Show types of fractions. Positive and negative fractions

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 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 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:

1. 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;
2. 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;
3. 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:

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:

1. Multiply the integer part by the denominator. The result can be quite big numbers, but we should not be embarrassed;
2. Add the resulting number to the numerator of the original fraction. Write the result in the numerator of an improper fraction;
3. Rewrite the denominator - again, no change.

Here are specific examples:

Task. Translate to 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:

1. Plus times minus equals minus. Therefore, if the numerator is a negative number, and the denominator is positive (or vice versa), feel free to cross out the minus and put it in front of the whole fraction;
2. "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:

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.

The numerator, and that by which it is divided is the denominator.

To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as an oblique "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.

To calculate the product of fractions, first multiply the numerator of one fractions to another numerator. Write the result to the numerator of the new fractions. Then multiply the denominators as well. Specify the final value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).

To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the steps, first “flip” the divisor, if it’s more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by the new denominator of the divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 × 5 = 5; 3 × 1 = 3).

Sources:

• Basic tasks for fractions

Fractional numbers allow you to express in different form exact value quantities. With fractions, you can perform the same mathematical operations as with integers: subtraction, addition, multiplication, and division. To learn how to decide fractions, it is necessary to remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution require reduction of the fractional part of the result.

You will need

• - calculator

Instruction

Look carefully at the numbers. If there are decimals and irregulars among the fractions, it is sometimes more convenient to first perform actions with decimals, and then convert them to the wrong form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part stands out, lead to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given values will become the new numerator fractions. To extract the whole part from the initially incorrect fractions, divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division becomes the new numerator, the denominator fractions while not changing. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation separately of integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 + (8/12 + 9/12) = 3 + 17/12 = 3 + 1 5/12 = 4 5 /12.

Rewrite them through the separator ":" and continue the usual division.

To get the final result, reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integer numbers above and below the line.

note

Don't do arithmetic with fractions that have different denominators. Choose a number such that when the numerator and denominator of each fraction are multiplied by it, as a result, the denominators of both fractions are equal.

Helpful advice

When recording fractional numbers the dividend is written above the line. This quantity is referred to as the numerator of a fraction. Under the line, the divisor, or denominator, of the fraction is written. For example, one and a half kilograms of rice in the form of a fraction will be written in the following way: 1 ½ kg rice. If the denominator of a fraction is 10, it is called a decimal fraction. In this case, the numerator (dividend) is written to the right of the whole part separated by a comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values ​​by dividing them by a single whole number. AT this example dividing by 2 is possible. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to do arithmetic with are in the same form.

Common fraction

quarters

1. 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

2. 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: .
3. 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: .
4. Transitivity of the order relation. For any triple of rational numbers a , b and c if a smaller b and b smaller c, then a smaller c, and 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.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
8. Commutativity of multiplication. By changing the places of rational factors, the product does not change.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
12. Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
13. Connection of the order relation with the operation of addition. To the left and right rational inequality you can add the same rational number. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. 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, i.e., establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An endless table is being compiled ordinary fractions, 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.

We encounter fractions in life much earlier than they begin to study at school. If you cut a whole apple in half, then we get a piece of fruit - ½. Cut it again - it will be ¼. This is what fractions are. And everything, it would seem, is simple. For an adult. For a child (and they begin to study this topic at the end of elementary school), abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must explain in an accessible way what proper fraction and incorrect, ordinary and decimal, what operations can be performed with them and, most importantly, why all this is needed.

What are fractions

Acquaintance with a new topic at school begins with ordinary fractions. They are easy to recognize by the horizontal line separating the two numbers - above and below. The top is called the numerator, the bottom is called the denominator. There is also a lower case spelling of improper and proper ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to apply the "two-story" form of the record. Why? Yes, because it is more convenient. A little later we will verify this.

In addition to the usual ones, there are also decimals. It is very easy to distinguish between them: if in one case a horizontal or slash is used, then in the other - a comma separating sequences of numbers. Let's see an example: 2.9; 163.34; 1.953. We deliberately used the semicolon as a delimiter to delimit the numbers. The first of them will be read like this: "two whole, nine tenths."

New concepts

Let's go back to ordinary fractions. They are of two kinds.

The definition of a proper fraction is as follows: it is such a fraction, the numerator of which is less than the denominator. Why is it important? Now we'll see!

You have several apples cut into halves. In total - 5 parts. How do you say: you have "two and a half" or "five second" apples? Of course, the first option sounds more natural, and when talking with friends, we will use it. But if you need to calculate how much fruit each will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from the point of view of mathematics, this will be clearer.

So, for naming regular and improper fractions, the rule is as follows: if an integer part (14/5, 2/1, 173/16, 3/3) can be distinguished in a fraction, then it is incorrect. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.

Basic property of a fraction

If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value will not change. Imagine: the cake was cut into 4 equal parts and they gave you one. The same cake was cut into eight pieces and given you two. Isn't it all the same? After all, ¼ and 2/8 are the same thing!

Reduction

Authors of problems and examples in math textbooks often try to confuse students by offering fractions that are cumbersome to write and can actually be reduced. Here is an example of a proper fraction: 167/334, which, it would seem, looks very "scary". But in fact, we can write it as ½. The number 334 is divisible by 167 without a remainder - having done this operation, we get 2.

mixed numbers

An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.

To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division above, above the line, and the whole part before the expression. Thus, we get two structural parts: whole units + proper fraction.

You can also carry out the reverse operation - for this you need to multiply the integer part by the denominator and add the resulting value to the numerator. Nothing complicated.

Multiplication and division

Oddly enough, multiplying fractions is easier than adding them. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2*3 / 3*5 = 2/5.

With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) \u003d 7 * 15 / 8 * 14 \u003d 15/16.

Addition of fractions

What if you need to perform addition or if they have different numbers in the denominator? It will not work in the same way as with multiplication - here one should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the same numbers should appear at the bottom of both fractions.

To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2*2)/(5*2) + 1/10 = 5/10 = ½.

How to choose which denominator to bring the terms to? This must be the smallest multiple of both denominators: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without a remainder.

Usage

What are improper fractions for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and that's it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use the wrong ones.

Let's take the following example: (2 + 3/17) / (37 / 68).

It would seem that there is nothing to cut at all. But what if we write the result of the addition in the first brackets as an improper fraction? Look: (37/17) / (37/68)

Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37 * 68) / (17 * 37).

Let's reduce the 37 in the numerator and denominator, and finally divide the top and bottom parts by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number, as long as we do it for the numerator and denominator at the same time.

So, we get the answer: 4. The example looked complicated, and the answer contains only one digit. This often happens in mathematics. The main thing is not to be afraid and follow simple rules.

Common Mistakes

When exercising, the student can easily make one of the popular mistakes. Usually they occur due to inattention, and sometimes due to the fact that the studied material has not yet been properly deposited in the head.

Often the sum of the numbers in the numerator causes a desire to reduce its individual components. Suppose, in the example: (13 + 2) / 13, written without brackets (with a horizontal line), many students, due to inexperience, cross out 13 from above and below. But this should not be done in any case, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when adding, no operations with one of the terms are allowed, only with the entire sum.

Children often make mistakes when dividing fractions. Let's take two regular irreducible fractions and divide by each other: (5/6) / (25/33). The student can confuse and write the resulting expression as (5*25) / (6*33). But this would have happened with multiplication, and in our case everything will be a little different: (5 * 33) / (6 * 25). We reduce what is possible, and in the answer we will see 11/10. We write the resulting improper fraction as a decimal - 1.1.

Parentheses

Remember that in any mathematical expression, the order of operations is determined by the precedence of operation signs and the presence of brackets. Other things being equal, the sequence of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.

It is the result of dividing one number by another. If they do not divide completely, it turns out a fraction - that's all.

How to write a fraction on a computer

Since standard tools do not always allow you to create a fraction consisting of two "tiers", students sometimes go for various tricks. For example, they copy the numerators and denominators into the Paint editor and glue them together by drawing between them horizontal line. Of course, there is a simpler option, which, by the way, provides a lot additional features that will be useful to you in the future.

Open Microsoft Word. One of the panels at the top of the screen is called "Insert" - click it. On the right, on the side where the icons for closing and minimizing the window are located, there is a Formula button. This is exactly what we need!

If you use this function, a rectangular area will appear on the screen in which you can use any mathematical symbols that are not available on the keyboard, as well as write fractions in classical form. That is, separating the numerator and denominator with a horizontal bar. You may even be surprised that such a proper fraction is so easy to write down.

Learn Math

If you are in grade 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) Will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, fractions cannot be dispensed with. Soon you will learn to calculate everything in your mind, without even writing expressions on paper, but more and more complex examples. Therefore, learn what a proper fraction is and how to work with it, keep up with the curriculum, do your homework on time, and then you will succeed.

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 of 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 unnecessary 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 "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 they began to write fractions 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 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 full stop 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 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 issues. It was necessary to divide bread, mark 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 the fractions of numbers has formed, developed and is successfully used today for a variety of needs, both practical and theoretical.