Decimal

A decimal system, also called base-ten, uses ten as its base for counting and representing numbers. This system is used all around the world to write both whole numbers and numbers with parts smaller than one. The most common way we write numbers today is called the Hindu–Arabic numeral system.

A decimal numeral is just a number written in this system. When we write numbers that have parts smaller than one, we use a special sign called a decimal separator, usually a dot (.) or a comma (,). For example, in "25.9703" the part after the dot tells us how much there is beyond the whole number 25.

Numbers that can be written exactly with a finite number of digits after the decimal point are called decimal fractions. These are special because they help us show measurements very accurately. By adding more digits after the decimal point, we can get closer to the true value. For example, in science, if something is measured as 1.32 milligrams, it means the actual amount is very close to this number.

Origin

Many old cultures used a system based on the number ten. They likely chose ten because people have ten fingers, which they used for counting. Examples include the Egyptian numerals, Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. These old systems made it hard to work with big numbers.

These problems were solved with the Hindu–Arabic numeral system. This system made it easier to work with whole numbers and numbers with parts after a decimal point, called decimal fractions. This created what we now call the decimal numeral system.

Decimal notation

Numbers are usually written using ten special symbols called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To show smaller parts of numbers, we use a dot called a decimal mark. In some places, the dot is used (like in 20.5), and in other places, a comma is used instead (like in 20,5).

When we write a number like 20.75, the part before the dot (or comma) tells us how many whole numbers there are — here it’s 20. The part after the dot shows the smaller parts — here it’s 0.75. This helps us write and understand both whole numbers and parts of numbers easily.

Decimal fractions

Decimal fractions are numbers that can be written as a fraction where the bottom number is a power of ten. For example, 0.8 is the same as 8/10, and 14.89 is the same as 1489/100.

Decimal numbers are used in science, engineering, and daily life because they can get very close to any number. For example, the decimal 3.14159 is very close to the value of π. When we measure things, we often use decimals to show how accurate our measurement is.

Infinite decimal expansion

Decimal numbers can keep going forever after the decimal point. For example, the number 0.3333... has the digit 3 repeating forever. This is called an infinite decimal expansion.

Some numbers, like fractions, have parts that repeat in their decimal expansions. For example, 1 divided by 81 equals 0.012345679 and then the same group of digits repeats forever. If a decimal expansion starts repeating, the number it represents is a rational number, which means it can be written as a fraction.

Decimal computation

Computers usually use a system called binary for their calculations, but they show numbers to people using a system called decimal, which is what we use every day. For example, when you write 123.1 in a computer program, it understands it as a decimal number even though it might store it differently inside.

Computers sometimes use special ways to store decimal numbers accurately, especially for important jobs like money calculations. This helps make sure that results are exact, which is important for keeping track of money correctly.

History

Many old cultures used numbers based on ten, maybe because humans have ten fingers. Early groups like the Indus Valley Civilisation and the Egyptians used ten for their weights and measures. The Greeks and Romans also used numbers based on ten. The scientist Archimedes made a special way to write big numbers using ten.

History of decimal fractions

Later, people started using parts of numbers. In China, math experts used tools to work with small parts as early as the 3rd century CE. Arab math experts later found good ways to write these fractions. In the 16th century, a man named Simon Stevin helped make the way we write decimals today. And in 1620, John Napier introduced the dot we use to separate whole numbers from parts, like in 3.14.

Natural languages

Many languages fit well with the decimal system. For example, in Chinese, eleven is said as "ten-one." We see similar patterns in Vietnamese, Japanese, Korean, and Thai. Even some languages in the Andes, like Quechua and Aymara, speak numbers in ways that match the decimal system.

Other bases

Main article: Positional notation

Not all cultures used ten as their base. For example, some Mesoamerican peoples like the Maya used a system based on twenty, maybe counting all their fingers and toes. Other groups, like the Yuki in California, used eight as their base, counting the spaces between fingers. Some used bases like four, five, six, or even fifteen, depending on their needs and traditions.