## Fractions and Decimals

This page is about converting a fraction (i.e. a ratio of two numbers, also called a rational number) into a decimal fraction and the patterns that occur in such a decimal fraction. It is interactive and you can use the calculators on this page to investigate fractions for yourself to many decimal places. No special knowledge beyond decimals and division is required.

## The Number of Fractions With the Same Denominator and Euler's Totient Function Φ(n)

The Calculator above computes the number of fractions with the same denominator by using a relatively simple formula which is why it is so fast. The number of fractions with the same denominator d is an important function in mathematics and in Number Theory in particular. Euler used it in his research and called it φ(d), the (Euler) Totient function. It has nothing to do with the golden mean number φ as that is a completely different use for the Greek letter phi!

## Why Do the Powers Patterns Ultimately Disappear?

We said earlier that all fractions when put into decimal form either terminate or recurr. This is true for our fractions whose decimal fractions correspond to a particular number series. They cannot go on for ever in the decimal digits! The reason lies in the fact that the numbers (the powers of 2, say) appear with two digits each. When we get to a power greater than 100, there will be an "overflow" into the 2-digit power before it. In fact what happens is that we do include every number from 0 upwards but the overflows eventually cause the decimal to get into a recurring sequence.

## Decimals in Maths (definition, Types, Properties, Conversion, and Examples)

In Mathematics, the numbers can be classified into different types, namely real numbers, natural numbers, whole numbers, rational numbers, and so on. Decimal numbers are among them. It is the standard form of representing integer and non-integer numbers. In this article, let us discuss in detail about “ Decimals”, its types, properties, and place value representation of decimal numbers with many solved examples.

## Fraction Calculator: 1/4-1/8

The calculator performs basic and advanced operations with fractions, expressions with fractions combined with integers, decimals, and mixed numbers. It also shows detailed step-by-step information about the fraction calculation procedure. Solve problems with two, three, or more fractions and numbers in one expression.