How to Use a Slide Rule. Computer Simulation 2 - Multiplication and Division

(1) To calculate A x B = C Click the top ruler and drag it until its left edge is over A on the bottom ruler (green line) . Click on the small black bar at the top, to move the vertical indicator(the hairline) until it reads B on the top ruler (red line). The answer C is given by the location of the vertical indicator on the bottom ruler (blue line). . x . = .

(2) To calculate C / B = A Reverse the A x B = C procedure Move the hairline to C on the bottom ruler (blue line). Move the top ruler until the hairline is over B on the top ruler (red line). The answer A is where the left edge of the top ruler touches the bottom ruler (green line). . / . = .

(3) Wraparound mode

For results larger than 10 and less than 100, for example with (click to set ruler) the bottom ruler is too short to show the answer.

Instead of using a longer ruler, mode (click to set ruler) is used.

Wraparound mode is entered when the top ruler is moved further left than the bottom ruler. The right edge of the top ruler is used to set the value of A, not the left edge. This is equivalent to setting the left edge of the virtual (light yellow) ruler (green line).

Similarly, setting the real hairline (black) at 2 on the top ruler (red line) is equivalent to setting the virtual hairline (blue) on 2 on the virtual (light yellow) ruler (magenta / pink line).

The answer 16, (cyan / light blue line) indicated by the virtual hairline (blue) on the bottom virtual (light yellow) ruler, is equivalent to reading the real hairline (black) on the gray virtual ruler. Which is equivalent to reading the real hairline (black) on the real bottom (yellow) ruler = 1.6 (blue line), and multiplying by 10.

To summarize, imagine the virtual hairline and rulers, wrapping round the right edge of the real rulers, and coming back at the left edge.

The whole procedure is a little weird, so play around a bit .Move the top ruler and the hairline, watch how the virtual hairline and lines change, to get used to how it all works. Come back another day and give it another shot. It gets easier. Really.

(4) Numbers of any size

We have seen how to multiply numbers with results up to 100 (1 to 10 normal mode, 10 to 100 wraparound mode). A slide rule can actually be used to multiply numbers of any size, to get results of any size, with some manual estimation by the user.

Consider:

• 1.6 x 3.2 = 5.12
• 16 x 3.2 = 51.2
• 1.6 x 32 = 51.2
• 16 x 32 = 512
• 160 x 32 = 5120
• 160 x 3.2 = 512
• 0.016 x 0.32 = 0.00512

The numbers in the result are the same, only the location of the decimal point is different.

Example 1

160 x 32000 can be rewritten in scientific notation as

1.6 x 10² x 3.2 x 10⁴

Which can be rewritten as

1.6 x 3.2 x 10⁶

by adding the powers of 10.

The result of 1.6 x 3.2 can be directly calculated using the slide rule, as 5.1 (the slide rule on this page is accurate to only 2 digits). The final answer is therefore:

1.6 x 3.2 x 10⁶ = 5.1 x 10⁶

(we can estimate that 1.6 x 3.2 is less than 10, so the answer is not 5.1 x 10⁷

Example 2

Multiplying: 467 x 529

Rewriting in scientific notation:

467 x 529 = 4.67 x 10² x 5.29 x 10² = 4.67 x 5.29 x 10⁴

Rounding to two signficant digits gives

4.7 x 5.3 x 10⁴

Using the slide rule (in wraparound mode),

4.7 x 5.3 = 2.5

Which is obviously wrong as the answer should be close to 4 x 5 = 20. So we need to shift the decimal point. (Remember also that when in wraparound mode, you need to multiply by 10)

4.7 x 5.3 = 25

So the answer is

467 x 529 = 25 x 10⁴ = 2.5 x 10⁵

How it works

Notes

• Rounding errors and the limited resolution of the computer monitor mean that (a) the exact integer number sometimes cannot be set using the hairline or the top ruler, and (b) sometimes the scales of the two rulers don't line up exactly. Real slide rules are more accurate but still suffer from the same errors, only to a smaller amount.
• The log scale causes the larger numbers to bunch up at the right hand end of the scale, resulting in larger inaccuracies.
• The smaller numbers on the left have the opposite problem - the resolution there is "wasted" without additional, smaller, divisions.

(This page uses a JavaScript Drag Drop library from http://www.walterzorn.com/dragdrop/dragdrop_e.htm)