3.01 Decimals and place value

Partitioning numbers means breaking numbers down in to two or more parts (or splitting them into smaller units). To do this, we must understand the value of each digit in a number.

There are different ways of partitioning a decimal number. For example, 1.23 is equal to

• 1 one, 2 tenths, and 3 hundredths (1+0.2+0.03 or 1+\dfrac{2}{10}+\dfrac{3}{100})

• 1 one and 23 hundredths (1+0.23 or 1+\dfrac{23}{100})

• 12 tenths and 3 hundredths (1.2+0.03 or \dfrac{12}{10}+\dfrac{3}{100})

By breaking up numbers into smaller components, we can more easily compare which numbers are larger. Understanding of this will also make addition and subtraction of decimals easier.

Using place values, we can compare the relative sizes of decimal numbers in a similar way to how we compare whole numbers - the difference being that we can't necessarily just look to see which number has a digit in the higher place value. While this method works for numbers greater than 1, it won't work for numbers less than 1 as there will always be a digit in the tenths column.

Consider the numbers 12.34 and 2.34. We can see that 12.34 is larger than 2.34 as it has a number in the tens column, and 2.34 doesn't.

Now consider 0.123 and 0.19, we can see they both have a 1 in the tenths column, so can we just compare 123 to 19? If so, 123 is definitely a larger number than 19, so 0.123 is clearly larger. But is it really?

Let's stop and think about what these numbers mean by representing them both in a place value table.

First let's write 0.123 in a place value table.

We can see that 0.123=\dfrac{1}{10}+\dfrac{2}{100}+\dfrac{3}{1000}.

We could also express this as \dfrac{12}{100}+\dfrac{3}{1000}=\dfrac{120}{1000}+\dfrac{3}{1000} or simply \dfrac{123}{1000}.

Now let's write 0.19 in a place value table.

Notice that we included a 0 in the thousandths column to show that there were no thousandths in the number, and also to make sure the two numbers we are comparing are the same length.

We can now see that 0.19=\dfrac{1}{10}+\dfrac{9}{100}, which we could express as \dfrac{10}{100}+\dfrac{9}{100}=\dfrac{100}{1000}+\dfrac{90}{1000} or simply \dfrac{190}{1000}.

This means we are actually comparing 123 thousandths to 190 thousandths, so in fact the number 0.19 is greater than the decimal number 0.123.

If we put both numbers in the same place value table it becomes obvious when we compare columns from left to right:

The second number has more hundredths, so it is the larger value.

Examples

Multiplying or dividing by powers of 10 (10, 100, 1000, and so on) is much easier than multiplying by other numbers, because our number system is base 10. Let's have a look at what this means.

Consider the number 253. We can write this as:

253=2\times 100+5\times 10+3\times 1

When we multiply 253 by 100, we can write it like this:

\begin{aligned} \left(2\times 100+5\times 10+3\times 1\right)\times 100 &=2\times 10\,000+5\times 1000+3\times 100\\ &=25\,300 \end{aligned}

Notice that when we multiplied by 100 the place value of the "5" changed from tens to thousands, since 10\times 100=1000. In fact, all of the digits moved up by two place values.

Multiplying by 10 increases the place of each digit by one place.

Dividing by 10 decreases the place of each digit by one place.

Multiplying or dividing by 100=10\times 10 moves the values by two places, multiplying or dividing by 1000=10\times 10\times 10 moves the values by three places, and so on.

You may have to add or take away zeros, and you may have to add or take away the decimal point, after you have finished moving the values. This depends on which columns the values end up in.

Examples