Mixed Fractions and Multiplication (with Variables)

Question

I stumbled over this expression: $3 \frac{1}{x^3}$. How should you interpret something like that?

While you could see that as implicitit multiplication ($3 * \frac{1}{x^3}$), you could also argue that $3 \frac{1}{x^3}$ is a mixed fraction ($3 + \frac{1}{x^3}$).

I think in situations with only numbers or only variables everything should be clear:

$3 \frac{1}{2} = 3 + \frac{1}{2} = 3.5$

$a \frac{b}{c} = a * \frac{b}{c}$

This should also be true: $3 \frac{b}{c} = 3 * \frac{b}{c}$.

But what do our conventions say to something like $3 \frac{1}{x^3}$ or $6 \frac{x}{3}$ or $\frac{1}{x^2}5$? Is there any written standard that you should generally follow? Multiplication or addition?

Answer 1

This can be written as $$\frac{3}{1}\cdot \frac{1}{x^3}=\frac{3\cdot 1}{1\cdot x^3}=\frac{3}{x^3}$$

Answer 2

When there is no operator between expressions the usual interpretation is multiplication, so $3\frac 1{x^3}=3\cdot \frac 1{x^3}=\frac 3{x^3}$

Mixed fractions are an exception to this, so $3\frac 13 \neq 3 \cdot \frac 13$. Instead it is $3 \frac 13=3+\frac 13$.

If there is any possibility of confusion one should supply the operator, but sometimes people do not. In that case you need to figure it out from context.

Answer 3

I have never seen mixed fractions used with variables and it definitely means multiplication.

The point of a mixed fraction is that you can better see the magnitude of a number (e.g. $2\frac 5{17}$ is obviously between $2$ and $3$ but can you tell at a glance with $\frac{39}{17}$?). One usually also requires that the fraction be proper For this reason, i.e. the numerator smaller than the denominator and both of them natural numbers.

With something like $3\frac{1}{x^3}$, none of this would be guaranteed. Writing it would be similar to writing $3.x^2$ to mean the number that you get by writing the digits of $x^2$ after the decimal point and $3$ in front of it.

As an anecdote, in my fourth semester at university, we had to take a course on numerical mathematics. At one point, a teaching assistant wrote a mixed fraction on the board and it took all of us student quite some time to even remember that that notation exists.