Dividing by 2 numbers at once, what is the answer?

Question

Let's say i have 4/1/5. or 4 divided by 1 divided by 5. Are there any rules that i am allowed to use to stop any mistakes?, for example this has 2 solutions, 4/5 , and 20.

Edit: Thanks for your responses. It seems the rule of thumb is to start from the beginning and follow through the numbers, while any brackets should be dealt with beforehand.

In which case something like 1/2/3/4 is in fact 1/24.

Edit2: There's a small bounty for whoever can prove that there solution is always the case, and any misuse of it always fails

Answer 1

In the USA and Canada, perhaps other places too, we use the BEDMAS system for interpreting expressions. First, we calculate the result of bracketed sub-expressions. Then exponentiation. Then from left to right, resolving divisions or multiplications as they show up. Then from left to right, resolving additions and subtractions as they show up.

So in the BEDMAS convention, we would deal with your 4/1/5 by doing the divisions left-to-right. 4/5.

Edit:

As the comments point out, some use PEDMAS instead. P for parenthesis instead of brackets ;)

Answer 2

The notation should not be allowed. You should always write $(4/1)/5$ or $4/(1/5)$ (or better, write the fraction vertically).

That being said, at least the way I was taught in school in the US, with no parentheses, you should perform divisions in the order written, so the answer is $4/5$.

Answer 3

The double fraction $a/b/c$ should not be allowed because it does not make sense : which operation should we do first ? To make sense, you must add parenthesis (this way, you specify which operation is done first). There are two ways to do it :

$$\frac{\Big( \displaystyle \frac{a}{b} \Big)}{c} = \frac{a}{bc} = (a/b)/c $$

and

$$\frac{a}{\Big( \displaystyle \frac{b}{c} \Big)} = \frac{ac}{b} = a/(b/c). $$

Therefore, $\displaystyle \frac{\Big( \displaystyle \frac{a}{b} \Big)}{c} \neq \frac{a}{\Big( \displaystyle \frac{b}{c} \Big)}$ in general. Be careful to the way you put parenthesis. In general :

$$\frac{\Big( \displaystyle \frac{a}{b} \Big)}{\Big( \displaystyle \color{red}{\frac{c}{d}} \Big)} = \frac{a}{b} \times \color{\red}{\frac{d}{c}} = \frac{ad}{bc}.$$

Answer 4

Generally speaking, a slash '/' is used to state a denominator. So, by convention a/b/c/d/e... should in most cases be interpreted as a/(b*c*d*e...). However, the statement is inherently ambiguous and ill-advised. The reader will, in almost all cases (be they living or machine) be the ultimate decider as to what it means. Therefore, never use such an expression if its interpretation may not be clear (especially if you know a better way) unless correctly understanding is not important.

Answer 5

Operator Associativity is a core concept of mathematical standard notation. This is a very well defined domain and not the kind of "convention voodoo" implied in many of the answers given here. It is a property of a defined operator that needs to be learned alongside its other properties and cannot be deduced.

Edit:

Most of the confusion here stems from the fact that division is really defined as the multiplicative inverse of a given algebra (which is associcative). Because this abstraction is too hard to grasp for most people the division operator is introduced (non-associative). This needed the inclusion of operator associativity (division is left-associative in school algebra [other systems can define it differently, but its always defined]). Using multiplicative inverse we get:

4 * (1/1) * (1/5)

where order doesnt matter (remember, multiplication is associative).

Edit2:

The multiplicative inverse is defined as that element in a ring that when you multiply the inverse with the original that the result must be the neutral element regarding multiplication. The neutral element is that element that satisfies that being multiplied with anything will result in the same thing. The neutral element of the ring of rational numbers is 1.

So given an element of any ring q, its multiplicative inverse i and the neutral element regarding multiplication e the following must hold:

q * i = e

q * e = q

From this simple rule we define the division operator and it must follow the same behaviour.

Answer 6

This isn't really a question of mathematics, but of communication. Some people here are quite sure how to interpret $4/1/5$ correctly, but they don't all agree. Importantly, if I read this expression, even if I know how to interpret it correctly, I don't know if the person writing it knew.

In other words, this kind of expression should be avoided. Either $(4/1)/5$ and $4/(1/5)$ are one hundred percent clear.

Answer 7

First you have to notice that you are dealing with notation. The problem is not about the meaning of the $/$ symbol. Nor about how to evaluate expressions. Here you are at the step when a sequence of symbols is given and should be interpreted to form a meaningful expression. Notice that while the commutative, distributive and associate laws are mathematical properties of the operations, the evaluation order has nothing to do with the operations involved.

You have three possibilities when parsing the string A/B/C:

1. the string should be interpreted as the expression $(A/B)/C$;
2. the string should be interpreted as the expression $A/(B/C)$;
3. the string is not well formed and should be left undefined.

The correct answer is 1. It is a natural convention to associate simbols on the left, following the order in which they are written. So this was recognized as the correct way to interpret this notation. As @hagenvonheitzen as pointed out, this is very natural if you replace division with subtraction... the use of the / symbol is a little bit less natural because we were used to other symbols when we were first taught division. Maybe A:B:C would be more natural. In the era of computers, the / symbol has been choosen for division in every programming language I know, like * has replaced $\times$. In every programming language the string 'A/B/C' is correctly interpreted from left to right. The shift from mathematics to informatics makes sense, since we are speaking of parsing which is a problem largely studied in this field.

About 2. To my knowledge there is nobody who has deliberately made this choice. This would be against intuition, against other convention and against the vast majority of people who as choosen the first interpretation. No way.

We can speak about 3. One could decide that A/B/C cannot be interpreted as a valid notation. Something like AB/C/. It seems that many people here agree with this answer. However think a bit about how to make it precise. If you make this choice, please tell me: what is the rule to follow to decide when the expression is valid and when it is not? Is such a rule easier to grasp than the rule of left associativity? Do you think that it would be good to teach such a rule at school? If you have common sense the answer should be: no.

Because here the point is not if you should or shouldn't use that expression. We all agree that whenever possible: you should'nt use it! The reason being that some people could interpret that notation in a wrong way. However if you are asked to give your interpretation, the answer is: associate from left to right.

addendum. A different story would be about the notation $A^{B^C}$ where most people agree that the better choice is to associate on the right: $A^{(B^C)}$. But also in this case it is good to make a choice and not let the notation be undefined.

addendum 2. A different story would also be the notation: $$ \frac{A}{\frac{B}{C}}. $$ In this case the interpretation is $A/(B/C)$ because the lower bar is shorter than the upper one. For this notation it makes sense to decide that the bar should always be longer than the expressions to which it applies because it cost nothing (non parenthesis are needed) and gives no ambiguity. As a consequence an expression where the two bars have equal length should be rejected as undefined.

Answer 8

It is a matter of notation convention. / is not really much of mathematical notation as it is a typewriter shorthand for a fraction.

But shorthands are not formalized as much as proper mathematical notation. As one interesting data point, see what the METAFONT program designed by none other than Donald Knuth does:

mf
This is METAFONT, Version 2.718281 (TeX Live 2013/Debian)
**\relax

*show 1/2/3/4;
>> 0.66667
*end
Transcript written on mfput.log.

So METAFONT indeed interprets this as (1/2)/(3/4). However, this high affinity for / only works for literal integers: x/y/w/t is interpreted as (((x/y)/w)/t) even when x=1, y=2, w=3, t=4.

Now this is a program written by a renowned mathematician. Morale? Don't rely on any specific interpretation by the reader when using / but rather parenthesize.

Answer 9

YouTube: The Order of Operations is Wrong

Take 8-2+1 for example. In reality, it actually equals (8/1)+(-2/1)+(1/1).

Applying the same logic to 4/1/5 you get:

 (4/1)/(1/1)/(5/1)
=(4/1)x(1/1)x(1/5)
=(4x1x1)/(1x1x5)
=4/5

With (P|B)EDMAS, 6/3/3 could be:

 6/(3/3)
=6/1
=6

or

 (6/3)/3
=2/3

whereas 6/3/3 actually equals

 (6/1)/(3/1)/(3/1)
=(6/1)x(1/3)x(1/3)
=(6x1x1)/(1x3x3)
=6/9
=2/3

Answer 10

According to the international standard ISO 80000,

(…) a solidus (/) shall not be followed by a multiplication sign or a division sign on the same line unless parentheses are inserted to avoid any ambiguity.

$\frac{a/b}{c} = \frac{a}{bc} = ({a/b})/c = a/({bc})$, not $a/b/c$

$\frac{a}{bc} = a/({b \cdot c})$, not $a/b \cdot c$

Answer 11

$4\div1\div5=\dfrac45$, because division (using the division sign) is done from left to right.

$4/1/5$ is more ambiguous and confusing, and should never be written. But if I had to give it a value, I'd go with the above rule and say $\dfrac45$, too.

But writing (4/1)/5 or 4/(1/5) (depending on what you mean) is always preferable to 4/1/5 (unless you don't actually mean fractions—like, if you mean the date or something).