### Calculating with numbers: Computing with integers

### Long division

The following examples explain how to carry out long division. They are in ascending order of degree of difficulty. With larger dividens you must repeat calculation steps more often.

We describe how to do this in steps:

The long division is a tree-stage rocket in this case: \[\begin{aligned} \begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{11}9 & \!\!\! \Bigm{\backslash} \end{array}\qquad\qquad & \begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{11}9 & \!\!\! \Bigm{\backslash} \; 3 \\ & \underline{9}\phantom{0} & \\ & \phantom{0}2\phantom{0} & \end{array} \qquad\quad & \begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{119} & \!\!\! \Bigm{\backslash} \; 39\;\;\blue{\leftarrow \textit{quotient}} \\ & \underline{9}\phantom{0} & \\ & \phantom{0}29 & \\ & \phantom{0}\underline{27} & \\ & \phantom{00}2 & \blue{\leftarrow \textit{remainder}}\end{array} \end{aligned} \]

- Start with writing the division as \(\textit{divisor} \Bigm/ \textit{dividend} \Bigm{\backslash}\)

In this case, this is \( 3 \Bigm/ 119 \Bigm{\backslash}\)

To the right of the backslash will appear the quotient digit by digit.

Below the dividend will grow with the computational details, with the remainder in the division at the end. - Search in \(119\) the shortest initial part of the digits forming a number greater than or equal to the divisor \(3\): that is \(11\) in this case.

You can indicate this by underling the starting part: \( 3 \Bigm/ \underline{11}9 \Bigm{\backslash}\) - Calculate the largest multiple of the divisor \(3\) that is less than or equal to the number \(11\): that is \(3\times 3 =9\). Write \(3\) as first digit after the backslash in the top line, and write \(9\) under the initial part \(11\) and then subtract this number. This step is visualised below. \[\begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{11}9 & \!\!\! \Bigm{\backslash} \; 3 \\ & \underline{9}\phantom{0} & \\ & \phantom{0}2\phantom{0} & \end{array}\]
- Underline now the next digit (here \(9\) ) in the dividend \(119\) and also concatenate this digit to the \(2\) on the bottom line, so that the number \(29\) is formed. Repeat now step three with this number.

Calculate the largest multiple of the divisor \(3\) that is less than or equal to the number \(29\): that is \(9\times 3 =27\). Write \(9\) as next digit in the top line, and write \(27\) under \(29\) and then subtract this number. On the bottom line you get \(2\) and this is the remainder of the division.

This step is visualised below. \[\begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{119} & \!\!\! \Bigm{\backslash} \; 39 \\ & \underline{9}\phantom{0} & \\ & \phantom{0}29 & \\ & \phantom{0}\underline{27} & \\ & \phantom{00}2 & \end{array}\] - The quotient equals \(39\) and the remainder equals \(2\).

The long division is a tree-stage rocket in this case: \[\begin{aligned} \begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{11}9 & \!\!\! \Bigm{\backslash} \end{array}\qquad\qquad & \begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{11}9 & \!\!\! \Bigm{\backslash} \; 3 \\ & \underline{9}\phantom{0} & \\ & \phantom{0}2\phantom{0} & \end{array} \qquad\quad & \begin{array}[t]{rrl} 3\;\Bigm/ \!\!\! & \underline{119} & \!\!\! \Bigm{\backslash} \; 39\;\;\blue{\leftarrow \textit{quotient}} \\ & \underline{9}\phantom{0} & \\ & \phantom{0}29 & \\ & \phantom{0}\underline{27} & \\ & \phantom{00}2 & \blue{\leftarrow \textit{remainder}}\end{array} \end{aligned} \]

The result of division with remainder is: \(119=39\times 3+2\) (you can use this to check the answer), that is \(119 : 3 = 39\textit{ remainder }2\).

This implies the following exact result of division in mixed notation: \(119 : 3 = 39\tfrac{2}{3}\).