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# Infallible Calculator?

You may have seen the claim on the Internet that millions of people get the answer to the equation  $6 \div 2\left(1 + 2\right)$  (or similar) wrong. You may have even read the explanations by some people about why the answer should be 9 or should be 1. Let me clear it up and explain why the answer is most definitely 1, but why the calculator (and others) give 9.

Firstly, the important thing to recognise is that this equation, like all others, is subject to the Order of Operations, which states that we must solve the contents of the brackets first. Thus our equation becomes  $6 \div 2(3)$ .

Next comes the tricky bit and where the errors occur and why there is argument from some.

When a number is outside of a bracket, this implies multiplication, so  $2\left(3\right)$  means two multiplied by three. Now, if we take this to mean our equation is  $6 \div 6$  then clearly the answer is  $1$ .

However, if we take it to mean  $6 \div 2 \times 3$  then, depending on whether we multiply or divide first determines whether our answer is  $1$  or  $9$ . Putting it into a calculator directly as  $6 \div 2 \times 3$  will result in an answer of nine and is why so many argue (incorrectly) that this is the correct answer.

The correct answer is most definitely  $1$  because of an often forgotten rule of mathematical operational precedence. Notice that there is no multiplication sign between the two and the contents of the bracket. It is mathematical convention that this implies that the two and the bracket contents are inseparable, however, when trying to solve using a calculator we are required to insert a multiplication sign. It is this insertion of the multiplication sign that leads to the calculator’s result of nine. If we follow mathematical convention fully and treat the two and bracket contents as inseparable, then we are really being asked to find the value of  $6 \div \left(2\left(1 + 2\right)\right)$  which is clearly equal to one.

Another good demonstration of where failing to follow the conventions of mathematics properly gives the wrong answer is in using your calculator to find the square of negative three, i.e.  $-3^{2}$ . If you use a modern calculator, it will most likely give you the answer of  $-9$ . Unfortunately, the correct answer should be POSITIVE nine, since  $-3 \times -3 = 9$ . The calculator is not faulty, but, rather it is applying the Order of Operations correctly, interpreting that we have asked for the negative value of three squared when in fact we want the square of negative three. Once again, we need to insert brackets (or parentheses) when entering into the calculator to make the meaning clear and so we should enter  $\left(-3\right)^{2}$ .

The morale of the story is that CALCULATORS ARE ONLY AS GOOD AS THE INSTRUCTIONS THEY ARE GIVEN. A key skill of any mathematician is the ability to estimate a result. For example, if asked for the value of  $\sqrt{57}$ , you should be able to estimate that the answer is between 7 and 8 (since  $7^{2} \textrm{ \textless } 57 \textrm{ \textless } 8^{2}$ . An even better estimate would be that the answer should be around  $7 \frac{1}{2}$ . Of course, if we check it with our calculator, we find that  $\sqrt{57} \approx 7.5498$ . If our calculator gave us a value smaller than seven or larger than eight, we would have known immediately that something was wrong.

If you want to learn more about the Order of Operations or Estimation, arrange a session today.

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