Studying the queen of all sciences - mathematics, at some point everyone is faced with fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is quite simple, it must be treated carefully, because in real life outside of school it will be very useful. So, let's refresh our knowledge of fractions: what is it, what is it for, what types of fractions are there and how to make various arithmetic operations.
Her Majesty the fraction: what is it
Fractions in mathematics are numbers, each of which consists of one or more parts of the unit. Such fractions are also called ordinary, or simple. As a rule, they are written as two numbers, which are separated by a horizontal or slash line, it is called a "fractional". For example: ½, ¾.
The top, or first of these numbers is the numerator (shows how many fractions of the number are taken), and the bottom, or second, is the denominator (demonstrates how many parts the unit is divided into).
The fractional bar actually functions as a division sign. For example, 7:9=7/9
Traditionally common fractions 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 together one delicious chocolate bar. When they were about to share dessert, they met a friend and decided to treat her as well. However, now it is necessary to correctly divide the chocolate bar, given that it consists of 12 squares.
At first, the girls wanted to share everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their girlfriend, not 1/3, but 1/4 chocolates. And since schoolgirls did not study fractions well, they did not take into account that in such a scenario, as a result, they would have 9 pieces that are very poorly divided into two. This rather simple example shows how important it is to be able to correctly find the 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 digits: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it is worth paying attention to the second.
A decimal is a positional notation of a fraction of a number, which is fixed in a letter 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 one, however, its denominator is always one followed by zeros - hence its name.
The number preceding the decimal point is the integer part, and everything after the decimal point is the fractional part. Any simple fraction can be converted to decimal. So, the decimal fractions indicated in the previous example can be written as ordinary ones: ¾ and ½.
It is worth noting that both decimal and ordinary fractions can be both positive and negative. If they are preceded by a "-" given fraction negative, if "+" - then positive.
Subspecies of ordinary fractions
There are such types of simple fractions.
Subspecies of the decimal fraction
Unlike a simple, decimal fraction is divided into only 2 types.
- Final - got its name due to the fact that after the decimal point it has a limited (final) 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 is infinite fraction 3,333…
Addition of fractions
Performing various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you learn 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 value because the numerator is greater than the denominator. It can and should be converted into the correct mixed one by dividing 17:12 = 1 and 5/12.
If mixed fractions are added, first the actions are performed with integers, and then with fractional ones.
If the example contains a decimal fraction and an ordinary one, it is necessary that both become 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 mixed fraction 3 and 1/10 or as incorrect - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator 1/2 by 5 in turn, it turns out 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper contractible fraction, we bring it to normal form, reducing it by 5: 7/2=3 and 1/2, or decimal - 3.5.
When adding 2 decimals, 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 task, you need to add 2 zeros to the first number and then add in turn: 3.500 + 3.005 = 3.505.
Subtraction of fractions
When subtracting fractions, it is worth doing the same as when adding: reduce to a common denominator, subtract one numerator from another, if necessary, convert the result into 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, multiplying both of 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 parts by 2 and the result is 3/10.
Multiplication of fractions
Division and multiplication of fractions - much more simple steps 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 just need to alternately multiply both numerators together, and then both denominators. Reduce the resulting result if the fraction is a reduced value.
For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. Such a 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 action, in fact it still comes down to multiplying them. To divide one fraction by another, you need to flip the second and multiply by the first.
For example, division of 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, it is worth writing this number as an improper fraction, and then dividing according to the same scheme. For example, 2/13:5 should be written as 2/13:5/1. Now you need to flip 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.
Sometimes you have to divide mixed fractions. You need to deal with them, as with integers: turn them into improper fractions, flip the divisor and multiply everything. For example, 8 ½: 3. Turning 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 should translate the wrong fraction into the right one - 2 integers 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 right.
Speaking of mathematics, one cannot help but remember fractions. Their study is given a lot of attention and time. Remember how many examples you had to solve in order to learn certain rules for working with fractions, how you memorized and applied the main property of a fraction. How many nerves were spent to find a common denominator, especially if there were more than two terms in the examples!
Let's remember what it is, and refresh our memory a little about the basic information and rules for working with fractions.
Definition of fractions
Let's start with the most important thing - definitions. A fraction is a number that consists of one or more unit parts. A fractional number is written as two numbers separated by a horizontal or slash. In this case, the upper (or first) is called the numerator, and the lower (second) is called the denominator.
It is worth noting that the denominator shows how many parts the unit is divided into, and the numerator shows the number of shares or parts taken. Often fractions, if they are correct, are less than one.
Now let's look at the properties of these numbers and the basic rules that are used when working with them. But before we analyze such a thing as "basic property rational fraction Let's talk about the types of fractions and their features.
What are fractions
There are several types of such numbers. First of all, these are ordinary and decimal. The first are the type of record already indicated by us using a horizontal or slash. The second type of fractions is indicated using the so-called positional notation, when the integer part of the number is indicated first, and then, after the decimal point, the fractional part is indicated.
It is worth noting here that in mathematics both decimal and ordinary fractions are used equally. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions, correct and wrong numbers. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than unity. In an improper fraction, on the contrary, the numerator is greater than the denominator, and it itself is greater than one. In this case, an integer can be extracted from it. In this article, we will consider only ordinary fractions.
Fraction properties
Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers are no exception. They have one important feature, with the help of which it is possible to carry out certain operations on them. What is the main property of a fraction? The rule says that if its numerator and denominator are multiplied or divided by the same rational number, we will get a new fraction, the value of which will be equal to the original value. That is, multiplying the two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, while they will be equal.
Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.
Operations
Although fractions seem more complex to us, they can also perform basic mathematical operations, such as addition and subtraction, multiplication and division. In addition, there is such a specific action as the reduction of fractions. Naturally, each of these actions is performed according to certain rules. Knowing these laws makes it easier to work with fractions, making it easier and more interesting. That is why further we will consider the basic rules and the algorithm of actions when working with such numbers.
But before we talk about such mathematical operations as addition and subtraction, we will analyze such an operation as reduction to a common denominator. This is where the knowledge of what basic property of a fraction exists will come in handy.
Common denominator
In order to reduce a number to a common denominator, you first need to find the least common multiple of the two denominators. That is smallest number, which is simultaneously divisible by both denominators without a remainder. The easiest way to find the LCM (least common multiple) is to write in a line for one denominator, then for the second and find a matching number among them. In the event that the LCM is not found, that is, these numbers do not have a common multiple, they should be multiplied, and the resulting value should be considered as the LCM.
So, we have found the LCM, now we need to find an additional multiplier. To do this, you need to alternately divide the LCM into denominators of fractions and write down the resulting number over each of them. Next, multiply the numerator and denominator by the resulting additional factor and write the results as a new fraction. If you doubt that the number you received is equal to the previous one, remember the main property of the fraction.
Addition
Now let's go directly to mathematical operations on fractional numbers. Let's start with the simplest. There are several options for adding fractions. In the first case, both numbers have the same denominator. In this case, it remains only to add the numerators together. But the denominator does not change. For example, 1/5 + 3/5 = 4/5.
If the fractions have different denominators, they should be reduced to a common one and only then the addition should be performed. How to do this, we have discussed with you a little higher. In this situation, the main property of the fraction will come in handy. The rule will allow you to bring the numbers to a common denominator. The value will not change in any way.
Alternatively, it may happen that the fraction is mixed. Then you should first add together the whole parts, and then the fractional ones.
Multiplication
It does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of the fraction. It is enough to first multiply the numerators and denominators together. In this case, the product of the numerators will become the new numerator, and the product of the denominators will become the new denominator. As you can see, nothing complicated.
The only thing that is required of you is knowledge of the multiplication table, as well as attentiveness. In addition, after receiving the result, you should definitely check whether this number can be reduced or not. We will talk about how to reduce fractions a little later.
Subtraction
Performing should be guided by the same rules as when adding. So, in numbers with the same denominator, it is enough to subtract the numerator of the subtrahend from the numerator of the minuend. In the event that the fractions have different denominators, you should bring them to a common one and then perform this operation. As with the analogous addition case, you will need to use the main property algebraic fraction, as well as skills in finding LCM and common divisors for fractions.
Division
And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause any particular difficulties even for those who do not understand how to work with fractions, especially to perform addition and subtraction operations. When dividing, the rule is to multiply by reciprocal. The main property of a fraction, as in the case of multiplication, will not be used for this operation. Let's take a closer look.
When dividing numbers, the dividend remains unchanged. The divisor is reversed, i.e. the numerator and denominator are reversed. After that, the numbers are multiplied with each other.
Reduction
So, we have already examined the definition and structure of fractions, their types, the rules of operations on given numbers, and found out the main property of an algebraic fraction. Now let's talk about such an operation as reduction. Reducing a fraction is the process of converting it - dividing the numerator and denominator by the same number. Thus, the fraction is reduced without changing its properties.
Usually, when performing a mathematical operation, you should carefully look at the result obtained in the end and find out whether it is possible to reduce the resulting fraction or not. Remember that the final result is always written as a fractional number that does not require reduction.
Other operations
Finally, we note that we have listed far from all operations on fractional numbers, mentioning only the most famous and necessary. Fractions can also be compared, converted to decimals, and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less frequently than those that we have given above.
conclusions
We talked about fractional numbers and operations with them. We also analyzed the main property. But we note that all these issues were considered by us in passing. We have given only the most well-known and used rules, we have given the most important, in our opinion, advice.
This article is intended to refresh the information you have forgotten about fractions rather than to give new information and "fill" your head with endless rules and formulas, which, most likely, will not be useful to you.
We hope that the material presented in the article simply and concisely has become useful to you.
Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - these are such simple and harmless mathematical objects that surpass the rest of the algebra course in their ability to “take out the brain”.
The main danger of fractions is that they occur in real life. In this they differ, for example, from polynomials and logarithms, which can be passed and easily forgotten after the exam. Therefore, the material presented in this lesson, without exaggeration, can be called explosive.
A numeric fraction (or simply a fraction) is a pair of integers written through a slash or horizontal bar.
Fractions written through a horizontal bar:
The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.
Usually fractions are written through a horizontal line - it's easier to work with them, and they look better. The number written on top is called the numerator of the fraction, and the number written on the bottom is called the denominator.
Any whole number can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the above example.
In general, you can put any whole number in the numerator and denominator of a fraction. The only restriction is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”
If the denominator is still zero, the fraction is called indefinite. Such a record does not make sense and cannot participate in calculations.
Basic property of a fraction
Fractions a /b and c /d are called equal if ad = bc.
From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4 because 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4 because 1 4 ≠ 3 5.
A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:
The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.
This is a very important property - remember it. With the help of the basic property of a fraction, many expressions can be simplified and shortened. In the future, it will constantly “emerge” in the form of various properties and theorems.
Incorrect fractions. Selection of the whole part
If the numerator is less than the denominator, such a fraction is called proper. Otherwise (that is, when the numerator is greater than or at least equal to the denominator), the fraction is called an improper fraction, and an integer part can be distinguished in it.
The integer part is written as a large number in front of the fraction and looks like this (marked in red):
To isolate the whole part in an improper fraction, you need to follow three simple steps:
- Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (in the extreme case, equal). This number will be the integer part, so we write it in front;
- Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting "stub" is called the remainder of the division, it will always be positive (in extreme cases, zero). We write it down in the numerator of the new fraction;
- We rewrite the denominator unchanged.
Well, is it difficult? At first glance, it may be difficult. But it takes a little practice - and you will do it almost verbally. For now, take a look at the examples:
A task. Select the whole part in the given fractions:
In all examples, the integer part is highlighted in red, and the remainder of the division is in green.
Pay attention to the last fraction, where the remainder of the division turned out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 \u003d 4 is a harsh fact from the multiplication table.
If everything is done correctly, the numerator of the new fraction will necessarily be less than the denominator, i.e. fraction becomes correct. I also note that it is better to highlight the whole part at the very end of the task, before writing the answer. Otherwise, you can significantly complicate the calculations.
Transition to improper fraction
There is also an inverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because improper fractions are much easier to work with.
The transition to an improper fraction is also done in three steps:
- Multiply the integer part by the denominator. The result can be quite big numbers, but we should not be embarrassed;
- Add the resulting number to the numerator of the original fraction. Write the result in the numerator of an improper fraction;
- Rewrite the denominator - again, no change.
Here are specific examples:
A task. Convert to an improper fraction:
For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is in green.
Consider the case when the numerator or denominator of a fraction contains a negative number. For example:
In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to take out minuses as a fraction sign.
This is very easy to do if you remember the rules:
- Plus times minus equals minus. Therefore, if there is a negative number in the numerator, and a positive number in the denominator (or vice versa), feel free to cross out the minus and put it in front of the whole fraction;
- "Two negatives make an affirmative". When the minus is in both the numerator and the denominator, we simply cross them out - no additional action is required.
Of course, these rules can also be applied in the opposite direction, i.e. you can add a minus under the fraction sign (most often - in the numerator).
We deliberately do not consider the case of “plus on plus” - with him, I think, everything is clear anyway. Let's take a look at how these rules work in practice:
A task. Take out the minuses of the four fractions written above.
Pay attention to the last fraction: it already has a minus sign in front of it. However, it is “burned” according to the rule “minus times minus gives plus”.
Also, do not move minuses in fractions with a highlighted integer part. These fractions are first converted to improper ones - and only then they begin to calculate.
Common fraction
quarters
- Orderliness. a and b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “<
», « >' or ' = '. This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a and b are related by the same relation as two non-negative numbers and ; if suddenly a non-negative, and b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
summation of fractions
- addition operation. For any rational numbers a and b there is a so-called summation rule c. However, the number itself c called sum numbers a and b and is denoted , and the process of finding such a number is called summation. The summation rule has next view:
.
- multiplication operation. For any rational numbers a and b there is a so-called multiplication rule, which puts them in correspondence with some rational number c. However, the number itself c called work numbers a and b and is denoted , and the process of finding such a number is also called multiplication. The multiplication rule is as follows:
.
- Transitivity of the order relation. For any triple of rational numbers a , b and c if a less b and b less c, then a less c, what if a equals b and b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
- Associativity of addition. The order in which three rational numbers are added does not affect the result.
- The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
- The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
- Commutativity of multiplication. By changing the places of rational factors, the product does not change.
- Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
- The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
- The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
- Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
- Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
- Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">
Additional properties
All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. Such additional properties lots of. It makes sense here to cite just a few of them.
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Set countability
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 suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers.
The simplest of these algorithms is as follows. An infinite table of ordinary fractions is compiled, on each i-th line in each j th column of which is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted , where i- the row number of the table in which the cell is located, and j- column number.
The resulting table is managed by a "snake" according to the following formal algorithm.
These rules are searched from top to bottom and the next position is selected by the first match.
In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions 1 / 1 are assigned the number 1, fractions 2 / 1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to unity of the greatest common divisor of the numerator and denominator of the fraction.
Following this algorithm, one 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, simply by 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 bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.
Insufficiency of rational numbers
The hypotenuse of such a triangle is not expressed by any rational number
Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates a deceptive impression that rational numbers can measure any geometric distances in general. It is easy to show that this is not true.
It is known from the Pythagorean theorem that the hypotenuse of a right triangle is expressed as the square root of the sum of the squares of its legs. That. isosceles hypotenuse length right triangle with a single leg is equal to, i.e., a number whose square is 2.
If we assume that the number is represented by some rational number, then there is such an integer m and such a natural number n, which, moreover, the fraction is irreducible, i.e., the numbers m and n are coprime.
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 clear. So there is a natural number k, such that the number m can be represented as 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, which means that the number n- exactly like m. But then they are not coprime, since both are divisible in half. The resulting contradiction proves that is not a rational number.
Fractions are considered to be one of the most difficult sections of mathematics to this day. The history of fractions has more than one millennium. The ability to divide the whole into parts arose in the territory ancient egypt and Babylon. Over the years, the operations performed with fractions became more complicated, the form of their recording changed. Each had its own characteristics in the "relationship" with this branch of mathematics.
What is a fraction?
When it became necessary to divide the whole into parts without extra effort, then there were fractions. 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 "break, divide." Since ancient times, little has changed in this sense. The modern definition is as follows: a fraction is a part or the sum of parts of a unit. Accordingly, examples with fractions represent a sequential execution of mathematical operations with fractions of numbers.
Today, there are two ways to record them. arose in different time: the former are more ancient.
Came from ancient times
For the first time they began to operate with fractions on the territory of Egypt and Babylon. The approach of the mathematicians of the two states had significant differences. However, the beginning was the same there and there. The first fraction was half or 1/2. Then came a quarter, a third, and so on. According to archaeological excavations, the history of the emergence of fractions has 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. They are the sum of several terms of the form 1/n. The numerator is always one, and the denominator is a natural number. Such fractions appeared, no matter how hard it is to guess, in ancient Egypt. When calculating all the shares, they tried to write them down in the form of such sums (for example, 1/2 + 1/4 + 1/8). Only 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 Rhinda Mathematical Papyrus, dated to 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 states.
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. The calculation technique in Babylon was more convenient and perfect than in Egypt. Mathematics in this country solved a much wider range of problems.
One can judge the achievements of the Babylonians today by the surviving clay tablets filled with cuneiform writing. Due to the characteristics of the material, they have come down to us in in large numbers. According to some in Babylon, a well-known theorem was discovered 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 category differed from the previous one by 60. This system was preserved in modern world to indicate time and angles. Fractions were also sexagesimal. For recording, special icons were used. As in Egypt, the fraction examples contained separate symbols for 1/2, 1/3, and 2/3.
The Babylonian system did not disappear with the state. Fractions written in the 60th system were used by ancient and Arabic astronomers and mathematicians.
Ancient Greece
The history of ordinary fractions has not been enriched much in ancient greece. The inhabitants of Hellas believed that mathematics should operate only with whole numbers. Therefore, expressions with fractions on the pages of ancient Greek treatises practically did not occur. However, the Pythagoreans made a certain contribution to this branch of mathematics. They understood fractions as ratios or proportions, and they also considered the unit to be indivisible. Pythagoras and his students built general theory fractions, learned how to carry out all four arithmetic operations, as well as comparing fractions by bringing 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 assa was called an ounce. There were 18 names for fractions. Here are some of them:
semis - half of the assa;
sextante--sixth of assa;
semi-ounce - half an ounce or 1/24 ass.
The inconvenience 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 ordinary fractions
In Antiquity, fractions were already written in a familiar way: one number over another. However, there was one significant difference. The numerator was below the denominator. For the first time, fractions began to be written in this way in ancient India. The Arabs began to use the modern way for us. But none of these 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 from about the 3rd century BC. The history of decimal fractions began with the Chinese mathematician Liu Hui, who proposed to use them when extracting square roots.
In the 3rd century AD, decimal fractions in China began to be used 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 form fraction records. Both integer and fractional parts were now written in one 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 them.
Decimals in Europe
A new kind of fractions began to appear in the works of European mathematicians from 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. The French scientist wrote the Mathematical Canon, which contained trigonometric tables. In them, Viet used decimal fractions. To separate the integer and fractional parts, the scientist used a vertical line, as well as different size font.
However, these were only special cases of scientific use. To solve everyday problems, decimal fractions in Europe began to be used somewhat later. This happened thanks to the Dutch scientist Simon Stevin at the end of the 16th century. He published the mathematical work The Tenth in 1585. In it, the scientist outlined the theory of using decimal fractions in arithmetic, in the monetary system, and to determine measures and weights.
Period, period, comma
Stevin also didn't use a comma. He separated the two parts of the fraction using a zero circled.
For the first time, a comma separated two parts of a decimal fraction only in 1592. In England, however, the dot was used instead. In the United States, decimal fractions are still written in this way.
One of the initiators of the use of both punctuation marks to separate integer and fractional parts was the Scottish mathematician John Napier. He made his proposal in 1616-1617. A comma was also used by a German scientist
Fractions in Russia
On Russian soil, the first mathematician who outlined 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 "calculating 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.
The division of the whole into parts was used when calculating the amount of tax in the XV-XVII centuries. The operations of addition, subtraction, division and multiplication with fractional parts were used.
The very word "fraction" appeared in Russia in the VIII century. It comes from the verb "to crush, divide into parts." Our ancestors used special words to name fractions. For example, 1/2 was designated as half or half, 1/4 - four, 1/8 - half an hour, 1/16 - half an hour, 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 of broken lines or with fractions”. Magnitsky gives operations with "broken" numbers, their different designations.
Today, fractions are still among the most difficult sections of mathematics. The history of fractions was also not simple. different peoples sometimes independently of each other, and sometimes borrowing the experience of their predecessors, they came to the need to introduce, master and use fractions of a number. The doctrine of fractions has always grown out of practical observations and thanks to pressing problems. It was necessary to divide the bread, mark equal plots 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 formed, developed and is successfully used today for a variety of needs, both practical and theoretical.
