Between any two numbers there are infinitely many numbers. This is because the number system is continuous so it has no gaps between numbers. This also applies to decimals, we can find a number between any two decimals no matter how close the decimals seem to be. Let’s look at how we can find a number between two decimals.
Worked example 1
Find a decimal between $0.5$0.5 and $0.6$0.6.
Think: To compare two decimals, we must start at the left-most digit. The first digit in the units column is the same: $0$0. So we move on to the digits in the tenths column which are $5$5 and $6$6. There is no whole number between $5$5 and $6$6. Since the decimals have the same number of decimal places, we can just add a zero to each decimal to make them easier to compare.
Do: So the decimals become $0.50$0.50 and $0.60$0.60. If we ignore the decimal point for a moment, we can come up with a number between $50$50 and $60$60, such as $51,52,53,....,59$51,52,53,....,59.
Therefore, a decimal between $0.50$0.50 and $0.60$0.60 is $0.52$0.52.
Worked example 2
Find a decimal between $0.42$0.42 and $0.5$0.5.
Think: To compare two decimals, we must start at the left-most digit. The first digit in the units column is the same: $0$0. So we move on to the digits in the tenths column which are $4$4 and $5$5. There is no whole number between $4$4 and $5$5. Since the decimals have a different number of decimal places, we can add a zero to the second decimal to make them easier to compare.
Do: So the decimals become $0.42$0.42 and $0.50$0.50. If we ignore the decimal point for a moment, we can come up with a number between $42$42 and $50$50, such as $43,44,45,....,49$43,44,45,....,49.
Therefore, a decimal between $0.42$0.42 and $0.50$0.50 is $0.45$0.45.
Worked example 3
Find a decimal between $1.2$1.2 and $2.001$2.001.
Think: To compare two decimals, we must start at the left-most digit. The first digit in the units column is the not the same. There is no whole number between $1$1 and $2$2. Since the decimals have a different number of decimal places, we can add two zeros to the first decimal to make them easier to compare.
Do: So the decimals become $1.200$1.200 and $2.001$2.001. If we ignore the decimal point for a moment, we can come up with a number between $1200$1200 and $2001$2001, such as $1350,1409,1500,1890,\dots$1350,1409,1500,1890,…. So you can choose any number between $1200$1200 and $2001$2001, and just put a decimal point after the first digit, like the original decimals have.
Therefore, a decimal between $1.200$1.200 and $2.001$2.001 is $1.409$1.409.
Worked example 4
Find the decimal exactly half-way between $0.85$0.85 and $1.4$1.4.
Think: To find a decimal that is exactly half-way between two decimal, we need to add them and divide them by two. To do this we must first add a zero to the second decimal so that they have the same number of decimal places.
Do: So the decimals become: $0.85$0.85 and $1.40$1.40.
Now we must add the decimals:
| $0.85+1.40$0.85+1.40 | $=$= | $2.25$2.25 |
Then divide the sum by two:
| $2.25\div2$2.25÷2 | $=$= | $1.125$1.125 |
So the decimal half-way between $0.85$0.85 and $1.4$1.4 is $1.125$1.125.