This simple math problem went viral (about five million views) on YouTube:

6÷3(1+2)

What is your answer and its justification?

But I'm started.

The short answer:

If you're a high school math teacher or textbook writer, the answer is 6.
If you're a Nobel laureate physicist or college physics textbook writer, the answer is 6/9 = 2/3.

A little explanation:

The math teacher part comes from my wife, a retired math teacher. To quote "I've been teaching order of operations for many years and know that multiplication and division have the same precedence and proceed from left to right."

For the physicist part see the details section below.

The question really pertains to whether multiplication (either in general or merely when implied) takes precedence over division in the order of operations.

Why the difference? I can only conjecture:

Abstract mathematicians like symmetry even more than physicists do. Having equal precedence for multiplication and division rules mimic those for addition and subtraction adds symmetry.

On the other hand, the idea of order of operations is a practical consideration--its purpose is to avoid the clutter of having many parentheses. Those who are not abstract mathematicians, or teachers, find giving precedence to multiplication is a way to avoid more parentheses than giving division an equal footing:

If you have a term in a given formula that varies directly with some factors and inversely with other factors, just use one slash and put all the direct factors on the left and all the inverse factors on the right. With precedence for multiplication you don't need parentheses. If one of the factors itself is a sum or difference it of course needs parentheses, but the multiplication rule at least avoids nested parentheses.

The real answer:

There is no solution as the puzzle does not specify the system to be used. It's like sending a rocket to Mars with part of the controls using English measure and another part using Metric, without any tags saying what the units were, as actually happened to the detriment of the lander.

Details:

Blogger Presh Talwalkar  goes with the high schoolers, in explaining the similar problem, 6÷2(1+2):

<blockquote>If you type 6÷2(3) into a calculator, Google or WolframAlpha, the input has to be parsed and then computed. All of these will first convert the parentheses into an implied multiplication. The expression becomes the following.

6÷2(3)
= 6÷2×3

According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right. First take 6 and divide it by 2, and then multiply by 3.

6÷2×3
= 3×3
= 9

This gets to the correct answer of 9.

This is without argument the correct answer of how to evaluate this expression according to current usage.

Some people have a different interpretation. And while it’s not the correct answer today, it would have been regarded as the correct answer 100 years ago.

</blockquote>

Talwalkar then goes on to describe the physicist's version: treating 6÷2×3, or rather 6/2(3), as 6/(2*3). Indeed the symbol ÷ is rarely used today, and the controversy would be better using the slash / instead.

He does give a reference to a Slate article that's more even-handed than "This is without argument...", but seeing that even-handedness depends on his readers' following the link.

He also gives a reference to a 1917 article on order of operations to paint its usage as "obsolete". Follow the link within the preceding sentence, for that usage.

From Wikipedia (order of operations):

<blockquote>Exceptions

... [a discussion of unary minus] ...

Similarly, there can be ambiguity in the use of the slash symbol / in expressions such as 1/2x.[6] If one rewrites this expression as 1 ÷ 2x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:

1 ÷ 2 × x = 1 × ½ × x = ½ × x.

With this interpretation 1 ÷ 2x is equal to (1 ÷ 2)x.[1][7] However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[8] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[nb 1]

</blockquote>

To quote from that Physical Review set of submission instructions:

<blockquote>(e) When slashing fractions, respect the following conventions.In mathematical formulas this is the acceptedorder of operations:
(1) raising to a power,
(2) multiplication,
(3) division,
(4) addition and subtraction.
According to the same conventions, parentheses indicate that the operations within them are to be performed before what they contain is operated upon. Insert parentheses in ambiguous situations. For example, do not write a/b/c; write in an unambiguous form, such as (a/b)/c or a/(b/c).</blockquote>

And, referring back to my College Physics text, by Sears and Zemansky, Addison-Wesley, 1960, part 2, page 728:

Capacitive reactance, X_C = 1/ωC, is inversely proportional both to the capacitance and the frequency.

spelling out in words what the formula says. In other parts of the book the usage is made without such explicit wording to explain it.

The TI-82 calculator gave precedence to multiplication but the TI-83 went back to give equal precedence to division.

https://www.youtube.com/watch?v=JTTGYLYtQZI is a video, giving the correct answer as "there is no solution".

So Please Excuse Just My Dear (Aunt Sally).

Edited on August 9, 2018, 10:28 am

Posted by Charlie on 2018-08-09 09:41:01