Positional notation

Positional notation, also called place-value notation, is a way of writing numbers where the value of each symbol depends on its position. In a positional system, each symbol has a place or position, and its value is linked to a power of a fixed base. The most common system we use today, the Hindu–Arabic numeral system, is a positional system in base ten. It uses ten numerical digits, from zero to nine, and each symbol's value is the digit multiplied by a power of ten.

Most early numeral systems, like Roman numerals, are based on the additive principle. In these systems, each symbol has a fixed value, and the total value is the sum of the symbols. For example, in Roman numerals, CCXXVIII means 100 + 100 + 10 + 10 + 5 + 1 + 1 + 1, which equals 228. In contrast, the Hindu–Arabic numeral 228 shows 2 representing 2 × 100, 2 representing 2 × 10, and 8 representing 8 × 1.

The Babylonian numeral system, base 60, was the first positional system ever developed. Its influence is still seen today in how we count time and angles, such as 60 minutes in an hour and 360 degrees in a circle. The Inca used a decimal positional system with knots in quipu cords to store numbers. The binary numeral system, which uses base two, is used in almost all computers and electronic devices because it is easy to implement with electronic circuits. Positional notation makes arithmetic much simpler than older numeral systems, which helped it spread quickly when it was introduced in western Europe.

History

Today, we use a system with ten symbols (0-9) to write numbers, which is very common. In the past, other systems were used. For example, the Babylonians used a system with base 60, but they did not have a true zero. Later, around 700 BC, they used a space or a symbol to show where numbers ended.

Before we used this place-value system, people used other ways to write numbers, like Roman numerals or Chinese numerals. Tools like the abacus helped people do math without needing to remember many rules.

The oldest known place-value system might be from China or from the Khmer people, and it dates back to at least the 8th century. These systems led to the Arabic numerals we use today.

After the French Revolution, France pushed to use the decimal system more widely. This helped spread the use of decimals around the world.

Counting rods and abacuses were used to show numbers in a place-value system. This made doing math easier because you did not need to remember many tables.

The oldest surviving place-value system is either from Chinese rod numerals, used from at least the early 8th century, or perhaps Khmer numerals, showing possible uses of place-value numbers in the 7th century. Khmer numerals and other Indian numerals come from the Brahmi numerals of about the 3rd century BC, whose symbols were not used positionally at the time. Medieval Indian numerals are positional, as are the derived Arabic numerals, recorded from the 10th century.

History of positional fractions

Main article: Decimal

Decimal fractions were first used by the Chinese in the 1st century BC. Later, they spread to other parts of the world. A mathematician in Damascus started using them in the mid-10th century. Other mathematicians also used them in later years.

The use of decimals for numbers smaller than one is often said to start with a book by Simon Stevin, but others also helped develop this idea. Mathematicians tried to avoid fractions by using very large numbers, and one of the first to do this was Regiomontanus. He used units that were small enough to express all needed quantities as whole numbers.

Mathematics

Base of the numeral system

In mathematical numeral systems, the base, also called the radix, is usually the number of unique digits, including zero, that the system uses to represent numbers. For example, the decimal system uses ten digits from 0 through 9, so its base is ten. When a number reaches 9, the next number is written as "10", meaning one group of ten and zero extra. In binary, the base is two, so after "1" comes "10", meaning one group of two and zero extra.

The highest digit in a system usually shows a value one less than the base. Different systems change only in the base they use. The base must be greater than 1, because a base of zero or one would not work well. Negative bases are rarely used.

In the decimal system, there are ten decimal digits. For example, the number 5305 in decimal can be shown as:

5305 = (5 × 10³) + (3 × 10²) + (0 × 10¹) + (5 × 10⁰)

In hexadecimal (base sixteen), there are sixteen digits from 0–9 and A–F. The number 14B9 in hexadecimal equals 5305 in decimal.

Notation

When we write numbers, we often show the base as a small number after the number. For example, 1111011₂ means the number is in binary (base two), and equals 123₁₀ in decimal. We can also say "base-2" or "base-10" when talking about different systems.

Each digit has a place value based on its position. For example, in the number 465 in base ten, it means:

4 × 100 + 6 × 10 + 5 × 1 = 465

But in base seven, the same digits would mean:

4 × 49 + 6 × 7 + 5 × 1 = 243

Digits and numerals

A digit is a single symbol used in numbers, like 0, 1, 2, and so on. A numeral is one or more digits together. Common digits today are the decimal digits from 0 to 9.

In different bases, the same digits can mean different numbers. For example, "23" in base eight equals 19 in decimal, but in base four it equals 11 in decimal. The digits themselves still mean the same thing — "2" always means two, and "3" always means three.

Computer science often uses binary (base two), octal (base eight), and hexadecimal (base sixteen). Binary uses only 0 and 1. Octal uses digits 0–7. Hexadecimal uses 0–9 and A–F, where A stands for ten, B for eleven, and so on up to F for fifteen.

Radix point

For numbers that are not whole, we use a point called the radix point. Places to the right of this point have smaller values. For example, 2.35 in decimal means:

2 × 1 + 3 × 0.1 + 5 × 0.01

Sign

To show negative numbers, we add a minus sign (−) before the number.

Base conversion

Changing a number from one base to another can be done by dividing the number by the new base and keeping the remainders. For example, turning A10B in hexadecimal into decimal gives 41227.

For numbers with parts after the radix point, we divide the part after the point by the new base and may need to approximate if the division does not end nicely.

Applications

Decimal system

Main article: Decimal representation

In the decimal (base-10) Hindu–Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 10⁰ (1), the second position 10¹ (10), the third position 10² (100), the fourth position 10³ (1000), and so on.

Fractional values are shown by a separator, which can be a period, a full stop, or a comma. Digits to the right of this separator are multiplied by 10 raised to a negative power. The first position to the right of the separator shows 10⁻¹ (0.1), the second position 10⁻² (0.01), and so on.

For example, the number 2674 in base-10 is:

(2 × 10³) + (6 × 10²) + (7 × 10¹) + (4 × 10⁰)

or

(2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).

Sexagesimal system

The sexagesimal or base-60 system was used for the integral and fractional parts of Babylonian numerals and other Mesopotamian systems, by Hellenistic astronomers using Greek numerals for the fractional part only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.

Modern time separates each position by a colon or a prime symbol. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°‍25′‍59″ (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second.^{[citation needed]} This contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments.

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.

In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.

Computing

In computing, the binary (base-2), octal (base-8) and hexadecimal (base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).

The octal numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.

Hexadecimal, decimal, octal, and a wide variety of other bases have been used for binary-to-text encoding, implementations of arbitrary-precision arithmetic, and other applications.

For a list of bases and their applications, see list of numeral systems.

Other bases in human language

Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 10², hundred, commerce developed a word for 12², gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling (GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.

The Maya civilization and other civilizations of pre-Columbian Mesoamerica used base-20 (vigesimal), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa.

Remnants of a Gaulish base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.

In English the same base-20 counting appears in the use of "scores". Although mostly historical, it is occasionally used colloquially. Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago".

The Irish language also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.

The Welsh language continues to use a base-20 counting system, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.

The Inuit languages use a base-20 counting system. Students from Kaktovik, Alaska invented a base-20 numeral system in 1994

Danish numerals display a similar base-20 structure.

The Māori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama-hokotahi, referring to a great warrior ("the one man equal to 20").

The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, with a 1/64 term thrown away (the system was called the Eye of Horus).

A number of Australian Aboriginal languages employ binary or binary-like counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.

North and Central American natives used base-4 (quaternary) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.

A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.

A base-8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for "new", newo-, suggesting that the number 9 had been recently invented and called the "new number".

Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for five is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region.

The Telefol language, spoken in Papua New Guinea, is notable for possessing a base-27 numeral system.

Non-standard positional numeral systems

Main article: Non-standard positional numeral systems

Some numeral systems have special rules that make them useful for solving certain problems. For example, balanced ternary uses a base of 3 but the digits are {1,0,T} where T stands for −1. This system helps solve balance problems, where you need to find the right weights to match an unknown object.

Another system is the factorial number system, which changes its base for each position. This system helps count arrangements and is connected to puzzles like the Towers of Hanoi.

Non-positional positions

Some old ways of writing numbers were not fully positional. The Babylonian sexagesimal numerals used a special method. In each place, they used two kinds of wedges: a thin vertical wedge for one and a wider wedge for ten. They could use up to five tens and nine ones in one place. Sometimes they used a special symbol to show that a place had no number.

Greek astronomers also used letters for numbers. They picked one letter from five to show numbers from 10 to 50 and another from nine letters to show numbers from 1 to 9. They even used a special symbol for zero.