Decimal

The decimal numeral system is the global standard for representing numbers.

It is also known as the base-ten positional system or the denary system. This system allows us to write both whole numbers, called integers, and parts of a whole, called non-integers. By using decimal notation, we can express almost any value with great detail. This system is an extension of the Hindu-Arabic numeral system. It provides a way to represent decimal fractions, which are numbers that can be written as a fraction with a denominator that is a power of ten.

Decimal notation relies on a specific mechanism called positional value. This means the value of a digit depends on its position within the number.

The system uses ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To separate whole numbers from fractional parts, we use a decimal separator. In many English-speaking countries, this is a dot. In other regions, a comma is used instead. The digits to the left of the separator form the integer part. The digits to the right form the fractional part.

For negative numbers, a minus sign is placed before the numeral.

There are different types of decimal expansions based on how they end. A terminating decimal is a number that has a finite number of non-zero digits. For example, the number 0.25 ends clearly. Other numbers result in an infinite decimal expansion. These expansions do not end and continue forever. Some infinite decimals are repeating decimals. These contain a sequence of digits that repeats indefinitely.

To show this, mathematicians use a vinculum, which is a horizontal line placed over the repeating digits. An infinite decimal represents a rational number if it eventually repeats or if it is finite.

History shows that many ancient civilizations used systems based on ten.

Some historians suggest this started because humans have ten fingers to count with. The Egyptians, Greeks, Romans, and Chinese all used base-ten systems in various ways.

However, these ancient systems often made large calculations very difficult. Only the most skilled mathematicians could multiply or divide large amounts. The introduction of the Hindu-Arabic system solved these problems for integers. Later, this system was extended to include the decimal fractions we use today.

Decimals are essential for scientific precision and approximation. Most measurements in the real world involve some level of uncertainty.

We use decimals to approximate real numbers to any desired level of accuracy. In science, the number of digits after the decimal separator indicates precision. For instance, a mass of 1.320 milligrams is more precise than 1.32 milligrams. The extra zero suggests the true mass is likely between 1.3195 and 1.3205 milligrams. A measurement of 1.32 milligrams suggests a wider range between 1.315 and 1.325 milligrams. By adding more digits, we can make approximation errors as small as we want.

An interesting fact involves the way we represent the same value. In pure mathematics, the numbers 4.69 and 4.690 are the same real number. However, the extra zero in 4.690 can be meaningful in specific contexts. Similarly, adding trailing zeros after a decimal mark does not change the value. You can also add zeros to the left of a number without changing its value. For example, 007.5 is the same as 7.5. In computing, the integer part might even be omitted if it is zero. This results in a notation like .5 instead of 0.5.

Decimals connect deeply to the study of rational numbers and limits. A decimal fraction is a rational number where the denominator is a power of ten. These can be expressed as fractions like 1/2, 1/4, or 1/5. However, some fractions like 1/3 cannot be written as a terminating decimal. They result in a repeating sequence. This connects to the mathematical concept of a limit. As we add more digits to a decimal, the difference between our approximation and the true value gets arbitrarily small. This allows decimals to bridge the gap between simple counting and complex real-world values.