Decimal

A decimal system, also called base-ten, denary, or decenary, 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, which arranges numbers in places like units, tens, hundreds, and so on.

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 approximate real numbers very accurately. By adding more digits after the decimal point, we can get closer and closer to the true value. For example, in science, if something is measured as 1.32 milligrams, it means the actual amount is likely between 1.315 and 1.325 milligrams, showing how precise the measurement is.

Origin

Many ancient civilizations used a system based on the number ten, likely because people have ten fingers and used them 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 very large numbers.

These problems were solved with the creation of the Hindu–Arabic numeral system, which made it easier to handle both whole numbers and numbers with parts after a decimal point, known as decimal fractions. This formed 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. Some fractions, like 1/3, cannot be written exactly with decimals because 3 is not a power of 10.

Decimal numbers are used in science, engineering, and daily life because they can get very close to any number, even if they cannot always be exact. 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 go on forever after the decimal point. For example, the number 0.3333... continues forever with the digit 3 repeating. This is called an infinite decimal expansion.

Some numbers, like fractions, have repeating parts 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 a group of digits, the number it represents is a rational number, which means it can be written as a fraction.

Decimal computation

Computers usually work with a system called binary to do their calculations, but they show numbers to people in a system called decimal, which is what we use in everyday life. 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 tasks like money calculations. This helps make sure that results are exact, which is crucial for keeping track of finances correctly.

History

Many ancient cultures used numbers based on ten, likely because humans have ten fingers. Early societies like the Indus Valley Civilisation and the Egyptians used decimal systems for their weights and measures. The Greeks and Romans also used numbers based on powers of ten. The famous scientist Archimedes even created a special way to write very large numbers using ten as a base.

History of decimal fractions

Later, people began using parts of numbers. In China, mathematicians used special tools to work with these smaller parts as early as the 3rd century CE. Arab mathematicians later developed ways to write these fractions clearly. In the 16th century, a man named Simon Stevin helped create the way we write decimals today. And in 1620, John Napier introduced the dot we use today to separate whole numbers from parts, like in 3.14.

Natural languages

Many languages around the world naturally fit with the decimal system. For example, in Chinese, numbers like eleven are said as "ten-one." Similar patterns appear in Vietnamese, Japanese, Korean, and Thai. Even some languages in the Andes, like Quechua and Aymara, have ways of speaking numbers 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, perhaps counting all their fingers and toes. Other groups, like the Yuki in California, used eight as their base, counting the spaces between fingers. Still others used bases like four, five, six, or even fifteen, depending on their needs and traditions.