Appendix:Hindu-Arabic script

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Arabic numerals and Hindu–Arabic numeral system

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The Hindu-Arabic numerical script refers to the group of Hindu-Arabic numerals, also called Arabic numerals or Hindu numerals, that are ten digits used as numerals in multiple languages.

The Hindu-Arabic digits are, in this order, from lower to higher: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The symbol 0 represents zero, a null quantity. The symbol 1 represents one unit and 2 represents two units. Each number represents a single unit higher when compared to the previous number. For example, 5 is one unit higher than 4.

It is possible to represent more numbers in Hindu-Arabic by restarting the count of digits while placing another digit at the left, effectively forming combinations of digits. A few numbers higher than 9 are, in this order, 10, 11 and 12. Other possible meaningful combinations of digits are 42, 851, and 9174623. A combination of digits also represents a number.

Here is zero, followed by the next one hundred fifteen numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115

The numerical value of each digit is multiplied by a power of ten depending on its position from right to left. For example, the numeral 639 is formed by a 9 in the first position which represents nine, a 3 in the second position which represents thirty, and a 6 in the third position which represents six hundred. Such a reasoning may be analyzed through exponential notation: ${\displaystyle 6\times 10^{2}+3\times 10^{1}+9=639}$, therefore the result is six hundred thirty nine.

It is also possible to represent real numbers between zero and one in Hindu-Arabic numerals. This is done by writing a symbol (see below, under "Separators", for discussion of what symbol is used) and then numerals in sequence representing multiples of successive powers of ${\displaystyle {\frac {1}{10}}=10^{-1}}$. Thus, for example, .78 represents ${\displaystyle 7\times 10^{-1}+8\times 10^{-2}}$.

Any real number can then be represented in Hindu-Arabic numerals, as the representation of an integer can be followed immediately by the representation of a number between zero and one that is added to it, as 3.42, which represents the number ${\displaystyle 3+.42}$.

The system of representing numbers from 0 to 9 is referred to as base-ten, but other bases can be used instead of ten. For example, in base four, a system comprising four digits, only 0, 1, 2 and 3 are used, and after them comes 10 (representing the number which in base ten is represented by 4), 11 (base ten's 5), 12 (base ten's 6), 13 (base ten's 7), 20 (base ten's 8), 21, 22, 23, 30, etc. Note that in base 4, 20 represents the number written in base ten as ${\displaystyle 2\times 4^{1}+0}$ and 312 represents base ten's ${\displaystyle 3\times 4^{2}+1\times 4+2=54}$. Usually, when using a system with more than ten digits, they are followed by Latin letters. A list of common numerical systems include:

• Two digits (0, 1): binary or base-2.
• Four digits (0, 1, 2, 3): quaternary or base-4.
• Eight digits (0, 1, 2, 3, 4, 5, 6, 7): octal or base-8.
• Ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9): decimal or base-10.
• Sixteen digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F): hexadecimal or base-16.

The Hindu-Arabic numerals can be used outside of positional notation as well. For instance, bijective numeration base ten, also known as ten-adic, has the empty string instead of any glyph to represent zero and uses nine Hindu-Arabic digits plus the first Latin letter representing ten (1, 2, 3, 4, 5, 6, 7, 8, 9, A).

The Hindu-Arabic numerals are widely known and widely used in many languages, including English. There are, however, multiple different typographical characteristics, commonly based on personal choice or on conventions of specific languages.

For integers with many digits, separator characters may be inserted between groups of digits for ease of reading. The choice of separator character, and the placement of the characters, varies somewhat from language to language, region to region, writer to writer, and context to context. One common approach is to insert commas, periods, apostrophes, or spaces between each three digits from right to left; for example, 71298754 might be written “71,298,754”, “71.298.754”, “71'298'754”, or “71 298 754”.

The symbol to be used as a separator usually varies from language to language:

• In English, commas or spaces are preferred, depending on region.
  • Many English writers do not use any separator for four-digit numbers: " 9998, 9999, 10,000 "
  • South Asian English writers typically separate numbers after every two digits, except for the last group which has three. For example, the above number would be written "7,12,98,754".
• In Romance languages including Italian, Portuguese, and Spanish, periods are preferred.
• In Switzerland, apostrophes are preferred.
• In Japanese and Chinese, commas are preferred.

For fractional numbers, a symbol known as a decimal mark is inserted after the whole number part. For numbers between zero and one, the leading zero is sometimes omitted, starting with just the decimal mark. This symbol is distinguished from the separator above for consistency within any one system. Unfortunately, this does not preserve consistency between systems, so "1,500" for instance could mean either fifteen hundred or one and a half depending.

• In English, period or commas are preferred, depending on region.
  • In the United States, a decimal point is used. Monetary values that split a dollar are written at least to the penny, for instance "$1.50" rather than "$1.5", although "$1.5 million" is acceptable.

The lining figures are digits written in uniform height and position, and text figures are digits varying in height and position.

In Chinese, Japanese and Korean, the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are known as halfwidth characters. There are larger variations known as fullwidth characters: ０, １, ２, ３, ４, ５, ６, ７, ８ and ９.

In multiple contexts, including algebra and chemistry, it is possible to write Hindu-Arabic numerals in:

• superscript: Small numbers that are positioned slightly above the baseline.
• subscript: Small numbers that are positioned slightly below the baseline.

For emphasis in comparison with the rest of a text, certain numbers in Hindu-Arabic script may be:

• bold: Digits written with thicker strokes.
• italicized: Digits written with inclined strokes.
• underlined: Digits written above a horizontal stroke.
• of a different color.

Although Hebrew text is read right-to-left, Hindu-Arabic numbers represent the same quantities in English and Hebrew: For example, 756 in a text of either language represents seven hundred fifty-six.

Values less than zero are prefixed with a minus sign (-). In bookkeeping, negative values can instead be represented with parentheses, for instance "(100)" indicating minus one hundred dollars, or in red instead of black.

The Hindu-Arabic numerals are found in Unicode from the following charts.