# Decimal to fraction calculator

In mathematics, fractions are written in two ways: ordinary and decimal. The first involves the use of a horizontal separator line. Above the line is the dividend, and below the line is the divisor. Such fractions look like 1/2, 3/7, 10/16 and so on.

The second kind of fractions is decimal, which is written lowercase, separating the integer value from the fractional value with a comma (or dot): 0.1; 1.3; 8.7. In such fractions, the divisor (denominator) is always equal to the number 10n, where n is an integer natural number. It can be 102 (100), 103 (1000), 104 (10000) and so on ad infinitum. Accordingly, any ordinary fraction, in the denominator of which is 10n, can be represented as a decimal: 2/10 = 0.2; 5/100 = 0.05.

## Decimal Properties

The main property of a decimal fraction is the possibility of endless "adding" zeros on the right side. That is, 0.5 = 0.50 = 0.500 = 0.500000. The value of the fraction does not change from this, which simplifies operations with numbers that have a different number of decimal places. So, to subtract the number 0.765 from 0.8, you need to add two zeros to the first number. We get: 0.800 − 0.765 = 0.035. We also refer the following to the properties of decimal fractions:

- The decimal is zero if the numerator/dividend is zero.
- Decimal is not possible if the denominator/divisor is zero.
- The fractions a/b and c/d are equal if a ⋅ d = b ⋅ c.

These rules apply to all fractions, including decimals, which can be represented as ordinary fractions. The name of the first one changes depending on how many decimal places.

The final (non-reducible) digit in a decimal is any last digit other than zero. Thus, the number 0.78900 can be reduced to 0.789, but the number 0.00789 cannot. Also, the former can always be represented as 789/1000 and the latter as 789/100000.

## History of origin and development

Decimal numbers represented in decimal form were known in Asia even before our era. For example, in ancient China in the 2nd century BC, there was a system of measures in which numbers were indicated as an integer and a fractional part. The second was separated from the first by a comma. Each decimal order had its own name, by analogy with our "tenths", "hundredths", "thousandths".

For example, the number 2.135436, according to ancient Chinese science, would be called "two chi, one cun, three shares, five ordinals, four hairs, three finest and six cobwebs." Subsequently, the Chinese mathematician Zu Chongzhi finalized this system and instead of the unit "chi" introduced "zhang", equal to "10 chi". The names of the orders shifted, and from the 5th century, decimal numbers began in China with "zhang" (an integer value). And after the comma followed "chi", "cun" and so on.

Decimal fractions were also widely used in Central Asia - starting from the 15th century. They were put into circulation by the Samarkand scientist Giyas-ad-din Jamshid ibn Masud al-Kashi, the grandson of the commander Tamerlane. In his book The Key of Arithmetic, al-Kashi described in detail the properties of decimal fractions, but denoted them differently. There were no commas separating integer and fractional parts, but there was a vertical line. The second option for writing such numbers is multi-colored ink. The integer part was written in black, and the fractional part was written in red.

After 150 years, decimal fractions were "rediscovered" in Europe by the Flemish engineer Simon Stevin in 1548. And their first European publication was the “Mathematical Canon” by the French scientist Francois Vieta, published in 1579 in Paris. The new system of measures became a successful alternative and began to replace the sexagesimal system that preceded it, which was widely used in astronomy, trigonometry and other sciences. By that time, the decimal fraction had not acquired its final form (with a comma). Its fractional part was alternately highlighted in small print, then in a different color, then underlined.

If in the XIV-XVI centuries the decimal fraction performed a secondary (applied) function and was used only in highly specialized calculations, then starting from the XVII century it became the main one, and almost completely replaced the sexagesimal. The reason for this was technological progress, requiring more and more complex and accurate calculations. The decimal fraction not only simplified them, but also accelerated them, allowing you to calculate the multiplication and division of complex numbers in a column. From mathematics and engineering, the decimal system gradually spread to agriculture and industry. By the 19th century, volumes, weights, lengths (and other measures) were commonly expressed as integer and fractional parts, using a separator. In Asian countries and in Russia it is a comma, and in Western countries it is a period.

Summing up, we can say that the decimal fraction played a significant role in the development of electronic computers. Due to its simplicity, it significantly accelerated binary calculations, which would require much higher computing power if we continued to use the sexagesimal system, and even more so - ordinary fractions.