Linear Number Line

First we will look at the properties of a normal number line. You will see below a number line with a mark for every integer, and a numbered mark every 5 integers. Notice how the steps between all numbers are equally spaced. The distance between 0 and 5 is the same as 95 and 100. Two identical number lines are shown below.

• Drag the slider and move the lower number line so that its 0 aligns with the 5 of the top.

By moving the bottom number line by 5, we have established a linear relationship between the two number lines. Note that for every pair of numbers, the upper number is always 5 greater than the lower. If we make the difference between the rules 20, every pair will again show the upper number is 20 greater than the lower. All pairs of normally spaced number lines have this property.

Non-Linear Number Line

Next we will look at number lines with a logarithmic scale, where the position of the markings for a number x is calculated by taking log base 10 of x. The integers 1 through 10 are shown, with markings indicating tenths. In this case, the numbers are not equally spaced, but instead are increasingly close together. Specifically, the numbers at the end of the scale are 10x closer than at the start.

• Drag the slider to move the lower number line so that its 1 lines up with the upper line's 2.

We have moved the bottom slider by 2x its original position, establishing a non-linear relationship between the two number lines. Note that for every pair of numbers, the upper number is always 2 times the lower. You can think of this as a 2 to 1 ratio. If we align the 3 on top with the 1 on the bottom, the difference between the rules will be 3 times. Every pair will have a 3 to 1 ratio.

The special spacing of these number lines transforms distance on the line into an act of multiplication. By placing two such number lines on a slide rule, we are able to perform a number of calculations, the most useful of which are multiplication and division.

Multiplication

In order to multiply two numbers, we need only travel down the rule the distance of the first number, plus the distance of the second number. In the case of 2.0 × 1.5:

1. Drag the 1 of the lower rule until it aligns with the first number, 2.0
2. Travel down the lower rule until you reach the second number, 1.5
3. The number across from it, on the upper rule is the answer, 3

Division

Division is the exact same process, but in reverse. In the case of 7.0 ÷ 5.0:

1. Align the first number 7.0 on the upper rule with the second number 5.0 on the lower rules.
2. Return to the 1 on the lower rule
3. The number across from it, on the upper rule is the answer, 1.4

Units and Significant Digits

You may wonder how to perform calculations with numbers that are either smaller or larger than the ones shown on the scale. When using a slide rule, numbers must be multiplied or divided by 10 until they match a number on the rule. This is essentially the same as moving the decimal point. Once you have calculated the answer, you must then adjust the decimal point in your answer to preserve the scale of the original numbers.

For example, 15 × 20 must be carried out with the numbers 1.5 and 2.0 on the rule. Once you have the answer 3.0, you must add back the required number of 10's places. When multiplying, you add the number of 10s. So in this case, 1 zero from the number 15, and 1 zero from the number 30, so you add 2 zeros to the answer 3.0 to get 300.

In the case of division, such as 400 ÷ 20, you perform the calculation using 4.0 and 2.0 to find the intermediate answer 2.0. When dividing, you subtract the number of 10s on the denominator from that of the numberator. So that would be two 10s from 400, minus one 10 from 20, so we end up adding back one 10's place to find the answer 20.

Watch Slide Rule Bezel

Many aviation watches have an integrated slide rule bezel in a circular layout. There are some differences, such as the scale beginning with 10, and the numbers looping around. Aside from the presence of various conversion indicators, both rules are identical, and you can perform multiplication and division using either rule as the "top" or "bottom". Instead of 1 being your reference point, you will use 10.

Care must be taken when calculation of the non-zero digits wraps clockwise or counter-clockwise past the 10 indicator, as this will adds a 10 to or remove a 10 from the answer. Often, the easiest solution is to use your intuition about the scale of the answer, as you need only determine if the correct number is 10x smaller or 10x larger than the alternatives.

You may also find a bezel has additional markings for carrying out common conversions between different units. The watch I based the slide rule below on has conversions for minutes and hours, liters and gallons, nautical/statute miles and kilometers, and pounds and kilograms. To perform the calculation, simply align the indicator on the bezel with the amount of that unit on the other bezel, and find the value pointed to by the corresponding indicator.