Finger binary
Encyclopedia
Finger binary is a system for counting
and displaying binary numbers
on the finger
s and thumb
s of one or more hand
s. It is possible to count from 0 to 31 (25−1) using the fingers of a single hand, or from 0 through 1023 (210−1) if both hands are used.
has two possible states (0 or 1) and each successive digit represents an increasing power of two
. The rightmost digit represents two to the zeroth power (i.e., it is the "ones digit"); the digit to its left represents two to the first power (the "twos digit"); the next digit to the left represents two to the second power (the "fours digit"); and so on. (The decimal number system is essentially the same, only that powers of ten are used: "ones digit", "tens digit" "hundreds digit", etc.)
It is possible to use anatomical digits
to represent numerical digit
s by using a raised finger to represent a binary digit in the "1" state and a lowered finger to represent it in the "0" state. Each successive finger represents a higher power of two. With palms oriented toward the counter's face, the values are:
if only the right hand is used, and:
if both hands are used.
The values of each raised finger are added together to arrive at a total number. In the one-handed version, all fingers raised is thus 31 (16 + 8 + 4 + 2 + 1), and all fingers lowered (a fist) is 0. In the two-handed system, all fingers raised is 1,023 (512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1) and two fists (no fingers raised) represents 0.
It is also possible to have each hand represent an independent number between 0 and 31; this can be used to represent various types of paired numbers, such as month
and day
, X-Y coordinates, or sports scores (such as for table tennis
or baseball
).
: raised means the number is negative, in a sign-magnitude system. Anywhere between -511 and +511 can be represented this way, using two hands. Note that, in this system, both a positive and a negative zero may be represented.
If a convention were reached on palm up/palm down representing positive/negative, you could maintain 2^10 - 1 in both positive and negative numbers (-1023 to +1023, with positive and negative zero still represented).
. (These are known as dyadic fractions.)
Using the left hand only:
Using two hands:
The total is calculated by adding all the values in the same way as regular (non-fractional) finger binary, then dividing by the largest fractional power being used (32 for one-handed fractional binary, 1024 for two-handed), and simplifying the fraction as necessary.
For example, with thumb and index finger raised on the left hand and no fingers raised on the right hand, this is (512 + 256)/1024 = 768/1024 = 3/4. If using only one hand (left or right), it would be (16 + 8)/32 = 24/32 = 3/4 also.
The simplification process can itself be greatly simplified by performing a bit shift operation: all digits to the right of the rightmost raised finger (i.e., all trailing zeros) are discarded and the rightmost raised finger is treated as the ones digit. The digits are added together using their now-shifted values to determine the numerator and the rightmost finger's original value is used to determine the denominator.
For instance, if the thumb and index finger on the left hand are the only raised digits, the rightmost raised finger (the index finger) becomes "1". The thumb, to its immediate right, is now the 2s digit; added together, they equal 3. The index finger's original value (1/4) determines the denominator: the result is 3/4.
and fractional values (i.e., rational number
s) can be represented by setting a radix point
somewhere between two fingers (for instance, between the left and right pinkies). All digits to the left of the radix point are integers; those to the right are fractional.
Instead, either decimal or vulgar fractions can be represented natively in finger binary. Decimal fractions can be represented by using regular integer binary methods and dividing the result by 10, 100, 1000, or some other power of ten. Numbers between 0 and 102.3, 10.23, 1.023, etc. can be represented this way, in increments of 0.1, 0.01, 0.001, etc.
Vulgar fractions can be represented by using one hand to represent the numerator and one hand to represent the denominator; a spectrum of rational numbers can be represented this way, ranging from 1/31 to 31/1 (as well as 0).
(base 3) could be used by having a fully raised finger represent 2, fully lowered represent 0, and "curled" (half-lowered) represent 1. This would make it possible to count up to 59,048 (310−1) on two hands. In practice, however, many people will find it difficult to hold all fingers independently (especially the middle and ring fingers) in more than two distinct positions.
Finger counting
Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in...
and displaying binary numbers
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
on the finger
Finger
A finger is a limb of the human body and a type of digit, an organ of manipulation and sensation found in the hands of humans and other primates....
s and thumb
Thumb
The thumb is the first digit of the hand. When a person is standing in the medical anatomical position , the thumb is the lateral-most digit...
s of one or more hand
Hand
A hand is a prehensile, multi-fingered extremity located at the end of an arm or forelimb of primates such as humans, chimpanzees, monkeys, and lemurs...
s. It is possible to count from 0 to 31 (25−1) using the fingers of a single hand, or from 0 through 1023 (210−1) if both hands are used.
Mechanics
In the binary number system, each numerical digitNumerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
has two possible states (0 or 1) and each successive digit represents an increasing power of two
Power of two
In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....
. The rightmost digit represents two to the zeroth power (i.e., it is the "ones digit"); the digit to its left represents two to the first power (the "twos digit"); the next digit to the left represents two to the second power (the "fours digit"); and so on. (The decimal number system is essentially the same, only that powers of ten are used: "ones digit", "tens digit" "hundreds digit", etc.)
It is possible to use anatomical digits
Digit (anatomy)
A digit is one of several most distal parts of a limb, such as fingers or toes, present in many vertebrates.- Names:Some languages have different names for hand and foot digits ....
to represent numerical digit
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
s by using a raised finger to represent a binary digit in the "1" state and a lowered finger to represent it in the "0" state. Each successive finger represents a higher power of two. With palms oriented toward the counter's face, the values are:
| Pinky Little finger The little finger, often called the pinky in American English, pinkie in Scottish English , or small finger in medicine, is the most ulnar and usually smallest finger of the human hand, opposite the thumb, next to the ring finger.-Muscles:There are four muscles that... | Ring Ring finger The ring finger is the fourth digit of the human hand, and the second most ulnar finger, located between the middle finger and the little finger. It is also called digitus medicinalis, the fourth finger, digitus annularis, digitus quartus, or digitus IV in anatomy.- Etymology :According to László A... | Middle Middle finger The middle finger or long finger is the third digit of the human hand, located between the index finger and the ring finger. It is usually the longest finger... | Index Index finger The index finger, , is the first finger and the second digit of a human hand. It is located between the first and third digits, between the thumb and the middle finger... | Thumb Thumb The thumb is the first digit of the hand. When a person is standing in the medical anatomical position , the thumb is the lateral-most digit... |
|
|---|---|---|---|---|---|
| Power of two | 24 | 23 | 22 | 21 | 20 |
| Value | 16 | 8 | 4 | 2 | 1 |
if only the right hand is used, and:
| Left hand | Right hand | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Thumb | Index | Middle | Ring | Pinky | Pinky | Ring | Middle | Index | Thumb | |
| Power of two | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |
| Value | 512 | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
if both hands are used.
The values of each raised finger are added together to arrive at a total number. In the one-handed version, all fingers raised is thus 31 (16 + 8 + 4 + 2 + 1), and all fingers lowered (a fist) is 0. In the two-handed system, all fingers raised is 1,023 (512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1) and two fists (no fingers raised) represents 0.
It is also possible to have each hand represent an independent number between 0 and 31; this can be used to represent various types of paired numbers, such as month
Month
A month is a unit of time, used with calendars, which was first used and invented in Mesopotamia, as a natural period related to the motion of the Moon; month and Moon are cognates. The traditional concept arose with the cycle of moon phases; such months are synodic months and last approximately...
and day
Day
A day is a unit of time, commonly defined as an interval equal to 24 hours. It also can mean that portion of the full day during which a location is illuminated by the light of the sun...
, X-Y coordinates, or sports scores (such as for table tennis
Table tennis
Table tennis, also known as ping-pong, is a sport in which two or four players hit a lightweight, hollow ball back and forth using table tennis rackets. The game takes place on a hard table divided by a net...
or baseball
Baseball
Baseball is a bat-and-ball sport played between two teams of nine players each. The aim is to score runs by hitting a thrown ball with a bat and touching a series of four bases arranged at the corners of a ninety-foot diamond...
).
Negative numbers and non-integers
Just as fractional and negative numbers can be represented in binary, they can be represented in finger binary.Negative numbers
Representing negative numbers is extremely simple, by using the leftmost finger as a sign bitSign bit
In computer science, the sign bit is a bit in a computer numbering format that indicates the sign of a number. In IEEE format, the sign bit is the leftmost bit...
: raised means the number is negative, in a sign-magnitude system. Anywhere between -511 and +511 can be represented this way, using two hands. Note that, in this system, both a positive and a negative zero may be represented.
If a convention were reached on palm up/palm down representing positive/negative, you could maintain 2^10 - 1 in both positive and negative numbers (-1023 to +1023, with positive and negative zero still represented).
Dyadic fractions
Fractions can be stored natively in a binary format by having each finger represent a fractional power of two:. (These are known as dyadic fractions.)Using the left hand only:
| Thumb | Index | Middle | Ring | Pinky | |
|---|---|---|---|---|---|
| Value | 1/2 | 1/4 | 1/8 | 1/16 | 1/32 |
Using two hands:
| Left Hand | Right Hand | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Thumb | Index | Middle | Ring | Pinky | Pinky | Ring | Middle | Index | Thumb |
| 1/2 | 1/4 | 1/8 | 1/16 | 1/32 | 1/64 | 1/128 | 1/256 | 1/512 | 1/1024 |
For example, with thumb and index finger raised on the left hand and no fingers raised on the right hand, this is (512 + 256)/1024 = 768/1024 = 3/4. If using only one hand (left or right), it would be (16 + 8)/32 = 24/32 = 3/4 also.
The simplification process can itself be greatly simplified by performing a bit shift operation: all digits to the right of the rightmost raised finger (i.e., all trailing zeros) are discarded and the rightmost raised finger is treated as the ones digit. The digits are added together using their now-shifted values to determine the numerator and the rightmost finger's original value is used to determine the denominator.
For instance, if the thumb and index finger on the left hand are the only raised digits, the rightmost raised finger (the index finger) becomes "1". The thumb, to its immediate right, is now the 2s digit; added together, they equal 3. The index finger's original value (1/4) determines the denominator: the result is 3/4.
Rational numbers
Combined integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
and fractional values (i.e., rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s) can be represented by setting a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
somewhere between two fingers (for instance, between the left and right pinkies). All digits to the left of the radix point are integers; those to the right are fractional.
Decimal fractions and vulgar fractions
Dyadic fractions, explained above, unfortunately have limited use in a society based around decimal figures. A simple non-dyadic fraction such as 1/3 can be approximated as 341/1024 (0.3330078125), but the conversion between dyadic and decimal (0.333) or vulgar (1/3) forms is complicated.Instead, either decimal or vulgar fractions can be represented natively in finger binary. Decimal fractions can be represented by using regular integer binary methods and dividing the result by 10, 100, 1000, or some other power of ten. Numbers between 0 and 102.3, 10.23, 1.023, etc. can be represented this way, in increments of 0.1, 0.01, 0.001, etc.
Vulgar fractions can be represented by using one hand to represent the numerator and one hand to represent the denominator; a spectrum of rational numbers can be represented this way, ranging from 1/31 to 31/1 (as well as 0).
Finger ternary
In theory, it is possible to use other positions of the fingers to represent more than two states (0 and 1); for instance, a ternary numeral systemTernary numeral system
Ternary is the base- numeral system. Analogous to a bit, a ternary digit is a trit . One trit contains \log_2 3 bits of information...
(base 3) could be used by having a fully raised finger represent 2, fully lowered represent 0, and "curled" (half-lowered) represent 1. This would make it possible to count up to 59,048 (310−1) on two hands. In practice, however, many people will find it difficult to hold all fingers independently (especially the middle and ring fingers) in more than two distinct positions.