Decimals are one way of expressing numbers that are smaller than one. Look at the columns to the right of the decimal point in the place value table below:
Examples
Question 1
Evaluate: What is the place value of the $4$4 in $23.47$23.47?
Think: We can see that the $4$4 is in the tenths column, so the place is tenths and the value of the digit is $4$4.
Do: We can multiply the place of the digit by the value of the digit to find the place value.
So the place value of the $4$4 is equal to:
$4\times\frac{1}{10}=\frac{4}{10}$4×110=410
We can also combine the place and the value to be written as:
$4$4 tenths
Or we can simply ignore all the other digits except $4$4 and see what its value is. If we do this, we get:
$00.40$00.40
So the place value of the $4$4 is either: $4$4 tenths or $\frac{4}{10}$410 or $0.4$0.4
We removed the unnecessary zeros from $00.40$00.40 to get $0.4$0.4 because removing those zeros did not change the value of the number.
Question 2
Evaluate: What is the place value of the $6$6 in $55.006$55.006?
Think: The $6$6 is in the thousandths column so the place is thousandths and the value of the digit is $6$6.
Do: We can multiply the place by the value to find the place value.
Similar to the previous example, we can see that the value of the $6$6 can be represented as either:
$6$6 thousandths or $\frac{6}{1000}$61000 or $0.006$0.006
Expanded notation
Writing numbers that include decimals in expanded notation is just like what you have done before using whole numbers. You just write the values off the place value table.
Examples
question 3
Evaluate: Write $6384.92$6384.92 in expanded notation
Think: Which columns do these numbers belong in on the place value table?
Do: We would write this as $6000+300+80+4+\frac{9}{10}+\frac{2}{100}$6000+300+80+4+910+2100
Question 4
Evaluate: Write $1059.004$1059.004 in expanded notation
Think: Which columns do these numbers belong in on the place value table?
Do: $1000+50+9+\frac{4}{1000}$1000+50+9+41000
Writing decimals
We can also use our knowledge of the place value table to change numbers that are written in words to numbers.
Examples
question 5
Evaluate: Write seventeen and four tenths as a decimal
Think: The tenths column is the first column after decimal point
Do: $17.4$17.4
question 6
Evaluate: Write six and seventy-seven hundredths as a decimal.
Think: $77$77 hundredths will be written in the first two columns after the decimal point
Do: $6.77$6.77
Decimals in pictures
We can use this knowledge about the place value table to interpret information about decimals from pictures.
Examples
In the picture above, $6$6 out of the $10$10, or $6$6 tenths of the squares are shaded. So in decimal form, we would write this as $0.6$0.6.
In the picture above, $43$43 out of hundred, or $43$43 hundredths of the squares are shaded. So in decimal form, we would write this as $0.43$0.43 .
Conversions using decimals
When we multiply or divide by numbers like $10$10, $100$100 or $1000$1000, we do so by increasing or decreasing the place value of each digit. The number of zeros in the number we are multiplying or dividing by tells us how many places each digit will move. When we are multiplying, the place values will increase and when we are dividing, the place values will decrease.
Examples
Question 7
Evaluate: $0.4\times100$0.4×100
Think: Multiplying by $100$100 means our number is getting bigger- the digits will increase by two place values.
| Hundreds | Tens | Units | . | Tenths | Hundredths |
|---|---|---|---|---|---|
| $0$0 | . | $4$4 | $0$0 | ||
 |
|||||
| $0$0 | $4$4 | $0$0 |
Do: $0.4\times100=40$0.4×100=40
Notice that we added an extra zero in the Hundredths column so that the units column would not be empty after moving the digits. Similarly, we removed the zero in the Hundreds column since removing it doesn't change the value of the number.
question 8
Evaluate: $34\div10$34÷10
Think: We need to decrease the place value of each digit by one place.
Do: $34\div10=3.4$34÷10=3.4
We can use this in everyday contexts as well.
Examples
question 9
Evaluate: Convert $16$16 cents into dollars.
Think: There are $100$100 cents in a dollar so we need to divide $16$16 by $100$100, which means we decrease the place value of each digit by two places.
Do: $16\div100=\$0.16$16÷100=$0.16
question 10
Evaluate: Convert $13$13 metres into kilometres.
Think: There are $1000$1000 metres in a kilometre so we need to divide $12$12 by $1000$1000, which means we decrease the place value of each digit by three places so that the number is $1000$1000 times smaller.
Do: $13\div1000=0.013$13÷1000=0.013km