Binary Finger Counting

Counting in Binary

How useful is binary? If it can encode anything, can it encode numbers? Letters? Of course it can! Let’s try counting in binary just like you first tried counting in kindergarten...on your fingers.

How high can you count on one hand? Five? Ha! You can do much better. Using the idea of dichotomous relationships and binary, Binary Finger Counting allows you to count to numbers much larger than five using only the five fingers you have on one hand. Consider each finger to be either “extended” or “not.” Each of your fingers then represents one bit. Here’s a comic that explains the entire process:

Notice that each finger doesn’t represent a separate number, it represents a dichotomy, or division into two classes (e.g., “the number is <16 (if not extended), or ≥16 (if extended)"). Each new finger doubles the number of values that can be represented.

Even better, let’s see binary numbers dance:

https://www.youtube.com/embed/OCYZTg3jahU

Now try it yourself. Use the cartoon or video to help you along, or if you prefer written instructions, here is a procedure (or algorithm) that explains how you can encode a number in the range 0-31 on one hand:

  • The pinky represents whether the number is <16 (if not extended), or ≥16 (if extended).
    • So, if it is 16 or higher, extend your pinky, subtract 16 from your number, and continue with Step 2.
    • If it is lower than 16, just skip ahead to Step 2.
  • The ring finger represents whether the remaining value is <8 (if not extended), or ≥8 (if extended).
    • So, if it is 8 or higher, extend your ring finger, subtract 8 from your number, and continue with Step 3.
    • If it is lower than 8, just skip ahead to Step 3.
  • The middle finger represents whether the remaining value is <4 (if not extended), or ≥4 (if extended).
    • So, if it is 4 or higher, extend your middle finger, subtract 4 from your number, and continue with Step 4.
    • If it is lower than 4, just skip ahead to Step 4.
  • The index finger represents whether the remaining value is <2 (if not extended), or ≥2 (if extended).
    • So, if it is 2 or higher, extend your index finger, subtract 4 from your number, and continue with Step 5.
    • If it is lower than 2, just skip ahead to Step 5.
  • The thumb represents whether the remaining value is <1 (if not extended), or =1 (if extended).
    • So, if it is 1, extend your thumb, subtract 1 from your number, and you are done (you should have 0 as a remaining value).
    • You are done (you should have 0 as a remaining value).

Notice that each finger doesn’t represent a separate number, it represents a dichotomy or division into two classes (e.g., “the number is <16 (if not extended), or ≥16 (if extended)"). Each new finger doubles the number of values that can be represented.

Check your counting using this Hand Counting Applet.

Challenge

How many values could this guy encode?

HINT: Think about what value the sixth finger would represent. Go back and look at the pattern for the values represented by your own five fingers, and then apply it to the sixth finger.