\times vs. \cdot vs. parentheses vs. nothing while writing fraction multiplications

Question

Is there a convention around writing multiplication of fractions. Let us take two examples to be specific.

Example A: $ 2 \times \frac{1}{2} $

Here are some ways to write this example:

• $ 2 \frac{1}{2} $ (does not seem like a good idea because this means $2.5$)
• $ 2 \times \frac{1}{2} $
• $ 2 \cdot \frac{1}{2} $
• $ (2) \left( \frac{1}{2} \right) $

Example B: $ \frac{1}{2} \times \frac{3}{4} $

Here are some ways to write this example:

• $ \frac{1}{2} \frac{3}{4} $
• $ \frac{1}{2} \times \frac{3}{4} $
• $ \frac{1}{2} \cdot \frac{3}{4} $
• $ \frac{1}{2} \left( \frac{3}{4} \right) $
• $ \left( \frac{1}{2} \right) \left( \frac{3}{4} \right) $

Is there a preferred way of writing such multiplications in each case?

Answer 1

It’s purely a matter of personal preference, and perhaps of emphasis in context.

For your cases, I would typically use $$2\left(\tfrac12\right)$$ and $$\tfrac12\left(\tfrac34\right),$$ but you are likely to get other answers from other people.

I prefer to omit the multiplication symbol unless that makes the expression appear ambiguous or confusing.

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Addendum:

I point out that you have omitted the “in-line” form $a/b$ which will often require parentheses on its own to avoid ambiguity. For example, $$\tfrac{x+1}{x-1}$$ is entirely distinct from $$x+1/x-1$$ and I should have written the latter as $$(x+1)/(x-1)$$ if the former were intended.

Answer 2

When you only have one kind of multiplication I recommend making it $\cdot$ (or $\times$ if aimed at a young audience, those who teach team or normal adults), but even then you only need to show it if there's a risk people won't know multiplication is intended, e.g. in distinguishing $2+\frac12$ from $2\times\frac12$.

Some kinds of "multiplication" have preferred symbols, such as scalar products on vectors ($\cdot$) or the "cross products" on $3$- or $7$-dimensional vector spaces ($\times$). When that happens, I recommend using whichever of these symbols is unused for a second kind of multiplication you need to discuss, or $\otimes$ (\otimes) if you need a third; but if you do have multiple operators on the go, make their meaning explicit in the text unless it's very standard, e.g. $a\cdot b\times c$ with vectors.

If you have closely related operators, such as the multiplication operations on multiple levels of the Cayley-Dickson construction, you might use subscripts (or hope the reader can disambiguate themselves). For example, the dimension-$2^n$ algebra with multiplication operator $\otimes_n$ gives way to the next viz. $$(p,\,q)\otimes_{n+1}(r,\,s)=(p\otimes_n r-s^\ast\otimes_n q,\,s\otimes_n p+q\otimes_n r^\ast).$$(To my mind $\otimes_n$ works much better for this than $\cdot_n$ or $\times_n$, contra the above advice for cases without subscripts, partly due to kerning issues.) But personally, I think in that case it looks even better to omit the operators, like so: $$(p,\,q)(r,\,s)=(pr-s^\ast q,\,sp+qr^\ast).$$