Decimal representation — Safekipedia Adventurer

A decimal representation is a way to write numbers using digits from 0 to 9, with a special symbol called the decimal separator. This symbol shows parts of the number that are smaller than one. For example, the number 123.45 has "123" before the separator and "45" after it. The numbers before the separator tell us how many whole units there are. The numbers after tell us parts of a whole, like halves, tenths, or hundredths.

Every nonnegative real number can be written in decimal form. Some numbers can be written in two different ways. For example, the number 0.999... (with endless 9s) is the same as 1.000... (with endless 0s). Decimal representations help us understand and work with numbers in many areas of math and everyday life.

Decimal representations let us express numbers as sums that go on forever. Each digit stands for a part of the whole, like adding up pieces that get smaller and smaller. This idea is important in many fields, from basic arithmetic to advanced mathematics. Decimal numbers are used everywhere, from measuring distances to calculating money. They are a key part of how we understand and use numbers.

Integer and fractional parts

The part of a decimal number before the decimal point is called the integer part. For example, in the number 123.45, the integer part is 123.

The part after the decimal point is called the fractional part. It shows values that are less than one. In 123.45, the fractional part is 0.45, which means 45 parts out of 100.

Finite decimal approximations

Any real number can be approximated very closely using numbers with finite decimal representations. For example, the number 1 can be shown as 1.000... or as 0.999..., where the dots mean the digits go on forever. Usually, we pick the form without endless 9s, like 1.000..., because it’s simpler.

When we write numbers with decimals, we use digits from 0 to 9. For whole numbers, we just write the number itself, like 5. For numbers between whole numbers, we use a decimal point and more digits, like 0.25 for a quarter. This way, we can express many numbers very precisely!

Main article: Decimal representation

a 0 + a 1 10 + a 2 10 2 + ⋯ + a k 10 k ≤ x . {\displaystyle a_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots +{\frac {a_{k}}{10^{k}}}\leq x.}

Types

Some numbers can be written with a decimal that ends. For example, 0.5 or 0.25 both end.

Other numbers have decimals that go on forever in a repeating pattern. For instance, one-third is written as 0.333… where the 3 repeats endlessly.

Numbers with decimals that go on forever without repeating are irrational numbers, like the square root of two, pi, or the number e. These cannot be expressed as a simple fraction.

Conversion to fraction

Further information: Fraction § Arithmetic with fractions

Some decimals have a part that repeats over and over. These decimals can be written as fractions. We split the number into three parts: the whole number, the part before the repeating digits, and the repeating digits. There is a special way to find a fraction that matches the decimal.

For example, the number ±8.123\overline{4567} can be changed into a fraction. If there are no repeating digits, we can think of them as zeros, which makes it easier to change the decimal to a fraction. This shows how decimals and fractions are related.