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.

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. The Babylonian numeral system, base 60, was the first positional system ever developed. The binary numeral system, which uses base two, is used in almost all computers and electronic devices.

History

Today, we use ten symbols (0-9) to write numbers. 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.

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.

The oldest surviving place-value system is either from Chinese rod numerals, used from at least the early 8th century, or perhaps Khmer numerals. Khmer numerals and other Indian numerals come from the Brahmi numerals of about the 3rd century BC. 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.

Mathematicians tried to avoid fractions by using very large numbers.

Mathematics

Base of the numeral system

In numeral systems, the base is the number of unique digits used to write numbers, including zero. For example, the decimal system uses ten digits, from 0 to 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 is usually one less than the base. Different systems change only in the base they use. The base must be greater than 1. Negative bases are rarely used.

In the decimal system, there are ten 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 from right to left shows a higher power of 10. The first position is 10⁰ (1). The next is 10¹ (10). Then 10² (100), and so on.

Fractional values use a separator, like a period or a comma. Numbers to the right of this separator are parts of 10 to a negative power. The first position shows 10⁻¹ (0.1). The next shows 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 old Babylonian numerals and by Hellenistic astronomers. It is still used today for time and angles, but only for minutes and seconds.

Modern time uses colons to separate hours, minutes, and seconds. For example, 10:25:59 means 10 hours, 25 minutes, and 59 seconds. Angles use similar signs. For example, 10°‍25′‍59″ means 10 degrees, 25 minutes, and 59 seconds. Only minutes and seconds use sexagesimal notation.

In the 1930s, Otto Neugebauer created a system for old numbers that uses modern decimal numbers from 0 to 59 in each position. It uses a semicolon (;) to separate whole numbers from parts, and a comma (,) to separate positions. For example, the mean synodic month is 29;31,50,8,20 days.

Computing

In computing, binary (base-2), octal (base-8) and hexadecimal (base-16) are common. Computers work with zeroes and ones, so powers of two are easy. Hexadecimal is short for binary—every 4 binary digits match one hexadecimal digit. In hexadecimal, the numbers after 9 are A, B, C, D, E, and F.

The octal system is another way to show binary numbers. It uses digits 0 to 7. Every 3 binary digits match one octal digit.

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

Other bases in human language

Base-12 systems (duodecimal or dozenal) are easy for multiplication and division. Twelve has many factors. English still uses "dozen" and "gross" for 12 and 144. The 12-hour clock and old British money used base-12.

The Maya civilization and others in Mesoamerica used base-20 (vigesimal). Some North American tribes also used base-20.

Old French used base-20 for numbers from 60 to 99. For example, sixty-five is soixante-cinq (sixty and five). English uses "scores" in old phrases like "four score and seven years ago".

The Irish language once used base-20. The Welsh language still uses base-20 for ages and dates.

The Inuit languages use base-20. Danish numerals also show base-20.

The Māori language has words showing base-20.

The binary system was used in ancient Egypt.

Some Australian Aboriginal languages use binary or binary-like counting.

North and Central American natives used base-4 (quaternary). Mesoamericans used a mix of base-4 and base-5 to make base-20.

Many cultures use base-5 (quinary), based on fingers. This is also a part of bases like base-10, base-20, and base-60.

The Yuki tribe of Northern California used base-8, counting on the spaces between fingers.

The Telefol language of Papua New Guinea uses base-27.

Non-standard positional numeral systems

Main article: Non-standard positional numeral systems

Some numeral systems have special rules that help solve 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.

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. They used two kinds of wedges: a thin 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 used letters for numbers. They picked one letter to show numbers from 10 to 50 and another to show numbers from 1 to 9. They even used a special symbol for zero.