Aids to calculation

In a positional system like our own, calculations with pencil or pen and paper are easy, but in a non-positional notation like that of the Romans they are much more difficult. To see this more clearly, consider the multiplication of 325 by 47. Using our numerals, the calculation is simple: <math display='block'> <mrow> <mfrac> <mtable columnalign='left'> <mtr> <mtd> <mn>325</mn> </mtd> </mtr> <mtr> <mtd> <mn>47</mn> </mtd> </mtr> </mtable> <mtable columnalign='left'> <mtr> <mtd> <mn>2275</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mn>1300</mn> </mrow> <mrow> <mn>15275</mn> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfrac> </mrow> </math>$$\frac{\begin{gathered} 325 \hfill \\ 47 \hfill \\ \end{gathered} }{\begin{gathered} 2275 \hfill \\ \frac{{1300}}{{15275}} \hfill \\ \end{gathered} }$$ Now try to multiply CCCXXV by XLVII. In the former system, one had to multiply every single figure contained in 325 with every figure in 47 (altogether 6 simple multiplications), to write the partial results in the right places, and to add them. If one tries to do the same thing with CCCXXV and XLVII, the first problem that arises is that XLVII cannot be decomposed into parts X + L + V + I + I, because the notation XL is subtractive. One can try writing XXXX instead of XL and try computing the product CCCXXV.XXXXVII by a method similar to today’s, multiplying every single component C -or- X or V contained in the first factor by every component X or V or I of the second factor. This method, however, will involve 42 single multiplications, followed by the addition of the results. A very cumbersome method!