# What is implicit multiplication

## Viral math puzzle: what to do if arithmetic symbols are missing?

This math problem made the rounds on the Internet.

There are two groups, one is sure the answer is 16 and the other is sure that the answer is 1. Some members of both groups accuse the other group of mathematical incomprehension and worse. And why can you get different results in a math problem and persist in sticking to your own result when you get an explanation of why you are wrong? The task is, in a sense, not a serious problem. No mathematician would write anything like that. You can immediately avoid the ambiguity of the arithmetic expression by using brackets:

**(8÷2)(2+2)** is clearly 16 and **8÷(2(2+2))** is clearly 1. Some people think of the parentheses in the original expression as follows: **(8÷2)(2+2)** and others like this: **8÷(2(2+2)),** and so the two groups come to different conclusions. The hardcore faction, for whom the result is 16, claims that they learned the following rules in school (Laub et al., Textbook of Mathematics, 1st and 2nd grade, 1965 edition):

"If first and second level arithmetic types occur in a calculation, do: *first* the bills in brackets, *on this* the second level arithmetic and *last* the first-stage arithmetic operations. To step *just* Calculation types of a level, so calculate from left to right in the given order. "Addition and subtraction are calculation types of the first level, multiplication and division are calculation types of the 2nd level. So these rules were learned in 1965.

In the follow-up textbook (Reichel et al., Das ist Mathematik 1, 2007 edition) it says: "The second level calculation types have priority over the first level calculation types. In short: 'Point calculation before line calculation'. However, the rule in brackets applies before the priority rule: What in Brackets is to be calculated first. "

In 2007, you no longer learned the rule that arithmetic operations of the same rank should be processed from left to right. Without this rule, however, the arithmetic expression is not unique.

### Bracket before point before line

One thing is clear: First of all, you have to calculate the expression in the brackets. Then the task becomes the calculation **8÷2(4)**

In expression **8÷2(2+2)** and in **8÷2(4) **but there is no multiplication sign! But hardly anyone would write it like that. The notation that you can omit multiplication signs in front of brackets is typically only learned when you learn algebra (i.e. expressions with variables). So let's first write multiplication signs in the appropriate place in the original exercise: **8÷2⋅(2+2)**

If you are to calculate that according to the rules from the 2007 school book, the task is not clearly set. It could **(8÷2)⋅(2+2)** or** 8÷(2⋅(2+2))** be meant. According to the system of rules from the school textbook of 1965, it is clear that **(8÷2)⋅(2+2) **is meant.

But there is another ambiguity. The omission of parentheses is an algebraic notation convention, and many people "feel" that the expression **2a** is to be read differently than the expression **2⋅a.** The feeling is that placing two symbols next to each other expresses a far stronger cohesive force than a multiplication sign. I suspect that a lot of people (perhaps even ignorant of the left-right rule) will be familiar with the spelling **8÷2⋅(2+2) **would calculate from left to right.

But the problem is a pseudo-problem, halfway serious mathematicians would never write anything like that; they would use fractions and write like this:

Such expressions also occur in programming languages, and in most of them the original expression without the multiplication sign would be a programming error.

### International "problem"

Incidentally, the problem was not only discussed in Austria and Germany. The British mathematician Hannah Fry was interviewed in the Daily Mail, and the mathematician Steven Strogatz wrote an article and a follow-up article in the New York Times. Both come to very similar conclusions as mine.

When I saw the expression for the first time, I thought to myself: This was written by someone who either lacks an understanding of the meaning and purpose of mathematical notation, or someone who deliberately wanted to lead others off the beaten track and possibly the heated discussions watched with pleasure on Twitter. (Erich Neuwirth, August 14, 2019)

**Erich Neuwirth** is a statistician, mathematician and computer science educator. Before retiring, he was the head of the IT Didactics Center at the University of Vienna.

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