What is a/b/c/d? That is: What is the correct order for multiple consecutive division operations?

Question

Unlike with addition, subtraction and multiplication where the order of operations will lead to an unambiguous result when the rules of BIDMAS (or PEMDAS) are applied, this does not hold true for division.

For example: 5 + 3 x 5 x 7 - 2 = 5 x 7 x 3 + 5 - 2 = -2 + 5 + 7 x 5 x 3

Of course, BIDMAS states we must group our multiplications together but within that grouping the order of multiplications makes no difference to the result. Furthermore, following the rules of BIDMAS and performing the multiplications first, the subsequent order of the arithmetic also makes no difference. Right?

In the above example, for instance, 3 x 5 x 7 = 105. So: 5 + 105 - 2 = -2 + 105 + 5, etc...

However, operation order DOES affect the result when dividing! For example:

(1/4)/27 does NOT equal 1/(4/27).

We know this, right!

I always took it for granted that the order of operations when presented with multiple divisions must be performed in the order it is presented; ie from left to right. That is what we were taught at kindergarten, right?

So:

a/b/c/d = ((a/b)/c)/d

Right?

Well that's what I believed all my life and took this for granted. But now my sanity is being challenged by Casio's most recent scientific calculators which do not obey this rule!!!!

If you were to type in a/b/c/d into a modern Casio scientific calculator using its fraction key for '/' rather than the division button, the calculator yields a/(b/(c/d))!!!!!

Using the division button instead of the fraction key gives a conflicting (but what I believe to be the correct) answer; that is:

a divided by b divided by c divided by d = ((a divided by b) divided by c) divided by d

I feel sure that the older Casio scientifics (pre 1991) gave the correct result following the rules as I have explained above. But I cannot get hold of one to check my theory.

So my question is: What is the correct order of operation with multiple consecutive divisions? That is, what is the order of division for:

a/b/c/d

as written without any parenthesis? Are the new Casio calculators giving the wrong answer or have I been living a lie?

PS: I am aware that the calculators give what I believe is the correct answer if the right cursor key is pressed after the keying in of each denominator. Bu that is a fudge. By using the right cursor, you are applying implicit parenthesis.

As inputted, either the calculators are giving the wrong answers or I have been living a lie. Can someone please confirm the order of division should indeed be from left to right!

PS: I have searched everywhere I could on the internet and in many maths text books but have not been able to find anything that addresses the ambiguity in the order of multiple consecutive division operations.

Answer 1

Operators of equal priority are evaluated left-to-right. For commutative operators, it makes no difference, but for the non-commutative ones, it does.

$$a+b-c+d=((a+b)-c)+d$$

$$a\times b\div c\times d=((a\times b)\div c)\times d$$

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The left-to-right ordering was naturally adopted on calculators in the early days (with all operators at the same priority) because it allows immediate execution and no "pending operation".

E.g.

$$3+5\times2-1\div 5=8\times2-1\div 5=16-1\div 5=15\div 5=3$$

vs.

$$3+5\cdot2-\frac15=\frac{64}5.$$

Even with priorities, the left-to-right rule remains.

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In mathematical notation, $a/b/c/d$ is not used. Instead, you would write something like

$$\frac{\dfrac ab}{\dfrac cd}$$ or $$\dfrac a{\dfrac b{\dfrac cd}}$$ or $$\dfrac{\dfrac{\dfrac ab}c}d$$ where the order is subtly indicated by the bar lengths (and vertical alignment).

Answer 2

While addition and subtraction is conventionally evaluated left-to-right, multiplication and division isn't. If we have a sequence of multiplications and divisions without parentheses, and there is a division which is not the rightmost operation, then the expression is inherently ambiguous.

Thus: $5\cdot 4\div 3$ is unambiguous. $5\div 4\cdot 3$ is ambiguous. There may be convention on this, but it is not universal, which means that any time you see such an expression you cannot be certain that the author followed the convention, which makes it ambiguous. Turn it around, and I recommend that any time you write such an expression you use fractions and brackets to clarify.

PS: see this youtube video by minutephysics for something more or less reflecting my view on this focus on rules for order of operations.

Answer 3

The easisest way to intrepret $``/x"$ is to turn it into $``\cdot \dfrac 1x"$. In which case $a/b/c/d = a \cdot \dfrac 1b \cdot \dfrac 1c \cdot \dfrac 1d = \dfrac{a}{bcd}$. This agrees with Wolfram Alpha's answer after identifying $b$ and $d$ as variables. In the same manner, $c/2\pi = c \cdot \dfrac 12 \cdot \pi= \dfrac{c\pi}{2}$. This too can be verifies using Wolfram Alpha.

The main reason for order of operation rules is to remove any ambiguity in notation. Very often, the human brain sees what it wants to see, not what it is supposed to see. Since computers came along, the rules for order of operation evolved into the rules for a computer compiler to use to parse a line of code. If there is any disagreement with what an expression means, then, rules or no rules, if you want to be understood clearly, write the expression so that it can be understood clearly.

In his book, Surely You're Joking Mr. Feynman, Richard Feynman describes how me developed new symbols for trigonometric and logarithmic functions.

I thought my symbols were just as good, if not better, than the regular 
symbols ­­it doesn't make any difference what symbols you use (sic) ­­ but I 
discovered later that it does make a difference. Once when I was explaining 
something to another kid in high school, without thinking I started to make 
these symbols, and he said, "What the hell are those?" I realized then that 
if I'm going to talk to anybody else, I'll have to use the standard symbols, 
so I eventually gave up my own symbols