Identity Elements

There are two signs, and they each go with the numbers that follow them. so we have $\div11$ and $\times22$ . But what sign goes with the $5$ ?

Since the question involves only multiplication and division, the sign must be a multiplication or division sign. But which is it? Well, we can write something at the beginning of any multiplication or division expression that will include an extra sign, but will not change its value. We can write $1\times$ in front of the expression:

$5\div11\times22=1\times5\div22$

We see that number 1 is called the $\textbf{identity element for multiplication}$ , because putting $1\times$ in front of an expression does not change its value. To rearrange the sample expression above, we simply start with $1$ , and then do the rest of the operations in any order we want. However, if you think about it, we cannot start by dividing, so we must choose either $\times5$ or $\times22$ as our first operation. Although either one will work, $22$ is a better choice than $\times5$ .

Here is why: We know that we will have to divide by 11 at some point, and the sooner the better. This way, we keep the running result as small as possible. Doing divisions as early as possible keeps the running result as small as possible. So the order $1\times22\div11\times5$ puts the pair $\times22\div11$ as close to the front as possible. The only other choices are $1\times22\times5\div11$ and $1\times5\times22\div11$ , and you will notice that the numbers become large in each of these orders if you calculate them from left to right. Therefore, the best arrangement for calculation is:

$5\div11\times22=22\div11\times5$ $=2\times5$ $=10$