# Quaternary numeral system

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Hindu–Arabic numeral system |

East Asian |

Alphabetic |

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Positional systems by base |

Non-standard positional numeral systems |

List of numeral systems |

**Quaternary** is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.

Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being 36), making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers (the next best being the primorial base six, senary).

Quaternary shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties.

## Relation to other positional number systems

× | 1 | 2 | 3 | 10 | 11 | 12 | 13 | 20 |

1 | 1 | 2 | 3 | 10 | 11 | 12 | 13 | 20 |

2 | 2 | 10 | 12 | 20 | 22 | 30 | 32 | 100 |

3 | 3 | 12 | 21 | 30 | 33 | 102 | 111 | 120 |

10 | 10 | 20 | 30 | 100 | 110 | 120 | 130 | 200 |

11 | 11 | 22 | 33 | 110 | 121 | 132 | 203 | 220 |

12 | 12 | 30 | 102 | 120 | 132 | 210 | 222 | 300 |

13 | 13 | 32 | 111 | 130 | 203 | 222 | 301 | 320 |

20 | 20 | 100 | 120 | 200 | 220 | 300 | 320 | 1000 |

Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Quaternary | 0 | 1 | 2 | 3 | 10 | 11 | 12 | 13 | 20 | 21 | 22 | 23 | 30 | 31 | 32 | 33 | |

Octal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | |

Hexadecimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | |

Binary | 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 | |

Decimal | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | |

Quaternary | 100 | 101 | 102 | 103 | 110 | 111 | 112 | 113 | 120 | 121 | 122 | 123 | 130 | 131 | 132 | 133 | |

Octal | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | |

Hexadecimal | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1A | 1B | 1C | 1D | 1E | 1F | |

Binary | 10000 | 10001 | 10010 | 10011 | 10100 | 10101 | 10110 | 10111 | 11000 | 11001 | 11010 | 11011 | 11100 | 11101 | 11110 | 11111 | |

Decimal | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | |

Quaternary | 200 | 201 | 202 | 203 | 210 | 211 | 212 | 213 | 220 | 221 | 222 | 223 | 230 | 231 | 232 | 233 | |

Octal | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | |

Hexadecimal | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 2A | 2B | 2C | 2D | 2E | 2F | |

Binary | 100000 | 100001 | 100010 | 100011 | 100100 | 100101 | 100110 | 100111 | 101000 | 101001 | 101010 | 101011 | 101100 | 101101 | 101110 | 101111 | |

Decimal | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

Quaternary | 300 | 301 | 302 | 303 | 310 | 311 | 312 | 313 | 320 | 321 | 322 | 323 | 330 | 331 | 332 | 333 | 1000 |

Octal | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 100 |

Hexadecimal | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 3A | 3B | 3C | 3D | 3E | 3F | 40 |

Binary | 110000 | 110001 | 110010 | 110011 | 110100 | 110101 | 110110 | 110111 | 111000 | 111001 | 111010 | 111011 | 111100 | 111101 | 111110 | 111111 | 1000000 |

### Relation to binary

As with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4, 8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2, 3 or 4 binary digits, or bits. For example, in base 4,

- 30210
_{4}= 11 00 10 01 00_{2}.

Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, quaternary does not enjoy the same status.

By analogy with *byte* and *nybble*, a quaternary digit is sometimes called a *crumb*.

## Fractions

Due to having only factors of two, many quaternary fractions have repeating digits, although these tend to be fairly simple:

Decimal base Prime factors of the base: 2, 5Prime factors of one below the base: 3Prime factors of one above the base: 11Other Prime factors: 7 13 |
Quaternary basePrime factors of the base: 2Prime factors of one below the base: 3Prime factors of one above the base: 11Other Prime factors: 13 23 31 | ||||

Fraction | Prime factors of the denominator |
Positional representation | Positional representation | Prime factors of the denominator |
Fraction |

1/2 | 2 |
0.5 |
0.2 |
2 |
1/2 |

1/3 | 3 |
0.3333... = 0.3 |
0.1111... = 0.1 |
3 |
1/3 |

1/4 | 2 |
0.25 |
0.1 |
2 |
1/10 |

1/5 | 5 |
0.2 |
0.03 |
11 |
1/11 |

1/6 | 2, 3 |
0.16 |
0.02 |
2, 3 |
1/12 |

1/7 | 7 |
0.142857 |
0.021 |
13 |
1/13 |

1/8 | 2 |
0.125 |
0.02 |
2 |
1/20 |

1/9 | 3 |
0.1 |
0.013 |
3 |
1/21 |

1/10 | 2, 5 |
0.1 |
0.012 |
2, 11 |
1/22 |

1/11 | 11 |
0.09 |
0.01131 |
23 |
1/23 |

1/12 | 2, 3 |
0.083 |
0.01 |
2, 3 |
1/30 |

1/13 | 13 |
0.076923 |
0.010323 |
31 |
1/31 |

1/14 | 2, 7 |
0.0714285 |
0.0102 |
2, 13 |
1/32 |

1/15 | 3, 5 |
0.06 |
0.01 |
3, 11 |
1/33 |

1/16 | 2 |
0.0625 |
0.01 |
2 |
1/100 |

## Occurrence in human languages

Many or all of the Chumashan languages originally used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16 (not 10). There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819.^{[1]}

The Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10.

## Hilbert curves

Quaternary numbers are used in the representation of 2D Hilbert curves. Here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected.

## Genetics

Parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0, 1, 2, and 3. With this encoding, the complementary digit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of the base pairs: A↔T and C↔G and can be stored as data in DNA sequence.^{[2]}

For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010 (= decimal 9156 or binary 10 00 11 11 00 01 00).

## Data transmission

Quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits.

## See also

- Conversion between bases
- Moser–de Bruijn sequence, the numbers that have only 0 or 1 as their base-4 digits

## References

- ↑ "Chumashan Numerals" by Madison S. Beeler, in
*Native American Mathematics*, edited by Michael P. Closs (1986), ISBN 0-292-75531-7. - ↑ "Archived copy" (PDF). Archived from the original (PDF) on December 14, 2010. Retrieved November 27, 2010.

## External links

- Quaternary Base Conversion, includes fractional part, from Math Is Fun
- Base42 Proposes unique symbols for Quaternary and Hexadecimal digits
- Visualization of nucleotide sequence, Visualization of numeral systems