When the internet argues about maths and calculators disagree

Livia Gamper

24.5.2019

Translation: machine translated

People shouting at each other online about every gugus is nothing new. But the fact that a discussion about a mathematical problem generates thousands of comments is. Not only people, but also calculators come up with different results. Why is that?

A mathematical problem has been haunting the internet since 2011. For so long, people have been arguing in forums and everyone knows better. Currently, the problem of a mathematical equation can be found again on various online platforms. In a thread on Facebook, more than 1000 people have been banging on about maths.

The corpus delicti on the Facebook page Word Porn:

At first glance, the equation 6÷2(1+2) appears to be quite simple. It contains a division, a multiplication and an addition in brackets. The parenthesis is dissolved and the result from the parenthesis is multiplied. That's how we all learnt it in primary school. Actually. Because it's not that simple after all.

While some say that the result is 9, others are sure that the equation is 1.

The problem

A simple equation like this actually only has one result. But this is debatable. The problem is the notation, which leads to a different result depending on the notation rules used. What is clear is that the equation can be rewritten as follows:

6÷2(1+2)=6÷2(3)
The term in the brackets must be calculated first.

But then the equation can be rewritten in two different ways: Either (6÷2)×3=9 or 6÷(2×3)=1

In the first equation, division is performed first, in the second, multiplication is performed first. Most people - and most calculators - solve the equation from left to right. This results in the solution 9, but there are certain notation rules which, when applied, make this solution incorrect. The process of applying notation rules is called parsing. Put simply, a parser interprets a character string.

The rule of implicit multiplication

The notation of the equation is ambiguous. It is either
(6÷2)×3=9 or 6÷(2×3)=1. There are two interpretations which state that the second notation, i.e. the result 1, is correct. The first is based on implicit and explicit multiplication.

In some textbooks, implicit multiplication takes precedence:
In mathematics, implicit multiplication is the notation 2(1+2), i.e. the notation expressed without an operator. An explicit multiplication would be the notation 2×(1+2).

This means that 6÷(2×3)=1 must be calculated. So if this parsing rule is applied, the result is 1. However, this rule can only be found in some and not all algebraic textbooks.

Why pocket calculators disagree

Some calculators also apply this rule when parsing - i.e. interpreting - the equation. Others do not. Even calculators from the same manufacturer calculate differently.

But it's not because of the Casio brand. The calculators from Texas Instruments do not achieve the same result either. The TI-80, TI-81, TI-82 and TI-85 models are programmed to prioritise implicit multiplication. The TI-83, TI-84 Plus, TI-89 and TI-92 models, on the other hand, do not, with them you get the result 9.

A second problem: different division signs

The problem is unfortunately not solved with the interpretations of implicit and explicit multiplication. This is because division can be represented using different notation. There is the obelus ÷ and the slash /. In the past, division denoted by ÷ was treated differently from division denoted by /. Instead of evaluating divisions and multiplications from left to right, ÷ would indicate that everything to the left of it should be divided by everything to the right of it. You can also think of it like writing everything on a fraction line.

With the obelus, i.e. the way the equation is posed in the forums, the result would again be 1. But none of the above rules is widely used or generally accepted.

The solution

. The internet will probably be arguing about the correct solution for a long time to come. Only someone is wrong. Namely the person who formulated the equation. Why is that?

The purpose of mathematical notation is to be unambiguous so that it is interpreted in the same way by everyone - this is clearly not the case here. The equation should be accompanied by a kind of instruction manual. <p

Livia Gamper

Editor

livia.gamper@digitecgalaxus.ch

Testing devices and gadgets is my thing. Some experiments lead to interesting insights, others to demolished phones. I’m hooked on series and can’t imagine life without Netflix. In summer, you’ll find me soaking up the sun by the lake or at a music festival.

These articles might also interest you