals) are "vinculi" (singular "vinculum"). But no one calls them that. Latin's former status as prestige language seems fading, the way the Sumerian language's prestige for Egypt and Mesopotamia seems to be long gone. $\endgroup$ Commented Jul 20, 2011 at 15:20
  • $\begingroup$ @paul garrett I know it wouldn't help in communication, but I just want to know for the sake of knowing. $\endgroup$ Commented Jul 20, 2011 at 15:23
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    $\begingroup$ Why the vote to close for too localized? This question is about the terminology of a widely used symbol... How is that localized? $\endgroup$ Commented Jul 20, 2011 at 16:00

5 Answers 5

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Jeff Miller calls it "bar notation" in his Earliest Uses of Symbols of Calculus (see below). The bar denotes an evaluation functional, a concept whose importance comes to the fore when one studies duality of vector spaces (e.g. such duality plays a key role in the Umbral Calculus).

The bar notation to indicate evaluation of an antiderivative at the two limits of integration was first used by Pierre Frederic Sarrus (1798-1861) in 1823 in Gergonne’s Annales, Vol. XIV. The notation was used later by Moigno and Cauchy (Cajori vol. 2, page 250).

Below is the cited passage from Cajori

enter image description here enter image description here

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  • $\begingroup$ That article of Sarrus can be found online here, but it does not contain the notation. $\endgroup$ Commented Oct 1 at 9:14
  • $\begingroup$ Maybe this article (credit to Cajori) was meant: numdam.org/item/AMPA_1821-1822__12__36_0 $\endgroup$ Commented Oct 1 at 9:23
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This may be called Evaluation bar. See, in particular, here (Evaluation Bar Notation:).

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  • $\begingroup$ Example 1: Page 2, third item from the end, here: math.uaa.alaska.edu/~afmaf/classes/math201/notes/IntParts/… $\endgroup$ Commented Jul 20, 2011 at 15:20
  • $\begingroup$ Example 2: Page 2, item 5, here: princeton.edu/~slynch/bayesbook/errata.pdf $\endgroup$ Commented Jul 20, 2011 at 15:28
  • $\begingroup$ Example 3: You can the sentence "and I couldn't figure out how to make the "evaluation bar" immediately to the right of the uv terms any bigger", here: physicsforums.com/showthread.php?t=376190 $\endgroup$ Commented Jul 20, 2011 at 15:35
  • $\begingroup$ I like this name. +1 just for that XD $\endgroup$ Commented Jul 20, 2011 at 15:39
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    $\begingroup$ the link doesn't work... $\endgroup$ Commented Feb 20, 2013 at 19:21
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In my calculus book, the vertical bar is called the "evaluation symbol", and this phrase is bolded when first mentioned. It makes sense, I suppose.

Copy paste from wikipedia: Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them.

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In the wikipedia article for the symbol no name for this particular use of it is mentioned, just that it is read as, simply, "evaluated at". It has a number of suggested names for the symbol from different situations though:

verti-bar, vbar, stick, vertical line, vertical slash, or bar, think colon, poley or divider line

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  • $\begingroup$ I saw that, which is another reason why I asked my question. $\endgroup$ Commented Jul 20, 2011 at 15:13
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There are several mathematical notations that use the vertical bar in a way similar to $(x^2+1)\big|_{x=4}$.

Examples include:

  • Conditional probability and expectation: $P(A \mid B)$, $E(x \mid B)$
  • Set-builder notation: $\{x \mid B\}$

In each case, the expression after the bar specifies a condition (a proposition) under which the expression before the bar is to be considered. A simple way to read the bar is as "when". For example:

  • $(x^2+1)\big|_{x=4}$: $x^2+1$ when $x=4$
  • $P(n<10 \mid n \text{ prime})$: The probability that $n<10$ when $n$ is prime.
  • $E(x \mid x^2 \leq 4)$: The expectation of $x$ when $x^2 \leq 4$".
  • $\{n \in \mathbb{N} \mid n \text{ divides } 100\}$: The set of natural numbers, where a number $n$ belongs when $n$ divides 100.

There is also fiber bundle restriction $E|_p$, which could also be interpreted this way with some more work.

Thus, a way to call this operation is conditioning (used in probability) or restriction (used in geometry/topology). As for the symbol itself, I would just continue to call it the vertical bar.

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