SafekipediaA decimal representation is a way to write numbers using digits from 0 to 9, with a special symbol called the decimal separator to show 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 digits before the separator tell us how many whole units there are, while the digits after tell us parts of a whole, like halves, tenths, or hundredths.
Every nonnegative real number can be written in decimal form, though some numbers have two different ways to appear. For instance, the number 0.999... (with endless 9s) is the same as 1.000... (with endless 0s). This shows how decimal representations help us understand and work with numbers in many areas of math and everyday life.
Decimal representations also let us express numbers as infinite sums. Each digit in the number 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, making them 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 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 finite decimal, meaning the digits stop after a certain point. These are rational numbers whose denominators are made up of only the factors 2 and 5. For example, 0.5 or 0.25 both end and fit this rule.
Other numbers have decimals that go on forever in a repeating pattern. These are also rational numbers. For instance, one-third is written as 0.333… where the 3 repeats endlessly. Numbers with non-repeating 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
Every decimal that ends in a repeating pattern can be turned into a fraction. We break the number into three parts: the whole number, the part before the repeating digits, and the repeating digits themselves. By using a special method, we can find a single fraction that equals the original decimal.
For example, the number ±8.123\overline{4567} can be converted to a fraction by treating the repeating part carefully. If there are no repeating digits, we can assume they are all zeros, which makes the conversion even simpler. This helps us understand how decimals and fractions are connected.
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