Slide Rule

We can use rulers to add together numbers.
  1. Set the mode to "n".
  2. Move the first slider or input a value in the box next to it so that the 0 on the lower ruler lines up with one of the numbers you want to add on the upper ruler.
  3. Move the second slider or input a value in the box next to it so that the red line lines up with the other number you want to add on the lower ruler.
  4. The green line will line up with their sum on the upper ruler.
  5. The corresponding addition fact will show up below.
Change the mode to 10n to replace the number n with the number 10n.
  1. Multiplying two powers of 10 is the same as adding their number of zeroes, so this rule will effectively allow us to multiply numbers.
  2. Move the first slider or input a value in the box next to it so that the 1 on the lower rule lines up with one of the numbers you want to multiply on the upper ruler.
  3. Move the second slider or input a value in the box next to it so that the red line lines up with the other number you want to multiply on the lower ruler.
  4. The green line will line up with their product on the upper ruler.
  5. The corresponding multiplication fact will show up below.
  6. Check "Draw interpolating lines" to draw lines at round integer values.
  7. Check "Label integer lines" to label these lines.
Change the mode to 2n to replace the number n with the number 2n. Finally, if you dare, change the mode to "zany" to get a ruler that also works as a slide rule, but for unusual reasons.
Mode: Draw interpolating lines Label integer interpolating lines
Blue line:
Red line:

Circular Slide Rule

We can also use circular rulers to add or multiply numbers.
To add together two numbers:
  1. Set the mode to "Addition".
  2. Move the first slider or input a value in the box next to it so that the 0 on the inner wheel lines up with one of the numbers you want to add on the outer wheel.
  3. Move the second slider or input a value in the box next to it so that the red line lines up with the other number you want to add on the inner wheel.
  4. The green line will line up with their sum on the outer wheel.
  5. The corresponding addition fact will show up below.
To multiply together two numbers:
  1. Set the mode to "Multiplication". This will replace n by kn where k is the tenth root of ten (about 1.26).
  2. Move the first slider or input a value in the box next to it so that the 0 on the inner wheel lines up with one of the numbers you want to multiply on the outer wheel.
  3. Move the second slider or input a value in the box next to it so that the red line lines up with the other number you want to multiply on the inner wheel.
  4. The green line will line up with their product on the outer wheel.
  5. The corresponding multiplication fact will show up below.

Mode:

Questions to ponder

  1. Don't forget the important fact that makes this work! xn * xm = xn+m. Using the fact that xn is just x multiplied by itself n times, convince yourself that this equation holds when n and m are whole numbers.
  2. Can you work out how to use the rulers to subtract? To divide?
  3. If the circular slide rule "overflows" (wraps around past the start), it still tells you something about the answer. In the case of addition, what does the circular slide rule tell you about the answer? What about in the case of multiplication?
  4. You may have run into problems entering exact values into these interactives. Sorry! You can enter exact values by typing them into the textboxes. What did you notice when you entered values that were almost the ones you wanted? For instance, if I multiply a number that's about 2 by a number that's about 3, what result will I get?

Explorations elsewhere

  1. Antiquark has virtualizations of various real world slide rules. Real world slide rules were able to perform most of the computations required by engineers, not just multiplication. Here is a list of various computations that can be performed. For example, very often engineers needed to calculate the sine of a number and then multiply the result by another number. The S scale allows them to calculate the sine of a number and directly sets up the result to be multiplied by another number.
  2. The Slide Rule Museum provides a history of computation devices.
  3. Numberphile has a video on the use of slide rules and circular slide rules.
  4. Numberphile also has a video on the use of log tables, which allow you to multiply numbers using the same principle.
  5. Exploring powers of 10:
    1. Wikipedia has a number of useful articles on powers of ten for Distance, Time, and Number .
    2. The 1977 video "Powers of Ten" shows both zooming out and zooming in and the corresponding powers of ten.