Calculating with numbers: Decimal numbers
Arithmetic operations with decimal numbers
Arithmetic operations with decimal numbers are closely related to arithmetic operations with natural numbers: you can calculate with natural numbers and add the decimal point in the right place afterwards.
When adding or subtracting decimal numbers you put the numbers in columns aligned on the decimal point, one below the other, and then you calculate columnwise as if they were natural numbers and insert the decimal point at the right place at the end of the calculation, i.e at the place where all decimal points are already aligned at the start of the computation.
When multiplying two decimal numbers, you first omit the decimal point in the calculation and put the numbers right aligned below each other. You then calculate with natural numbers and put the decimal point at the right place afterwards; to determine the number of decimal places of the final result, you only need to add the number of decimal places of the two numbers you started with.
Division of decimal numbers is done by a continued long division, where you first multiply the divisor and the dividend by a power of 10 so that the new dividend is a natural number. When dividing decimal numbers, the remainder only plays a role in rounding at the end.
Examples illustrate the arithmetic operation with decimal numbers.
The correct answer is \(122.8\).
You learned in elementary school to add such numbers, but perhaps not in the most convenient and systematic way. The best recipe is as follows:
- Place the numbers right-aligned below each other
- Add the units (digit in the right column): \(8+8+9+6+7=38\), and write of this number only \(8\) on the bottom line and move \(3\) to the top of the next column (the tens).
- Add the tens, i.e. add the middle digits of a three-digit number, together with the additional \(3\): \(3+1+5+2+7+4=22\). Again you write \(2\) on the bottom line and move \(2\) to the top of the next column on the left.
- Add the hundreds together with the extra \(2\): \(2+6+3+1=12\). The outcome at the bottom line and you're done with the calculation of the outcome \(1228\).
The following diagram illustrates the steps in the calculation; we remove at the end the extra digits added on the top.
\[\begin{aligned}
&\textit{tenths:}\qquad \\ \begin{array}[t]{rl} & \\ 61.\blue{8} & \\ 5.\blue{8} & \\ 32.\blue{9} & \\ 7.\blue{6} & \\ 14.\blue{7} & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ \blue{3.8} & \end{array} \quad &\longrightarrow\qquad \begin{array}[t]{rl} \phantom{6}3\phantom{.8} & \\ 61.8 & \\ 5.8 & \\ 32.9 & \\ 7.6 & \\ 14.7 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ .8 & \end{array}
\\ \\
&\textit{units}\qquad \\
\begin{array}[t]{rl} \phantom{6}\blue{3}\phantom{.8} & \\ 6\blue{1}.8 & \\ \blue{5}.8 & \\ 3\blue{2}.9 & \\ \blue{7}.6 & \\ 1\blue{4}.7 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ \blue2\blue{2}.8 & \end{array} \quad &\longrightarrow\qquad \begin{array}[t]{rl} 23\phantom{.8} & \\ 61.8 & \\ 5.8 & \\ 32.9 & \\ 7.6 & \\ 14.7 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ 2.8 & \end{array}
\\ \\
&\textit{tens:}\qquad \\
\begin{array}[t]{rl} \blue{2}3\phantom{.8} & \\ \blue{6}1.8 & \\ 5.8 & \\ \blue{3}2.9 & \\ 7.6 & \\ \blue{1}4.7 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ \blue{12}2.8 & \end{array} \quad &\longrightarrow\qquad \begin{array}[t]{rl} \phantom{.8} & \\ 61.8 & \\ 5.8 & \\ 32.9 & \\ 7.6 & \\ 14.7 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ 122.8 & \end{array}
\end{aligned}\]
Experienced persons do not write the additional numbers at the top of the columns anymore, but they memorise then and add them immediately in their brains. They write:
\[\begin{aligned} 8+8+9+6+7=38,\quad &\blue{8\text{ written, }3\text{ memorised.}} \\ \blue{3}+1+5+2+7+4=22, \quad &\blue{2\text{ written, }2\text{ memorised.}} \\ \blue{2}+6+3+1=12.\quad &\blue{\text{ready after placing the decimal point; outcome}=122.8.}\end{aligned}\]
