Are there implicit parentheses around fractions?

Question

I was just looking at a simple math question that came across my social media feed and noticed something I must have always ignored in the past.

In this equation $x = 3 \div \frac{1}{3}$ there seems to be an implicit set of parentheses around $\frac{1}{3}$ which yields a result of $x= 9$.

This seems strange to me because I look at $\frac{1}{3} = 1 \div 3$ and wonder why we can't rewrite it as $x = 3 \div \frac{1}{3} = 3 \div 1 \div 3 = 1$

Is omitting the the parentheses just shorthand that I never realized I was using; $\frac{1}{3} \neq 1 \div 3$, but instead $\frac{1}{3} = (1 \div 3)$?

Should fractions come at the front of (F)PEMDAS/(F)BEDMAS?

Or is it just that the definition of a fraction entails that it is an individual unit and expressing $\frac{1}{3}$ as $(1 \div 3)$ doesn't really make sense either?

Answer 1

The mnemonics BEDMAS and PEMDAS have some utility, but they're not perfect. They work relatively well for parsing computer code, where the operands and operators are all written in one linear string. Even then, neither mnemonic literally describes the correct order of application of operations in all cases. Just look at the ambiguity between DM in one mnemonic and MD in the other.

As soon as we start arranging things in two dimensions we have the effect of implicit parentheses. There should be no doubt that $\frac 2{3-1}$ is meant to be evaluated like $2/(3-1)$ and not like $2/3-1.$ When we write $e^{2x}$ it means the multiplication $2x$ is performed before the exponentiation, regardless of the occurrence of E before M in the mnemonics.

The mnemonic really should be something like PE(MD)(AS), and I would say it only applies when we have no two-dimensional clues to say what operation applies first, such as in the expression $3 - 1 \times 3$ (which is $3 - (1\times 3),$ not $(3-1)\times 3)$) or in the expression $3x^2$ (which is $3\times(x^2),$ not $(3 \times x)^2$).

The two-dimensional clues I would take from a horizontal fraction bar would be that it always has implicit parentheses like this: $$ \left(\frac{( \cdots )}{( \cdots )} \right).$$

Even these implicit parentheses can be unclear in some cases, so we simply should not write expressions like $$ \frac{1}{\frac AB}, $$ which require carefully looking at the relative sizes of typefaces and slightly different lengths of horizontal lines.

Answer 2

Since $\frac13=1\div3$, we can write $3\div\frac13$ as $3\div(1\div3)$. But this is not the same thing as $(3\div1)\div3$, since division is not associative. In other words, you don't get the same thing if you compute first $b\div c$ and then you compute $a\div(b\div c)$ and if you compute first $a\div b$ and then you compute $(a\div b)\div c$.

It's just like subtraction:$$12-(5-2)=12-3=9,$$but$$(12-5)-2=7-2=5.$$

Answer 3

Yes, you can certainly say that there are implicit parentheses, since $3 \div \frac{1}{3}$ equals $3 \div (1 \div 3)$, not $(3 \div 1) \div 3$. It helps to think of $\frac{1}{3}$ as a number in its own right, an individual unit as you have suggested. It can be visualised as the proportion of the circle below that has been shaded:

When you divide $3$ by $\frac{1}{3}$, you are asking 'what number, when multiplied by $\frac{1}{3}$ gives you $3$?' $$ \frac{1}{3} \times \, ? \, = 3 $$ Here, the $?$ must be equal to $9$, since $$ \frac{1}{3} \times 9 = 3 \, . $$ and so $3 \div \frac{1}{3}=9$. The number $3$ can be visualised as shading in $3$ circles. Then, you are trying to think of how many sectors of area $\frac{1}{3}$ can be fit into $3$. If we count them up, we get $9$. Alternatively, note that $\frac{1}{3} \times 3 = 1$, and $1 \times 3 = 3$. Hence, $$ \left(\frac{1}{3} \times 3\right) \times 3 = 3 \, . $$ With multiplication, we are allowed to regroup the brackets*: $$ \frac{1}{3} \times \left(3 \times 3 \right) = 3 \, , $$ and so $$ \frac{1}{3} \times 9 = 3 \, . $$

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*The technical term for an operation where you are allowed to regroup the brackets is associative, as outlined in José Carlos Santos' answer. For instance, addition is associative: $$ (3+2)+7 = 3+(2+7) \, . $$ In general, $$ (a+b) + c = a+(b+c) \, . $$ Multiplication is also associative. Be careful, though: division and subtraction are not associative.