Music and Math Project

Calculating in the Land of Twelve

Examples

If you want to figure out what the sum \[ 2\mathrm{X}_{12} + 41_{12} \] is, you could first translate back to numbers represented in the decimal system, do the calculation and translate back. This translation is a little cumbersome and in fact it is unnecessary.

Sums of small numbers

This brings us back to those early days in school, in which we first look at sums of numbers that are smaller than, in this case, twelve. If the result is larger than twelve, you first determine what you need to fill up to twelve, and then you add what you have left.

A sum of a big number and a small number

Suppose you want to calculate the sum of a big and a small number, for instance \(30\mathrm{X}_{12} + 8_{12}\). First, you see that if you add \(2_{12}\) to \(30\mathrm{X}_{12}\), you reach the next multiple of twelve, namely \(310_{12}\). Because \(8_{12}-2_{12} = 6_{12}\), you still need to add \(6_{12}\). So the result is \[ 30\mathrm{X}_{12} + 8_{12} = 30\mathrm{X}_{12} + 2_{12} + 6_{12} = 310_{12} + 6_{12} = 316_{12}. \]

Addition in columns

If you want to add two or more big numbers, you can write the numbers on top of each other and add in columns. \[ \begin{array}{cccc} 5& \mathrm{Y} & 6 &\\ 2& 3 & 8 & + \\ \hline & & ? & \end{array} \] Start by adding the numbers in the rightmost column. Remeber that we are working with base \(12\)! So \(6_{12} + 8_{12} = 12_{12} \). This means we can put \(2\) on the rightmost spot in the last line, and we need to remember the \(1\), so we put it in the column to the left. \[ \begin{array}{cccc} & 1 & & \\ 5 & \mathrm{Y} & 6 &\\ 2 & 3 & 8 & +\\ \hline & ? & 2 \end{array} \] We now add the numbers in the second column from the right. The result is \( 1_{12} + \mathrm{Y}_{12} + 3_{12} = 13_{12} \). So we put a \(3\) on the last line and again remember a \(1\). \[ \begin{array}{cccc} 1 & 1 & & \\ 5 & \mathrm{Y} & 6 &\\ 2 & 3 & 8 & +\\ \hline ? & 3 & 2 \end{array} \] The numbers in the leftmost column now add up to \(8_{12}\). So we put an \(8\) on the last line, and we found the result! \[ \begin{array}{cccc} 1 & 1 & & \\ 5 & \mathrm{Y} & 6 &\\ 2 & 3 & 8 & +\\ \hline 8 & 3 & 2 \end{array} \]

Exercises

Calculate the following sums:
  1. \(9\mathrm{Y} + 52\)
  2. \(153 + 4\mathrm{XX}\)
  3. \(\mathrm{X}07 + 659\)
  4. \(46 + 5\mathrm{Y} + 30 + 81\)
  5. \(2240 + \mathrm{YX}75\)