Multiplication – Napier’s method

John Napier of Merchiston, 8th Laird of Merchistoun, 1550 to 1617. Scottish mathematician, physicist, astronomer and astrologer. From Crabb's Historical Dictionary published 1825.
Credit: Ken Welsh/Bridgeman
Portrait of Scottish mathematician, John Napier (1550 – 1617)

This method uses a grid to multiply numbers.

Multiplying a two-digit number with another two-digit number would require a {2} by {2} grid, for example {43}\times{26}.

Multiplying {264}\times{53} would require a {3} by {2} grid.

Follow the steps below to see how Napier’s method is used to calculate {43}\times{26}.

  • The first step is to draw a {2} by {2} grid.
  • The second step is to draw a diagonal in each box. The diagonal line separates the tens and the units. Always write the tens above the diagonal line in each box.
  • Start by multiplying {4} and {2} to fill the left box on the top row.
  • {4}\times{2}={8}
  • Write {0} and {8} to show that there are no tens.
Multiplication diagram - Napiers' method: 4 x 2 = 08
  • Now multiply {3} and {2} to fill the right box on the top row.
  • {3}\times{2}={6}
Multiplication diagram - Napiers' method: 3 x 2 = 06
  • Next, multiply {4} and {6} to fill the left box on the bottom row.
  • {4}\times{6}={24}
  • Remember to put the {2} above the diagonal.
Multiplication diagram - Napiers' method: 4 x 6 = 24
  • Complete the grid by multiplying {3} and {6} to fill the right box on the bottom row.
  • {3}\times{6}={18}
Multiplication diagram - Napiers' method: 3 x 6 = 18

After completing the grid, add the columns along the diagonals, starting at the bottom-right. We sometimes need to carry over from one diagonal to the next. To get the answer, read the totals down the left and to the right.

Multiplication diagram - Napiers' method showing that 43 x 26 = 1,118

{43}\times{26}={1,118}

Question

Use Napier’s method to calculate {264}\times{53}.

Draw a {3}\times{2} box and follow the guidelines in the previous example.

Multiplication diagram - Napiers' method showing that 264 x 53 = 13,992

{264}\times{53}={13,992}

Did you notice that there’s no need to carry over in this example?