1.02 Place value

Place value and decimals

Decimals are numbers with digits to the right-hand side of the decimal point on the place value table. With whole numbers each place value is ten times the previous place value if we move from right to left. So $10$10 ones make up one ten and $10$10 tens make up one hundred. This pattern continues through to tenths ($10$10 tenths makes up one unit), hundredths ($10$10 hundredths makes up one tenth), thousandths and many more!

The tenths column is the largest column after the decimal point. A digit in the tenths column indicates a greater quantity than the same digit in the hundredths column, which is greater than the thousandths column, and so on. In other words $\frac{8}{10}$810​  is bigger than $\frac{8}{100}$8100​ which is bigger than $\frac{8}{1000}$81000​ or $0.8>0.08>0.008$0.8>0.08>0.008. So as with whole number place values, the further to the left a column is, the larger its place value.

This means that for two decimals that are the same, but have different digits in the tenths column, then the larger decimal is the one with larger digit in the tenths column.

Worked examples

Example 1

Which decimal is bigger $0.87$0.87 or $0.23$0.23?

Think: We can think of $0.87$0.87 as $\frac{87}{100}$87100​ and $0.23$0.23 as $\frac{23}{100}$23100​. If we write them out like we say them aloud, "$87$87 hundredths" and $23$23 hundredths", then we can see one decimal has more hundredths than the other.

Do: So $0.87$0.87 is bigger.

Reflect: Another way we can work out which is bigger, is to compare the digits in the tenths column. $0.87$0.87 has an $8$8 in the tenths column, while $0.23$0.23 only has $2$2 in the tenths column. The tenths column is the largest column that the two decimals differ, so we can tell that $0.87$0.87 is the bigger decimal.

 

Example 2

Which decimal is bigger $0.3$0.3 or $0.15677$0.15677?

Think: In this case, saying the number aloud doesn't help us yet, "$3$3 tenths" and "$15677$15677 hundred thousandths". This is because the names are not the same. We could try to convert them to be numbers of the same name, this would give us $0.30000$0.30000 which is "$30000$30000 hundred thousandths" and "$15677$15677 hundred thousandths" - but maybe there is an easier way.  

As we just saw above, if we look at the value in the columns and compare them from left to right we can find the bigger (or smaller decimal).

One number has a $3$3 in the tenths column and the other has a $1$1, so the one with the $3$3 is bigger.

Do: So $0.3$0.3 is the bigger decimal.

Reflect: We scan each of the decimals from left to right to find the first time the digits of the decimals are different. This is because the further to the left a column is, the larger its place value.

 

Example 3

Which number is smaller $0.13$0.13 or $0.121$0.121?

Think: The tenths columns both have $1$1's so we need to look at the hundredths columns – one has a $3$3 and one has a $2$2 so the decimal with the $2$2 is smaller.

Do: So $0.121$0.121 is smaller.

Reflect: We can still do this question by making them have the same name, that being "$130$130 thousandths" versus "$121$121 thousandths". Here we can see that the $121$121 thousandths is smaller. 

 

Example 4

Which number is bigger $0.55$0.55 or $0.552$0.552?

Think: The numbers in the tenths and hundredths columns are the same in both numbers, so now we will look at the thousandths. $0.55$0.55 is the same size as $0.550$0.550 so it has a $0$0 in the thousandths column, while $0.552$0.552 has a $2$2 in the thousandths column.

Do: So $0.552$0.552 is bigger.

Reflect: And as before we can write the decimals in the following way, "$550$550 thousandths" versus "$552$552 thousandths". So here the $552$552 thousandths is bigger.

 

Ordering decimals

Once we know how to tell the size of numbers with decimals, we can arrange them in different orders.

Ascending order means from smallest to biggest. For example: $-1.3$−1.3, $1.2$1.2, $3.4$3.4. $6.8$6.8, $10$10

Descending order means from biggest to smallest. For example: $8.5$8.5, $6.2$6.2, $3$3, $0.2$0.2, $-4.1$−4.1

$0.24>0.22$0.24>0.22 is how we write $0.24$0.24 is greater than $0.22$0.22

$0.12<0.123$0.12<0.123 is how we write $0.12$0.12 is less than $0.123$0.123

Practice questions

Question 1

Question 2

Question 3

Question 4