While studying the queen of all sciences - mathematics, at some point everyone comes across fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is not at all complicated, it must be treated carefully, because in real life It will be very useful outside of school. So, let's refresh our knowledge about fractions: what they are, what they are for, what types they are and how to perform various arithmetic operations with them.
Her Majesty fraction: what is it
In mathematics, fractions are numbers, each of which consists of one or more parts of a unit. Such fractions are also called ordinary or simple. As a rule, they are written in the form of two numbers that are separated by a horizontal or slash line, it is called a “fractional” line. For example: ½, ¾.
The upper, or first, of these numbers is the numerator (shows how many parts are taken from the number), and the lower, or second, is the denominator (demonstrates how many parts the unit is divided into).
The fraction bar actually functions as a division sign. For example, 7:9=7/9
Traditionally, common fractions are less than one. While decimals can be larger than it.
What are fractions for? Yes for everything, because in real world Not all numbers are integers. For example, two schoolgirls in the cafeteria bought one delicious chocolate bar together. When they were about to share dessert, they met a friend and decided to treat her too. However, now it is necessary to correctly divide the chocolate bar, considering that it consists of 12 squares.
At first, the girls wanted to divide everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their friend, not 1/3, but 1/4 of the chocolate. And since the schoolgirls did not study fractions well, they did not take into account that in such a situation they would end up with 9 pieces, which are very difficult to divide into two. This fairly simple example shows how important it is to be able to correctly find a part of a number. But in life there are many more such cases.
Types of fractions: ordinary and decimal
All mathematical fractions are divided into two large categories: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it’s worth paying attention to the second.
Decimal is a positional notation of a fraction of a number, which is written in writing separated by a comma, without a dash or slash. For example: 0.75, 0.5.
In fact, a decimal fraction is identical to an ordinary fraction, however, its denominator is always one followed by zeros - hence its name.
The number preceding the comma is an integer part, and everything after it is a fraction. Any simple fraction can be converted to a decimal. So, indicated in the previous example decimals can be written as usual: ¾ and ½.
It is worth noting that both decimal and ordinary fractions can be either positive or negative. If they are preceded by a “-” sign, this fraction is negative, if “+” is a positive fraction.
Subtypes of ordinary fractions
There are these types of simple fractions.
Subtypes of decimal fraction
Unlike a simple fraction, a decimal fraction is divided into only 2 types.
- Final - received this name due to the fact that after the decimal point it has a limited (finite) number of digits: 19.25.
- An infinite fraction is a number with an infinite number of digits after the decimal point. For example, when dividing 10 by 3, the result will be an infinite fraction 3.333...
Adding Fractions
Carrying out various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you understand the basic rules, solving any example with them will not be difficult.
For example: 2/3+3/4. The least common multiple for them will be 12, therefore, it is necessary that this number be in each denominator. To do this, we multiply the numerator and denominator of the first fraction by 4, it turns out 8/12, we do the same with the second term, but only multiply by 3 - 9/12. Now you can easily solve the example: 8/12+9/12= 17/12. The resulting fraction is an incorrect unit because the numerator is greater than the denominator. It can and should be transformed into a correct mixed one by dividing 17:12 = 1 and 5/12.
When mixed fractions are added, operations are performed first with whole numbers, and then with fractions.
If the example contains a decimal fraction and an ordinary fraction, it is necessary to make both simple, then bring them to the same denominator and add them. For example 3.1+1/2. The number 3.1 can be written as a mixed fraction of 3 and 1/10 or as an improper fraction - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator of 1/2 by 5 alternately, you get 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper reducible fraction, we bring it into normal form, reducing it by 5: 7/2 = 3 and 1/2, or decimal - 3.5.
When adding 2 decimal fractions, it is important that there are the same number of digits after the decimal point. If this is not the case, you just need to add required amount zeros, because in decimal fractions this can be done painlessly. For example, 3.5+3.005. To solve this problem, you need to add 2 zeros to the first number and then add one by one: 3.500+3.005=3.505.
Subtracting Fractions
When subtracting fractions, you should do the same as when adding: reduce to common denominator, subtract one numerator from the other, and, if necessary, convert the result to a mixed fraction.
For example: 16/20-5/10. The common denominator will be 20. You need to bring the second fraction to this denominator by multiplying both its parts by 2, you get 10/20. Now you can solve the example: 16/20-10/20= 6/20. However, this result applies to reducible fractions, so it is worth dividing both sides by 2 and the result is 3/10.
Multiplying fractions
Dividing and multiplying fractions are much simpler operations than addition and subtraction. The fact is that when performing these tasks, there is no need to look for a common denominator.
To multiply fractions, you simply need to multiply both numerators one by one, and then both denominators. Reduce the resulting result if the fraction is a reducible quantity.
For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. This fraction can be reduced by 4, so the final answer in the example is 5/18.
How to divide fractions
Dividing fractions is also a simple operation; in fact, it still comes down to multiplying them. To divide one fraction by another, you need to invert the second and multiply by the first.
For example, dividing the fractions 5/19 and 5/7. To solve the example, you need to swap the denominator and numerator of the second fraction and multiply: 5/19x7/5=35/95. The result can be reduced by 5 - it turns out 7/19.
If you need to divide a fraction by a prime number, the technique is slightly different. Initially, you should write this number as an improper fraction, and then divide according to the same scheme. For example, 2/13:5 should be written as 2/13: 5/1. Now you need to turn over 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.
Sometimes you have to divide mixed fractions. You need to treat them as you would with whole numbers: turn them into improper fractions, reverse the divisor and multiply everything. For example, 8 ½: 3. Convert everything into improper fractions: 17/2: 3/1. This is followed by a 3/1 flip and multiplication: 17/2x1/3= 17/6. Now you need to convert the improper fraction to the correct one - 2 whole and 5/6.
So, having figured out what fractions are and how you can perform various arithmetic operations with them, you need to try not to forget about it. After all, people are always more inclined to divide something into parts than to add, so you need to be able to do it correctly.
Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.
Correct A fraction is called in which the modulus of the numerator is greater than the modulus of the denominator. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.
The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:
The main property of a fraction
If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,
Reducing fractions to a common denominator
To bring two fractions to a common denominator, you need:
- Multiply the numerator of the first fraction by the denominator of the second
- Multiply the numerator of the second fraction by the denominator of the first
- Replace the denominators of both fractions with their product
Operations with fractions
Addition. To add two fractions you need
- Add the new numerators of both fractions and leave the denominator unchanged
Example:
Subtraction. To subtract one fraction from another, you need
- Reduce fractions to a common denominator
- Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged
Example:
Multiplication. To multiply one fraction by another, multiply their numerators and denominators:
Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:
Numerator and denominator of a fraction. Types of fractions. Let's continue looking at fractions. First, a small disclaimer - while we are considering fractions and corresponding examples with them, for now we will only work with its numerical representation. There are also fractional letter expressions (with and without numbers).However, all “principles” and rules also apply to them, but we will talk about such expressions separately in the future. I recommend visiting and studying (remembering) the topic of fractions step by step.
The most important thing is to understand, remember and realize that a FRACTION is a NUMBER!!!
Common fraction is a number of the form:
The number located “on top” (in this case m) is called the numerator, the number located below (number n) is called the denominator. Those who have just touched on the topic often have confusion about what they call it.
Here's a trick on how to forever remember where the numerator is and where the denominator is. This technique is associated with verbal-figurative association. Imagine a jar with muddy water. It is known that as water settles, clean water remains on top, and turbidity (dirt) settles, remember:
CHISS melt water ABOVE (CHISS litel top)
Grya Z33NN water is BELOW (ZNNNN amenator is below)
So, as soon as the need arises to remember where the numerator is and where the denominator is, we immediately visually imagined a jar of settled water with Pure water, and below dirty water. There are other memory tricks, if they help you, then good.
Examples ordinary fractions:
What does the horizontal line between numbers mean? This is nothing more than a division sign. It turns out that a fraction can be considered as an example of the action of division. This action is simply recorded in this form. That is, the top number (numerator) is divided by the bottom (denominator):
In addition, there is another form of notation - a fraction can be written like this (through a slash):
1/9, 5/8, 45/64, 25/9, 15/13, 45/64 and so on...
We can write the above fractions like this:
The result of division is how this number is known.
We figured it out - THIS IS A FRACTION NUMBER!!!
As you have already noticed, in a common fraction the numerator can be less than the denominator, it can be greater than the denominator, and it can be equal to it. There are many important points, which are intuitively understandable, without any theoretical refinements. For example:
1. Fractions 1 and 3 can be written as 0.5 and 0.01. Let's jump ahead a little - these are decimal fractions, we'll talk about them a little lower.
2. Fractions 4 and 6 result in the integer 45:9=5, 11:1 = 11.
3. Fraction 5 results in one 155:155 = 1.
What conclusions suggest themselves? Next:
1. The numerator when divided by the denominator can give a finite number. It may not work, divide with a column 7 by 13 or 17 by 11 - no way! You can divide endlessly, but we’ll also talk about this below.
2. A fraction can result in a whole number. Therefore, we can represent any integer as a fraction, or rather an infinite series of fractions, look, all these fractions are equal to 2:
More! We can always write any integer as a fraction - the number itself is in the numerator, the unit is in the denominator:
3. We can always represent a unit as a fraction with any denominator:
*These points are extremely important for working with fractions during calculations and transformations.
Types of fractions.
And now about the theoretical division of ordinary fractions. They are divided into right and wrong.
A fraction whose numerator is less than its denominator is called a proper fraction. Examples:
A fraction whose numerator is greater than or equal to the denominator is called an improper fraction. Examples:
Mixed fraction(mixed number).
A mixed fraction is a fraction written as a whole number and proper fraction and is understood as the sum of this number and its fractional part. Examples:
A mixed fraction can always be represented as improper fraction and vice versa. Let's move on!
Decimal fractions.
We have already touched on them above, these are examples (1) and (3), now in more detail. Here are examples of decimal fractions: 0.3 0.89 0.001 5.345.
A fraction whose denominator is a power of 10, such as 10, 100, 1000, etc., is called a decimal. It is not difficult to write the first three indicated fractions in the form of ordinary fractions:
The fourth is mixed fraction(mixed numbers):
The decimal fraction has the following form - withthe whole part begins, then the separator of the whole and fractional parts is a dot or comma and then the fractional part, the number of digits of the fractional part is strictly determined by the dimension of the fractional part: if these are tenths, the fractional part is written as one digit; if thousandths - three; ten thousandths - four, etc.
These fractions can be finite or infinite.
Examples of ending decimal fractions: 0.234; 0.87; 34.00005; 5.765.
The examples are endless. For example, the number Pi is an infinite decimal fraction, also – 0.333333333333…... 0.16666666666…. and others. Also the result of extracting the root of the numbers 3, 5, 7, etc. will be an infinite fraction.
The fractional part can be cyclic (it contains a cycle), the two examples above are exactly like this, and more examples:
0.123123123123…... cycle 123
0.781781781718...... cycle 781
0.0250102501…. cycle 02501
They can be written as 0,(123) 0,(781) 0,(02501).
The number Pi is not a cyclic fraction, like, for example, the root of three.
In the examples below, words such as “turning over” a fraction will sound - this means that the numerator and denominator are swapped. In fact, such a fraction has a name - reciprocal fraction. Examples of reciprocal fractions:
A small summary! Fractions are:
Ordinary (correct and incorrect).
Decimals (finite and infinite).
Mixed (mixed numbers).
That's all!
Sincerely, Alexander.
Common fraction
Quarters
- Orderliness. a And b there is a rule that allows you to uniquely identify one and only one of three relationships between them: “<
», « >" or " = ". This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relationship as two integers and ; two non-positive numbers a And b are related by the same relationship as two non-negative numbers and ; if suddenly a non-negative, but b- negative, then a > b.
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- Adding Fractions Addition operation. a And b For any rational numbers there is a so-called summation rule c summation rule. At the same time, the number itself called amount a And b numbers and is denoted by , and the process of finding such a number is called summation
.
- . The summation rule has the following form: Addition operation. a And b For any rational numbers Multiplication operation. multiplication rule summation rule c summation rule. At the same time, the number itself , which assigns them some rational number amount a And b work and is denoted by , and the process of finding such a number is also called multiplication
.
- . The multiplication rule looks like this: Transitivity of the order relation. a , b And summation rule For any triple of rational numbers a If b And b If summation rule less a If summation rule, That a, and if b And b, and if summation rule less a, and if summation rule equals
- . 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
- Associativity of addition. The order in which three rational numbers are added does not affect the result.
- Presence of zero. There is a rational number 0 that preserves every other rational number when added.
- The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
- Commutativity of multiplication. Changing the places of rational factors does not change the product.
- Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
- Availability of unit. Any rational number has an inverse rational number, which when multiplied by gives 1.
- Distributivity of multiplication relative to addition. The multiplication operation is coordinated with the addition operation through the distribution law:
- Connection of the order relation with the operation of addition. To the left and right parts rational inequality you can add the same rational number.
- /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0"> Axiom of Archimedes. a Whatever the rational number a, you can take so many units that their sum exceeds
.
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Additional properties
All other properties inherent in rational numbers are not distinguished as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proven based on the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense to list only a few of them here.
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Countability of a set
Numbering of rational numbers To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers. The simplest of these algorithms looks like this. An endless table of ordinary fractions is compiled, on each i-th line in each To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers. j i the th column of which the fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are denoted by , where
- the number of the table row in which the cell is located, and
- column number.
The resulting table is traversed using a “snake” according to the following formal algorithm. These rules are searched from top to bottom and the next position is selected based on the first match. In the process of such a traversal, each new rational number is associated with another
Following this algorithm, we can enumerate all positive rational numbers. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers by simply assigning to each rational number its opposite. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.
The statement about the countability of the set of rational numbers may cause some confusion, since at first glance it seems that it is much more extensive than the set of natural numbers. In fact, this is not so and there are enough natural numbers to enumerate all rational ones.
Lack of rational numbers
The hypotenuse of such a triangle cannot be expressed by any rational number
Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates the misleading impression that rational numbers can be used to measure any geometric distances. It is easy to show that this is not true.
From the Pythagorean theorem we know that the hypotenuse of a right triangle is expressed as the square root of the sum of the squares of its legs. That. length of the hypotenuse of an isosceles right triangle with a unit leg is equal to, i.e., a number whose square is 2.
If we assume that a number can be represented by some rational number, then there is such an integer m and such a natural number n, that , and the fraction is irreducible, i.e. numbers m And n- mutually simple.
If , then 
, i.e. m 2 = 2n 2. Therefore, the number m 2 is even, but the product of two odd numbers is odd, which means that the number itself m also even. So there is a natural number k, such that the number m can be represented in the form m = 2k. Number square m In this sense m 2 = 4k 2, but on the other hand m 2 = 2n 2 means 4 k 2 = 2n 2, or n 2 = 2k 2. As shown earlier for the number m, this means that the number n- even as m. But then they are not relatively prime, since both are bisected. The resulting contradiction proves that it is not a rational number.
Fractions are still considered one of the most difficult areas of mathematics. The history of fractions goes back more than one thousand years. The ability to divide a whole into parts arose in the territory ancient egypt and Babylon. Over the years, operations performed with fractions have become more complex, and the form of their recording has changed. Each had its own characteristics in its “relationship” with this branch of mathematics.
What is a fraction?
When it became necessary to divide a whole into parts without extra effort, then fractions appeared. The history of fractions is inextricably linked with the solution of utilitarian problems. The term “fraction” itself has Arabic roots and comes from a word meaning “to break, divide.” Little has changed in this sense since ancient times. The modern definition is as follows: a fraction is a part or sum of parts of a unit. Accordingly, examples with fractions represent sequential execution mathematical operations with fractions of numbers.
Today there are two ways to record them. arose in different time: the first ones are more ancient.
Came from time immemorial
For the first time they began to operate with fractions in Egypt and Babylon. The approach of the mathematicians of the two countries had significant differences. However, the beginning was made in the same way in both cases. The first fraction was half or 1/2. Then a quarter arose, a third, and so on. According to archaeological excavations, the history of the origin of fractions goes back about 5 thousand years. For the first time, fractions of a number are found in Egyptian papyri and on Babylonian clay tablets.
Ancient Egypt
Types of ordinary fractions today include the so-called Egyptian ones. They represent the sum of several terms of the form 1/n. The numerator is always one, and the denominator is a natural number. It’s hard to guess that such fractions appeared in ancient Egypt. When calculating, we tried to write down all shares in the form of such amounts (for example, 1/2 + 1/4 + 1/8). Only the fractions 2/3 and 3/4 had separate designations; the rest were divided into terms. There were special tables in which fractions of a number were presented as a sum.
The oldest known reference to such a system is found in the Rhind Mathematical Papyrus, dating from the beginning of the second millennium BC. It includes a table of fractions and math problems with solutions and answers presented as sums of fractions. The Egyptians knew how to add, divide and multiply fractions of a number. Fractions in the Nile Valley were written using hieroglyphs.
The representation of a fraction of a number as a sum of terms of the form 1/n, characteristic of ancient Egypt, was used by mathematicians not only in this country. Until the Middle Ages, Egyptian fractions were used in Greece and other countries.
Development of mathematics in Babylon
Mathematics looked different in the Babylonian kingdom. The history of the emergence of fractions here is directly related to the features of the number system inherited ancient state inherited from its predecessor, the Sumerian-Akkadian civilization. Calculation technology in Babylon was more convenient and more advanced than in Egypt. Mathematics in this country decided much larger circle tasks.
The achievements of the Babylonians today can be judged by the surviving clay tablets filled with cuneiform. Thanks to the peculiarities of the material, they have reached us in large quantities. According to some, a well-known theorem was discovered in Babylon before Pythagoras, which undoubtedly testifies to the development of science in this ancient state.
Fractions: The History of Fractions in Babylon
The number system in Babylon was sexagesimal. Each new digit differed from the previous one by 60. This system was preserved in modern world to indicate time and angles. Fractions were also sexagesimal. Special icons were used for recording. As in Egypt, examples with fractions contained separate symbols for 1/2, 1/3 and 2/3.
The Babylonian system did not disappear along with the state. Fractions written in the 60-digit system were used by ancient and Arab astronomers and mathematicians.
Ancient Greece
The history of ordinary fractions has been little enriched in ancient Greece. The inhabitants of Hellas believed that mathematics should operate only with integers. Therefore, expressions with fractions were practically never found on the pages of ancient Greek treatises. However, the Pythagoreans made a certain contribution to this branch of mathematics. They understood fractions as ratios or proportions, and the unit was also considered indivisible. Pythagoras and his disciples built general theory fractions, learned to carry out all four arithmetic operations, as well as comparing fractions by reducing them to a common denominator.
Holy Roman Empire
The Roman system of fractions was associated with a measure of weight called "ass". It was divided into 12 shares. 1/12 of an ace was called an ounce. There were 18 names for fractions. Here are some of them:
semis - half an assa;
sextante - the sixth part of the ass;
seven ounce - half an ounce or 1/24 ass.
The disadvantage of such a system was the impossibility of representing a number as a fraction with a denominator of 10 or 100. Roman mathematicians overcame the difficulty by using percentages.
Writing common fractions
In Antiquity, fractions were already written in a familiar way: one number over another. However, there was one significant difference. The numerator was located below the denominator. They first started writing fractions this way in ancient india. The modern method was used by the Arabs. But none of the named peoples used a horizontal line to separate the numerator and denominator. It first appears in the writings of Leonardo of Pisa, better known as Fibonacci, in 1202.
China
If the history of the emergence of ordinary fractions began in Egypt, then decimals first appeared in China. In the Celestial Empire they began to be used around the 3rd century BC. The history of decimal fractions began with the Chinese mathematician Liu Hui, who proposed their use in extracting square roots.
In the 3rd century AD, decimal fractions began to be used in China to calculate weight and volume. Gradually they began to penetrate deeper and deeper into mathematics. In Europe, however, decimals came into use much later.
Al-Kashi from Samarkand
Regardless of Chinese predecessors, decimal fractions were discovered by the astronomer al-Kashi from ancient city Samarkand. He lived and worked in the 15th century. The scientist outlined his theory in the treatise “The Key to Arithmetic,” which was published in 1427. Al-Kashi suggested using new uniform writing fractions. Both the integer and fractional parts were now written on the same line. The Samarkand astronomer did not use a comma to separate them. He wrote the whole number and the fractional part different colors using black and red ink. Sometimes al-Kashi also used a vertical line to separate.
Decimals in Europe
A new type of fractions began to appear in the works of European mathematicians in the 13th century. It should be noted that they were not familiar with the works of al-Kashi, as well as with the invention of the Chinese. Decimal fractions appeared in the writings of Jordan Nemorarius. Then they were used already in the 16th century by a French scientist who wrote the “Mathematical Canon,” which contained trigonometric tables. Viet used decimal fractions in them. To separate the whole and fractional parts, the scientist used a vertical line, as well as different size font.
However, these were only special cases of scientific use. Decimal fractions began to be used in Europe somewhat later to solve everyday problems. This happened thanks to the Dutch scientist Simon Stevin in late XVI century. He published the mathematical work "Tenth" in 1585. In it, the scientist outlined the theory of using decimal fractions in arithmetic, in the monetary system and for determining weights and measures.
Dot, dot, comma
Stevin also did not use a comma. He separated the two parts of the fraction using a zero surrounded by a circle.
The first time a comma separated two parts of a decimal fraction was in 1592. In England, however, they began to use a dot instead. In the United States, decimals are still written this way.
One of the initiators of the use of both punctuation marks to separate the integer and fractional parts was the Scottish mathematician John Napier. He made his proposal in 1616-1617. The German scientist also used the comma
Fractions in Rus'
On Russian soil, the first mathematician to explain the division of the whole into parts was the Novgorod monk Kirik. In 1136, he wrote a work in which he outlined the method of “counting years.” Kirik dealt with issues of chronology and calendar. In his work, he also cited the division of the hour into parts: fifths, twenty-fifths, and so on.
Dividing the whole into parts was used when calculating the amount of tax in the 15th-17th centuries. The operations of addition, subtraction, division and multiplication with fractional parts were used.
The word “fraction” itself appeared in Rus' in the 8th century. It comes from the verb “to split, to divide into parts.” Our ancestors used special words to name fractions. For example, 1/2 was designated as half or a half, 1/4 as a quarter, 1/8 as a half, 1/16 as a half and so on.
The complete theory of fractions, not much different from the modern one, was presented in the first textbook on arithmetic, written in 1701 by Leonty Filippovich Magnitsky. "Arithmetic" consisted of several parts. The author talks about fractions in detail in the section “On numbers broken or with fractions.” Magnitsky gives operations with “broken” numbers and their different designations.
Today, fractions are still among the most difficult branches of mathematics. The history of fractions has not been simple either. Different nations sometimes independently of each other, and sometimes borrowing the experience of predecessors, they came to the need to introduce, master and use fractions of numbers. The study of fractions has always grown out of practical observations and thanks to pressing problems. It was necessary to divide bread, mark out equal plots of land, calculate taxes, measure time, and so on. Features of the use of fractions and mathematical operations with them depended on the number system in the state and on general level development of mathematics. One way or another, having overcome more than one thousand years, the section of algebra devoted to fractions of numbers has been formed, developed and is successfully used today for a variety of needs, both practical and theoretical.