Does $x/yz$ mean $x/(yz)$ or $(x/y)z$?

Question

When people write $x/yz$, do they usually mean $x/(yz)$ or $(x/y)z$?

For example, from Wikipedia

If $p\geq 1/2$ , then $$ \Pr\left[ X>mp+x \right] \leq \exp(-x^2/2mp(1-p)) . $$

Thanks!

Answer 1

A few weeks ago I did a search for this construction. I found three things:

1. It was much less frequently used than I expected; I think this is because mathematicians are concerned that it might be unclear.
2. The few examples I did find invariably meant $x/(yz)$, not $(x/y)z$.
3. Complicated examples are extremely uncommon.

Here are some examples I found:

• In the proof of theorem 4.2 at the bottom of the page there is $u_n/n\alpha_n$. (Alberto Torchinsky, Real-Variable Methods in Harmonic Analysis)
• Page 3 of Walter Rudin's Real and Complex Analysis uses $y/2\pi$, and the bottom of page 99 has $\delta = \eta/2k$. However, on page 88 of the same book, Rudin writes $1/(2\pi)$, even though $1/2\pi$ always means $1/(2\pi)$, never $(1/2)\pi$.
• Page 12 of George Simmons' Differential Equations says that the negative reciprocal of $rd\theta/dr$ is $-dr/rd\theta$. Page 45 mentions the use of $1/xy$ as an integrating factor. In a rare complicated example, he writes $(2n)!/(n!)^2 2^n$ (meaning $(2n)!\over(n!)^2 2^n$) on page 222.
• I found no examples at all where $x/yz$ was used to mean $(x/y)z$.

However, Wikipedia articles are sometimes written by blockheads. From looking at related versions of the same formula, for example here (page 4) or here (theorem 1.5) I guess that that was the case here, and that the $p(1-p)$, if it belongs there at all, should have been in the numerator. I think you should be wary of taking the Wikipedia claims at face value, and instead refer to a source that is written by someone with a reputation.

You should also be wary of taking my claims at face value, since I know absolutely nothing about Chernoff bounds.

Answer 2

Because multiplication is expressed by the omission of an operator, simply by juxtaposing two factors, it should be treated as having a higher precedence than an operator whose presence is visible as an explicit glyph.

That is to say $abc/xyz$, should mean $(a\times b\times c)/(x\times y\times z)$. Basically we would like $abc/xyz$ to just be an in-line shorthand for ${abc}\over{xyz}$ with minimal fuss.

The interpretation $((((ab)c)/x)/y)/z)$ means we cannot do this. Expressing ${abc}\over{xyz}$ with a slash becomes cumbersome.

In computer programming languages, it is common for division and multiplication to have the same precedence and left-to-right associativity. But the situation there is different because multiplication is an explicit symbol like *. For instance in C a * b / c * d simply means (((a * b) / c) * d).

But if multiplication is notated by mere juxtaposition, a strong case can be made that it this juxtaposition should also denote "tighter binding" (higher precedence than any binary operator).

Mathematics notation is richly visual and two dimensional. Juxtaposition relationships matter. For instance the way an exponent binds with its base.

Answer 3

The American Physical Society gives multiplication a higher order than division in the slash notation:

When slashing fractions, respect the following conventions. In mathematical formulas this is the accepted order of operations:

1. raising to a power,
2. multiplication,
3. division,
4. addition and subtraction.

(Styling applied) (Source: Page 6 of Section IV of Physical Review Style and Notation Guide, retrieved 2012.10.14)

This means that (for them) $$ x/yz = x/(yz) $$ However, keep in mind that this may represent only the view of one group of scientists, I don't know of any established standard.

Answer 4

By the usual order of operations, it means $(x/y)z$. But people do often mean $x/(yz)$ instead, so you have to work out the intention from the context. :(

Answer 5

I was told once by a senior engineer that the last explicit symbol is what counts, it cancels the action of the previous one. So, for example, xyz/abc means $(x*y*z)/(a*b*c)$ and xyz/ab*c means $(x*y*z)/(a*b)*c$, as expressed by any standard computer language.

I was a little upset by that implicit convention, but soon found out that it worked well and was able to understand all old handbooks, where equations were written straight in a single line.

I suppose aesthetics is what counts here. I have the feeling that our need for explicit symbols came with a generation which had computer programming lessons at college, so a relatively new thing starting at what, the 60's, the 70's?

Answer 6

The link in the answer above by "@dualed" to the American Physical Society seems to be a dead link. Here is a current link, with some more information:

Section E.2.e of the Style Guide of the American Physical Society states that multiplication has higher precedence than division, implying that one should read $x/yz$ as $x/(yz)$.

I know I have read something similar from the American Mathematical Society in the past, but I am unable to find a current citation for corroboration.

I know of no programming language that treats x/y*z like x/(y*z). Rather, programming languages commonly treat x/y*z like (x*z)/y. Mathematica is one of these programming languages:

Wolfram Language 13.0.0 Engine for Mac OS X ARM (64-bit)
Copyright 1988-2021 Wolfram Research, Inc.
In[1]:= FullForm [ x / y z ]
Out[1]//FullForm= Times[x, Power[y, -1], z]
In[2]:= x / y z === x z / y
Out[2]= True

I know of a case, which shall remain anonymous, where this anomaly caused a failure in a robotic controller. An engineer wrote mathematics and a programmer transcribed it verbatim into code. It was in a corner case not caught in testing.

Some software companies mandate that all arithmetic and logical expressions be fully parenthesized.