Counting on Your Fingers

When you learned to count in elementary school, you probably started by counting on your fingers. At that time, you were told that you have 10 fingers, which allows you to count from 1 to 10. Unfortunately, your parents and teachers lied to you: they probably left out the number 0, and you can actually count as high as 35, not 10.

Video Lecture

The Number Zero

When you first learned to count, you probably were taught to start at the number 1. Using your fingers, you would stick out one finger. Note that which finger you used to represent the number 1 wasn’t standardized: it was whichever finger your parents or teachers taught you, or which one you preferred to use yourself. We’ll get to that detail in a moment, but first we need to address the number zero.

If you think about how you were taught to count on your fingers, you probably recall being able to count a total of 10 different numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. However, what happens if you don’t extend any of your fingers? If no fingers are up, isn’t that the number zero?

Speaking of the number zero, zero is a distinct number from the other 10. Therefore, even using your elementary school way of counting on your fingers, you can actually count 11 different numbers, including zero. This observation is extremely important in computing, as we often are tasked with representing information using the least amount of storage space possible. Every distinct value therefore counts.

Also, as you probably noticed when programming, we normally count from zero in computing, not from one. The reason why we do this has to do with the way we arrange lists, or arrays, of data in memory. In many programming languages (such as C), an array variable really just points to the memory location where the first element of the array is stored. To get to the second element, we start at the first element and add 1 unit of element size to the first element. For the third element, we add 2 units of element size to the first element. Thus, the first element in an array is at index 0, since we add no extra offset to the storage location. The second and third elements end up at indexes 1 and 2, respectively.

Disciplined Counting

An important key to being able to count to 35 on your fingers is to be disciplined about the way that you count. To count the number zero is easy: just don’t extend any fingers. To count the number 1, extend the thumb on your right hand. The number 2 would result in extending your right thumb and right index finger. Continue the process of extending the next finger, in order, until you have reached the number 5.

Effective Fingering

At the number 6, put down all your fingers on your right hand (including the thumb), and extend your left thumb. For the number 7, keep your left thumb extended, and extend your right thumb. For the number eight, you would extend your left thumb, right thumb, and right index finger. Continue counting this way until all the fingers on your right hand are extended, which will happen at the number 11.

Once again, close all your fingers (and thumb) on your right hand, and extend your left index finger along with your already extended left thumb. This configuration counts the number 12. Opening your right thumb and right fingers in order will enable you to count 13, 14, 15, 16, and 17.

If you continue this process, you will have your left thumb and 2 additional fingers extended to count 18. When you finally run out of fingers (and thumbs) on both hands, you will have counted to 35.

Digits

In medicine, the fingers, thumbs, and toes are called digits. So-called “digital” examinations normally involve the use of the doctor’s finger inserted into various bodily openings, not any type of computer technology.

Fortunately in computing, “digits” instead represent numerical values from zero up through the equivalent of the maximum number we could count on our right hand in this example. Thus, the digits that we would use to represent the number of fingers and thumbs extended on a single human hand would be 0, 1, 2, 3, 4, and 5. We would not use 6, 7, 8, or 9 in this case, for reasons that we’ll see later.

Keeping Count

As we count to 35 using our fingers, we can see which fingers are up. However, if we wanted to write down the count and track our fingers, we would need to think about how to do it. One way would be to draw our hands, showing which fingers are up. This approach works, but it gets extremely tedious quickly. Fortunately, we can use a simple shorthand notation to track the fingers on each hand. It looks like this:

(LR)₆

L is the number of fingers we have up on our left hand (including the thumb), and R is the number of fingers we have up on our right hand (also including the thumb). Therefore, if we had 2 fingers up on our left hand and 1 finger up on our right hand, we would write (21)₆. Using this notation, we can express the count in a compact form, shown in Table 1.

The Number 6

Notice that the subscript number 6 follows the closing parenthesis in each entry of the count. This number is always one higher than the number of fingers we have on our right hand. It is also equal to the value of each finger on our left hand. Thus, (10)₆ = 6, since each finger on our left hand is “worth” 6. Following this logic, our left hand is counting multiples of 6 (6, 12, 18, 24, and 30), while our right hand counts multiples of 1. As we’ll see in the next part of the lesson, we can extend this idea to hands that have more or fewer fingers. When we use a different-sized hand, the subscript will change from 6 to whatever is one more than the number of fingers per hand.

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