How to correctly do a division using a slash?

Question

What's the correct way to do division using a slash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$?

Basically, if I place a slash, will I divide by what's directly behind the slash $(b)$, the term that's behind the slash $(bc)$, or everything after the slash $(bc+d)$?

Answer 1

We use the order of operations:

1. parentheses/brackets

2. exponents

3. division, multiplication applied left to right

4. addition, subtraction applied left to right

$$a/bc = \frac ab \cdot c$$

Since division and multiplication have equal precedence, we apply division only for $a/b$, then multiply by $c$.

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Since parentheses have highest precedence, we compute, first, $bc$, then divide $a$ by $(bc)$.

$$a/(bc) = \frac a{bc}$$

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$$a/bc + d = \frac ab \cdot c + d$$ $$a/(bc) + d = (a/(bc)) + d = \frac a{bc} + d$$

$$a/(bc + d) = \dfrac{a}{(bc) + d}$$

When in doubt, use parentheses! The few extra keystrokes it takes makes certain how the expression is to be evaluated is worth sparing you from ambiguity or unintended calculations, etc.

Answer 2

Use parentheses. Often it is clear what is meant, but not always. Computer languages I know are explicit that $a/b*c=(a/b)c$ . Definitely the $+d$ gets added at the end.

Answer 3

It is a long-standing convention of mathematics that $a/bc$ means $a/(bc).$ By reducing the clutter of parentheses, this convention, along with writing $\sin 2x$ rather than $\sin(2x)$ and so on, makes mathematics more readable for humans. However, spacing is important here, and you need to check that the LaTeX output is right in this respect if you use these conventions. For example, $a\!/\!b\,c$ suggests $(a/b)c$ rather than $a/bc$. Notice that, in this simple example as in other cases, the default settings of LaTeX respect the conventions, as Donald Knuth doubtless intended.

Quite the opposite is the case for computers. They need to parse their input strings character by character, and are unfazed by unnecessarily cluttered notation.

In real-life mathematics, a fine balance has to be struck. Too much emphasis on ever-present nailing down of meaning in notation will daunt and slow the reader (perhaps to a grinding halt), while eloquently spare notation may baffle the reader as to what interpretation is intended.