Calculating with numbers: Decimal numbers
Ordering of decimal numbers
Decimal numbers, which are actually fractions, lie on the number line; see the figure below in which you can move the red dot to point to the decimal number on the number line. Sometimes you do not see decimal notation: that is when you point to a natural number.
You can see that a positive decimal number is larger the more the number is to the right on the number line:
- \(2\) is greater than \(1\) and you write \(2>1\).
- \(1.72\) is greater than \(1.68\) and you write \(1.72>1.68\)
- \(0.5\) is less than \(0.6\) and you write \(0.5<0.6\)
In order to compare two positive decimal numbers, we write them aligned to the decimal point below each other and optionally add zeros to decimal places to get an equal number of decimal places in the numbers and optionally zeros at the front of a number to make the number of digits of the two numbers equal to each other. Next, we compare the two decimal numbers digit by digit from left to right. At the first differing digit, the number with the lowest digit at that place is the smallest number.
Examples
\(4.321<12.34\) because addition with zeros \[\begin{aligned}\blue{0}4.321\\ 12.34\green{0}\end{aligned}\] gives \(\blue{0}<1\).
\(12.34<12.43\) because for the first different digits from left to right, \(3<4\) applies.
