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Along with decimal fractions, whose denominators can only be multiples of ten, ordinary fractions are widely used in mathematics and related sciences. It is written as a/b, where a is the numerator and b is the denominator. The first can be equal to any number, and the second can be any number except zero.

The concept of a fraction

A fraction is an expression represented as a dividend/numerator and divisor/denominator. The horizontal/slanted line separating them is called the vinculum/solidus, and can be written lowercase: a/b. Depending on the modular relationship between the dividend and the divisor, there are regular and improper ordinary fractions. In the first, the numerator module is greater than the denominator module, and in the second, vice versa.

Accordingly, if you divide a larger number by a smaller one, you will inevitably get a rational number - greater than one. Examples of such improper fractions are 6/5, 8/7, 11/3, and so on. Reduction in them is impossible, and they are recorded in their original form. If necessary, they can be calculated by obtaining an integer and a fractional part: in the remainder or as a decimal fraction.

There are also mixed and compound fractions. The first is written as a non-negative integer and a proper fraction, and the second is written as an expression containing several slashes/horizontal lines.

History of fractions

The English name fraction comes from the Latin fractura, but ordinary fractions were invented long before the formation of the Roman Empire. So, the splitting of numbers was used by the ancient Egyptians - about 4000 years ago. This is indicated by such archaeological finds as the Rinda mathematical papyrus, the Akhmim wooden tablet and the Egyptian mathematical leather scroll dated from the 20th to the 17th centuries BC.

Other studies indicate that number splitting was also done in ancient Babylon, more than 3,000 years ago. It was the Babylonians who introduced the division of a degree into 60 minutes, and minutes into 60 seconds. The number 60, in addition to itself and one, is divisible by 10 more numbers without a remainder: from 2 to 30. Accordingly, not decimal, but sexagesimal fractions were used in Babylon.

The system of sexagesimal fractions gradually migrated from ancient Babylonian to ancient Greek mathematics, and it is reliably known that it was used already in the 1st century AD: the ancient Greek scientists Diophantus of Alexandria and Heron of Alexandria. They wrote fractions in "alphabetic" form and in "inverted" form. That is, the numerator was at the bottom, and the denominator was at the top (without a dividing line). Due to the fact that the ancient Greeks understood the number as a set of units, they rarely used ordinary fractions in arithmetic, but sometimes used them to denote incommensurable quantities.

Similar studies were carried out in ancient China: from the 10th to the 2nd centuries BC. Initially, the Chinese used only ordinary fractions, and decimals were introduced only in the 3rd century AD - after the invention of the suanpan (算盤) counting board. The ancient Hindus also made a significant contribution to mathematics. It is they who own the modern form of an ordinary fraction - the numerator and denominator, separated by a horizontal line. Europeans began to use this system much later - only in the XII-XVI centuries, borrowing it from the Arabs, who, in turn, learned this knowledge from the Indians.

The first European thinker to use ordinary fractions (in the form in which they exist now) was Leonardo of Pisa, better known by his nickname Fibonacci. In 1350, decimal fractions began to be used in Europe in calculations - thanks to the French scientist Immanuel Bonfils, and from 1585 they became the main ones, replacing the outdated sexagesimal system.

Practical use of fractions

Today, ordinary fractions are used everywhere: from the exact sciences (for writing formulas) to everyday life. For example:

In cartography. The scale is always indicated as a natural fraction: 1/50000, 1/1000000. Instead of the "/" sign, a colon ":" is often written, but it means division, not enumeration.
In geography. For example, according to textbooks, Eurasia occupies about 1/3 of the land, and the Pacific Ocean - 1/2 of the world's oceans.
In medicine. When prescribing drugs, doctors rarely indicate their amount in grams, and use a more convenient fractional system of measures: 1/3 bottle, 1/2 tablet.

Ordinary fractions are used even in sports competitions: everyone knows such expressions as "a quarter of the final" or "one-half of the final". Despite the fact that decimal fractions are widely used in electronic computing devices, the common fraction has not lost its relevance. And in the exact sciences, it is simply impossible to do without it, since a significant part of the formulas, one way or another, contains expressions like a / b.

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Like decimals, fractions lend themselves to basic mathematical operations. Addition, subtraction, multiplication, division, raising to a power, extracting roots - these operations can be performed with any expressions represented as a / b, where b is not equal to zero. In addition, a decimal can always be represented as a common fraction, and a common fraction can almost always be converted to a decimal.

Arithmetic with fractions

Comparing fractions

The basic operation that precedes addition and subtraction is the comparison of two common fractions. In the case of the expressions 1/3 and 2/3, the answer is obvious: the first is 2 times greater than the second, but how to compare when the denominators of the numbers differ? In this case, an operation known as reduction to a common denominator is performed.

If we take the fractions 1/4 and 3/7, their common denominator will be the number 4 ⋅ 7, or 28. Accordingly, we cross-multiply 1 ⋅ 7 and 3 ⋅ 4 and get the numerators 7 and 12 with the same denominator 28. So the first less than the second: 7/28 < 12/28, and 1/4 < 3/7. And their ratio is 7/12.

Addition and subtraction

In order to add two ordinary fractions, they also need to be reduced to a common denominator, and then add the first dividend to the second, leaving the divisor unchanged. For example, if you add the fractions 1/3 and 2/4, we get: 1/3 + 2/4 = 4/12 + 6/12 = 10/12. Simplify and get: 5/6.

Subtraction occurs according to a similar algorithm: first, a common denominator is found, and then the second is subtracted from the first numerator. If we take the fractions 3/4 and 1/5, we get: 3/4 - 1/5 = 15/20 - 4/20 = 11/20.

Multiplication and division

To multiply fractions, reduction to a common denominator is not required, dividends and divisors are simply multiplied together. In general, the formula looks like this: (a/b) ⋅ (c/d) = (a ⋅ c)/(b ⋅ d). For example, take the fractions 5/6 and 1/4: (5/6) ⋅ (1/3) = (5 ⋅ 1)/(6 ⋅ 3) = 5/18. If an ordinary fraction is multiplied by a natural number, then only the numerator is involved in the operation. That is: (1/5) ⋅ 3 = 1 ⋅ 3/5 = 3/5.

The division of ordinary fractions is carried out according to the reverse scheme. So, to divide the fraction a/b by the fraction c/d, you need to multiply a/b by d/c. Substituting natural numbers instead of variables, we get: (1/3) / (2/11) = (1/3) ⋅ (11/2) = (1 ⋅ 11)/(3 ⋅ 2) = 11/6, or 1 (5/6), where the remainder is indicated in brackets. The main condition is that the denominator of the first fraction and the numerator of the second must not be equal to zero, since division by 0 is impossible.

Exponentiation and root extraction

These operations in the case of ordinary fractions are the simplest, since they do not require finding a common denominator or reversing fractions. The general formula for exponentiation is: (a/b)ⁿ = aⁿ/bⁿ. Substituting numbers instead of variables, we get: (2/3)² = 2²/3² = 4/9.

The extraction of the root is carried out similarly, according to the formula: √(a/b) = √a/√b. If we substitute 9 for a and 16 for b, we get: √(9/16) = √9/√16 = 3/4. A prerequisite for identities is a denominator other than zero (b ≠ 0): both for the exponentiation formula and for the root extraction formula.

Converting decimals to common fractions and common fractions to decimals

Any decimal fraction can be represented as a common fraction. To do this, you need to represent its fractional part as a natural number and divide it by the appropriate power of 10. For example, the fractions 0.234 and 0.65 can be represented as 234/1000 and 65/100, respectively. Moreover, in both cases they will sound the same: two hundred and thirty-four thousandths and sixty-five hundredths. You can also convert whole numbers with fractional remainders. For example, 72.86 is represented as 72 (86/100), where 72 is the integer part, and 86/100 is the fractional part.

Unfortunately, the reverse conversion - from a common fraction to a decimal - is not only more difficult, but also not always possible. At least - if you need a beautiful number with a small number of decimal places. A simple operation is only if the denominator is a multiple of 10. For example, the fraction 4/100 can be easily turned into 0.04, and 345/1000 into 0.345. But how to convert if the separator line is the number 7, 11, 14, and so on?

For example, let's take the fraction 19/44. To represent it in decimal form, you need to divide 19 by 44 in a column, putting a comma after 19 and adding zeros to it on the right until the desired order (accuracy) of calculation is reached. The calculation will lead to the fact that the number will be 0.43181818 ... (18 in the period). We write the number as 0.43 (18) - an infinite periodic decimal fraction and round it if necessary.

You can quickly count two-digit and three-digit numbers in a column, but when the number of characters exceeds 4-5, the procedure takes a lot of time and mental effort. Therefore, today, to convert ordinary fractions to decimals (and vice versa), special online calculators are used, where it is enough to enter known values band instantly get the correct result.

Fraction to decimal calculator