# Finger counting

by Admin Updated: July 5, 2018 |

## Use finger counting with bases other than ten and positional notation to manage large numbers

In modern Western societies we tend to discourage finger counting (dactylonomy) in children and view it as a sign of poor mental acuity (and education) in adults. Advanced finger counting takes some practice to master but it has significant benefits including improved mental agility, communication and multi-tasking (counting while doing something else).

Finger counting: We have ten fingers so we can use them to count to ten. End of story. Since we can easily count to ten in our minds, using our fingers to do it shows we are probably dumb.

Occasionally, we need to count *& retain* more than ten items and, in the absence of any other counting devices, running out of fingers is even more dumber.

### Positional number system

In normal finger counting, a finger represents a count of one, no matter which finger is used.

Using positional notation different fingers represent different amounts, e.g. thumb counts 1, index finger counts 2, so 3 can be represented by finger & thumb together, i.e. we can now count from zero to 3 using only two digits.

Positional dactylonomy typically works with both palms facing you. In order to telegraph the number to someone else, either turn your back and hold your hands up the same or rotate your palms and cross your wrists.

### Base or radix

The base is one more than the maximum number that can be counted in a single register, e.g. if a finger/digit represents zero or 1, the base is 2 and the value of counts in each successive register is b^{n} where b is the base and n is register number starting at zero.

This serves to avoid redundant overlap, e.g. if we decided the second finger could represent 3, we’d be able to count to 6 (all three digits) but we’d have two ways of representing 3 (redundancy).

If instead that second finger represents 4 (=2^{2}), we can now count to b^{(n+1)}-1 = 7 with no redundancy.

In fact, we have invented/discovered binary and we can now count to 1023 using two hands.

Be careful with 132 when using binary dactylonomy.

### Tally methods with redundancy

The problem with positional systems is that each register requires a pointer.

For example, if we could do finger counting in base 3, we could count to 59,048 - but representing three states (0, 1, 2) with each finger is difficult. If we use other fingers as pointers, we lose the use of them for counting.

If we count using 10 fingers and then use something external, e.g. a pebble, to record how many times we do this completely, we are using a non-positional tally method.

For example, if we count 10 fingers and have three pebbles, we can count to 40 (all the pebbles @ 10 plus all 10 fingers).

In a normal tally method, the maximum number that can be counted is r×q + r = r(q+1), where r is the register max and q is the number of tally markers.

### Sumerian (a.k.a. Babylonian) finger counting

In this method each finger bone or joint is counted (3 per finger) and the thumb is used as a pointer for a total of 12 on one hand.

The digits of the other hand can tally singly for a total of 5×12 = 60, or they can use the same partitioning method for 12×12 = 144.

### Tally methods *without* redundancy

Technically, the use of separate tally markers allows them to be +1.

For example, counting to 10 and using a marker for 11 has no redundancy. However, working with multiples of 11 brings a different set of challenges.

Instead, dropping the register to nine fingers and using a thumb as the [first] tally enables us to count to 19 with two hands (still better than 10) and we can continue to tally in multiples of 10.

If we count 9 fingers plus thumb and have three pebbles, we can count to 49 (all the pebbles @ 10 plus thumb @ 10 plus 9 fingers).

### Chisanbop

Chisanbop is a Korean finger-counting method that uses the same method as Japanese and Chinese abacuses.

It is a tally method without redundancy, where the fingers of one hand count 0-4 and the thumb is a sub-marker representing 5.

With this method we only need one hand to count to 9 and additional markers can continue to represent multiples of 10.

The maximum number that can be counted this way is (r+1)q + r = r(q+1) + q and, interestingly, if reducing r by 1 frees up a digit for q the new max will be r(q+2)-1 which will always be larger provided r-q > 1, i.e. optimal counting (to 11) is achieved for 5 digits when r=3 & q=2 but no improvement is achieved by going to r=2 & q=3.

Using Chisanbop on the other hand enables counting 9×10 + 9 = 99.

### Benefits of finger counting

- Recruits spatial function.
Counting using fingers (also visual & auditory representations) can help train our brains to process numbers more effectively.

- Allows communication over distance.
Whether placing orders on a trading floor or ordering drinks in a noisy bar, fingers can be a useful way to communicate numbers accurately.

- Allows retention of count without memorization.
A count on fingers can be maintained despite moderate levels of distraction and held during moments of interruption.

### More info

“Finger counting: The Debate Continues!” is at mathsinsider.com.

“If you can only count up to 10, you’re doing it wrong” at tpenguinltg.wordpress.com has some illustrated descriptions.

“What does the way you count on your fingers say about your brain?” at theguardian.com has several interesting links.