Multiplication and division by 10

What happens to a number when it is multiplied by ten? Let’s use the number 14,9 [14.9], for example. Let’s start by placing it in a positional system. The digit one represents one ten, the digit four represents four units, and the nine shows that we have nine tenths. When you multiply something by ten, each digit becomes ten times as big.

The number “one” here started out with a value of ten. When it becomes ten times as big, its new value becomes one hundred. That moves it to the hundreds position, one step to the left. The four started out showing how many units we have. When it becomes ten times bigger, it also moves one step to the left, now showing how many tens we have.

In the same way, the nine tenths become nine units. 14,9 · 10 = 149 [14.9 · 10 = 149] There is another way of thinking of multiplication by ten: “moving the decimal comma” [point] one step to the right. This gives the same result. But it’s more accurate to say that the digits are moving to the left, while the decimal comma [point] stays in place. When a number is multiplied by a hundred, each digit increases in value one hundred times.

This means that each digit is moved two steps to the left. From tens to thousands. From units to hundreds. And from tenths to tens. Now it’s empty in the units position.

Then we need to write a zero there. 14,9 · 100 = 1 490 [14.9 · 100 = 1,490] So how do you divide by ten or by a hundred? Let’s take the number 6,4 [6.4] divided by one hundred, for example. The number 6,4 [6.4] has a four in the tenths position, and a six in the units position. If you divide something by one hundred, each digit value becomes one hundredth of what it was before.

The four tenths become four thousandths. This moves the digit two steps to the right. The six units become six hundredths, also moving two steps to the right. Now we have zero units and zero tenths. These zeroes need to be written in.

The result is 0,064 [0.064] To summarise: When you multiply a number by ten, each digit in the number is moved one step to the left in the positional system. If you multiply by one hundred, the digits move two steps, one thousand moves them three steps, and so on. Dividing by ten means that the digits move one step to the right, dividing by a hundred moves them two steps, and so on. And it’s not the decimal comma [point] that’s moving, but the digits that get new places in the positional system.