What an ordinary fraction looks like. Ordinary fraction
This article pro ordinary fractions. Here we will get acquainted with the concept of the share of the whole, which will lead us to the definition of ordinary fraction. Further we will stop on the adopted designations for ordinary fractions and give examples of fractions, let's say about the numerator and denominator of the fraction. After that, we will give the definition of correct and incorrect, positive and negative fractions, as well as consider the situation of fractional numbers on the coordinate beam. In conclusion, we list the main steps with fractions.
Navigating page.
Founding
First introduce the concept of a share.
Suppose that we have some object compiled from several completely identical (that is, equal to) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange consisting of several equal lobes. Each of these equal parts constituting a whole subject, called fraction of the whole or simply share.
Note that the shares are different. Let's explain it. Let we have two apples. We cut the first apple into two equal parts, and the second - on 6 equal parts. It is clear that the proportion of the first apple will differ from the share of the second apple.
Depending on the number of shares that make up a whole subject, these shares have their own names. We will understand names. If the subject is two shares, any of them is called one second share of a whole object; If the subject is three shares, any of them is called one third share, and so on.
One second share has a special name - half. One third share is called third, and one quadruple share - quarter.
For brief recording, the following were introduced designations of Share. One second share is referred to as or 1/2, one third share - like 1/3; One fourth share - like 1/4, and so on. Note that the record with a horizontal feature is used more often. To secure the material, we give another example: the record indicates one hundred and sixty-seventh fraction of the whole.
The concept of shares naturally spreads from items by magnitude. For example, one of the measurement measures is a meter. To measure lower lengths than meter, you can use the meter shares. This can use, for example, half meter or tenth or thousandth meters. Similarly, the shares of other values \u200b\u200bare used.
Ordinary fractions, definition and examples of fractions
To describe the number of shares are used ordinary fractions. Let us give an example that will allow us to approach the definition of ordinary fractions.
Let the orange consists of 12 fractions. Each share in this case represents one twelfth share of the whole orange, that is,. Two shares are denoted by, three shares - like, and so on, we denote by 12 stakes as. Each of the above records is called an ordinary fraction.
Now give general definition of ordinary fractions.
The voiced definition of ordinary fractions allows you to bring examples of ordinary fractions: 5/10, 21/1, 9/4 ,. But records 
Not suitable for the voiced definition of ordinary fractions, that is, are not ordinary fractions.
Numerator and denominator
For convenience in ordinary fraction distinguish numerator and denominator.
Definition.
Numerator Ordinary fraction (M / N) is a natural number m.
Definition.
Denominator Ordinary fraction (m / n) is a natural number N.
So, the numerator is located on top above the fraction (left of the inclined line), and the denominator is from below below the fraction (to the right of the inclined line). For example, we give an ordinary fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.
It remains to discuss the meaning concluded in the numerator and denominator of the ordinary fraction. An indicator of the fraction shows, one object consists of many fractions, the numerator in turn indicates the number of such fractions. For example, denominator 5 fractions 12/5 means that one object consists of five pieces, and the numerator 12 means that 12 such fractions are taken.
Natural number as a fraction with denominator 1
An indicator of ordinary fraction can be equal to one. In this case, we can assume that the subject of weathering, in other words, is something. The numerator of such a fraction indicates how much items are taken. In this way, ordinary fraction The species M / 1 has the meaning of the natural number M. So we substantiated the validity of the equality M / 1 \u003d m.
I rewrite the last equality: m \u003d m / 1. This equality gives us the possibility of any natural number M representing in the form of an ordinary fraction. For example, the number 4 is a fraction 4/1, and the number 103 498 is the fraction 103 498/1.
So, any natural number M can be represented as an ordinary fraction with a denominator 1 as M / 1, and any ordinary fraction of the form M / 1 can be replaced with a natural number M.
Damn fraction as a sign of division
The representation of the initial object in the form of n shares is nothing more than dividing on n equal parts. After the subject is divided into N share, we can divide it equally between N people - everyone will receive in one share.
If we have initially m identical objects, each of which is divided into N share, then these M objects we can equally divide between n people, distribute to each person in one share of each of the objects. At the same time, each person will have M shares 1 / n, and M shares 1 / N gives an ordinary fraction M / N. Thus, the ordinary fraction M / n can be used to designate division M of objects between n people.
So we received a clear connection between ordinary fractions and division (see the general idea of \u200b\u200bdividing natural numbers). This connection is expressed as follows: damage fraction can be understood as a sign of division, that is, m / n \u003d m: n.
Using an ordinary fraction, you can record the result of dividing two natural numbersFor which the division is not performed. For example, the result of dividing 5 apples for 8 people can be written as 5/8, that is, everyone will get five eighth apple shares: 5: 8 \u003d 5/8.
Equal and unequal ordinary fractions, fraction comparison
Enough natural action is comparison of ordinary fractionsBut it is clear that 1/12 orange is different from 5/12, and 1/6 of the apple share is the same as another 1/6 share of this apple.
As a result of the comparison of two ordinary fractions, one of the results is obtained: the fractions are either equal or not equal. In the first case we have equal ordinary fractions, and in the second - unequal ordinary fractions. We give the definition of equal and unequal ordinary fractions.
Definition.
equalif equality A · d \u003d b · c.
Definition.
Two ordinary fractions A / B and C / D not equalIf the equality A · d \u003d b · C is not performed.
Let us give a few examples of equal fractions. For example, an ordinary fraction of 1/2 is equal to 2/4, as 1 · 4 \u003d 2 · 2 (if necessary, see the rules and examples of multiplication of natural numbers). For clarity, you can imagine two identical apples, the first cut in half, and the second - on 4 stakes. It is obvious that the two fourth shares of the apple make up 1/2 share. Other examples of equal ordinary fractions are fractions 4/7 and 36/63, as well as a pair of fractions 81/50 and 1,620/1,000.
And ordinary fractions 4/13 and 5/14 are not equal, since 4 · 14 \u003d 56, and 13 · 5 \u003d 65, that is, 4 · 14 ≠ 13 · 5. Another example of unequal ordinary fractions are the fractions 17/7 and 6/4.
If, when comparing two ordinary fractions, it turned out that they are not equal, it may be necessary to know which of these ordinary fractions less another, and what - more. To find out, a rule of comparison of ordinary fractions is used, the essence of which is reduced to bringing compared fractions to the general denominator and the subsequent comparison of the numerators. Detailed information on this topic is collected in the article Comparison of fractions: Rules, examples, solutions.
Fractional numbers
Each fraction is a record fractional number. That is, the fraction is just a "shell" of a fractional number, its appearance, and all the selligent load is contained in the fractional number. However, for brevity and convenience, the concept of a fraction and fractional number is combined and said simply fraction. It is appropriate to rephrase the famous saying: we speak fraction - mean a fractional number, we say a fractional number - we mean the fraction.
Fraction on the coordinate beam
All fractional numbers that correspond to ordinary fractions have their own unique place on, that is, there is a mutually unique correspondence between the fractions and points of the coordinate beam.
So that on the coordinate beam to get to the point corresponding to the fraction M / N, from the beginning of the coordinates in the positive direction, to postpone M segments, the length of which is 1 / N share of a single segment. Such segments can be obtained by separating a single segment to n equal parts, which can always be made using a circulation and a ruler.
For example, we show the point M on the coordinate beam corresponding to the fraction 14/10. The length of the segment with the ends at the point O and the point close to it marked with a small stroke is 1/10 share of a single segment. The point with the coordinate 14/10 was removed from the origin at a distance of 14 of these segments.
Equal fractions correspond to the same fractional number, that is, equal to the fractions are the coordinates of the same point on the coordinate beam. For example, one point corresponds to 1/2, 2/4, 16/32, 55/110 coordinate on the coordinate beam, since all recorded fractions are equal (it is located at a distance of half a single segment, peculiar from the beginning of the reference in the positive direction).
On the horizontal and directed to the right coordinate beam point, the coordinate of which is a big fraction, is the right point, the coordinate of which is a smaller fraction. Similarly, a point with a smaller coordinate lies to the left of the point with the greater coordinate.
Right and incorrect fractions, definitions, examples
Among ordinary fractions distinguish right I. incorrect fractions . This separation is based on a comparison of the numerator and denominator.
Let us give the definition of the right and wrong ordinary fractions.
Definition.
Proper fraction - this is an ordinary fraction, the numerator of which is less than the denominator, that is, if M Definition.
Improper fraction - This is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if M≥N, then the ordinary fraction is incorrect. Let us give a few examples of the correct fractions: 1/4, 32 765/909 003. Indeed, in each of the recorded ordinary fractions, the numerator is less than the denominator (if necessary, see the article comparing natural numbers), so they are correct by definition. But examples of incorrect fractions: 9/9, 23/4 ,. Indeed, the numerator of the first of the recorded ordinary fractions is equal to the denominator, and in the other fractions the numerator more denominator. There is also a definition of correct and incorrect fractions, based on the comparison of fractions with a unit. Definition. rightif it is less than one. Definition. Ordinary fraction is called wrongIf it is either equal to one, or more than 1. So ordinary fraction 7/11 - correct, as 7/11<1
, а обыкновенные дроби 14/3
и 27/27
– неправильные, так как 14/3>1, a 27/27 \u003d 1. Let's think about how ordinary fractions with a numerator, superior or equal to the denominator, deserve such a name - "wrong". For example, take the wrong fraction 9/9. This fraction means that the nine share of the subject is taken, which consists of nine shares. That is, from the existing nine fractions we can make a whole subject. That is, the wrong fraction 9/9 in essence gives a whole subject, that is, 9/9 \u003d 1. In general, incorrect fractions with a numerator equal to the denominator denote one whole subject, and such a fraction can replace the natural number 1. Now consider incorrect fractions 7/3 and 12/4. It is quite obvious that from these seven third fractions we can make two whole objects (one whole subject is 3 shares, then it will take 3 + 3 \u003d 6 pieces to compile two whole objects) and one third share will remain. That is, the wrong shot 7/3 in essence means 2 items and another 1/3 share of such an item. And from twelve fourth fractions we can make three whole objects (three subjects of four stakes in each). That is, the fraction 12/4 in essence means 3 entire objects. The considered examples lead us to the following conclusion: incorrect fractions, can be replaced by either natural numbers when the numerator shares aimed at the denominator (for example, 9/9 \u003d 1 and 12/4 \u003d 3), or the sum of the natural number and correct fraction when the numerator It is not divided by a denominator (for example, 7/3 \u003d 2 + 1/3). Perhaps this is exactly what the wrong fraction has been deserved. "Wrong". Separate interest is caused by the representation of the wrong fraction in the form of the sum of the natural number and the correct fraction (7/3 \u003d 2 + 1/3). This process is called the allocation of a whole part of incorrect fraction, and deserves a separate and more attentive consideration. It is also worth noting that there is a very close relationship between incorrect fractions and mixed numbers. Each ordinary fraction corresponds to a positive fractional number (see the article positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary fractions 1/5, 56/18, 35/144 - positive fractions. When it is necessary to highlight the positivity of the fraction, then it is put in front of it plus, for example, +3/4, +72/34. If before an ordinary shot, put a minus sign, then this entry will correspond to a negative fractional number. In this case, you can talk about negative fractions. Let us give a few examples of negative fractions: -6/10, -65/13, -1/18. Positive and negative fractions M / N and -M / N are opposite numbers. For example, the fractions 5/7 and -5/7 are opposite fractions. Positive fractions, as well as positive numbers in general, denote the addition, income, change of any value in the direction of magnification, etc. Negative fractions meet the flow, debt, a change in any value towards the decrease. For example, a negative fraction of -3/4 can be interpreted as a debt, the value of which is 3/4. On the horizontal and directed right, negative fractions are located to the left of the beginning of the reference. The points of the coordinate direct whose coordinates are the positive fraction M / N and the negative fraction of -M / N are located at the same distance from the origin, but on different sides of the point O. It is worth saying about the fractions of the type 0 / n. These fractions are equal to the number zero, that is, 0 / n \u003d 0. Positive fractions, negative fractions, and also fractions 0 / N are combined into rational numbers. One action with ordinary fractions is a comparison of fractions - we have already considered higher. Four more arithmetic actions with fractions - Addition, subtraction, multiplication and division of fractions. Let us dwell on each of them. The general essence of action with fractions is similar to the essence of the corresponding actions with natural numbers. We draw an analogy. Multiplication of fractions It can be considered as an action in which the fraction fraction is located. For explanation, we give an example. Let us have 1/6 of the apple and we need to take 2/3 parts from it. The part we need is the result of the multiplication of fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which is in a particular case equal to a natural number). Further, we recommend to study the information of the article multiplication of fractions - rules, examples and solutions. Bibliography. One of the most difficult sections of mathematics to this day are the fractions. The story of fractions is not one millennium. The ability to share the whole on the part arose in the territory of Ancient Egypt and Babylon. Over the years, the operations done with the fractions have become complicated, the form of their record has changed. Everyone had their own features in the "relationship" with this section of mathematics. When it became necessary to share the integer on the part without extra effort, then the fractions appeared. The story of fractions is inseparable associated with the solution of utilitarian problems. The term "fraction" itself has Arab roots and comes from the word denoting "breaking, divided". Since ancient times, little has changed in this sense. The current definition sounds as follows: the fraction is part or the sum parts of the unit. Accordingly, examples with fractions are a sequential performance of mathematical operations with groups of numbers. Today there are two ways to write them. There were at different times: the first are more ancient. For the first time to operate with fractions began on the territory of Egypt and Babylon. The approach of mathematicians of the two states had significant differences. However, the beginning and there and there was an equally. The first fraction was half or 1/2. Further there was a quarter, a third and so on. According to archaeological excavations, the history of fractions has about 5 thousand years. For the first time, the shares of the number are found in Egyptian papyrus and on Babylonian clay signs. Types of ordinary fractions today include and the so-called Egyptian. They represent the sum of several terms of the form 1 / n. The numerator is always a unit, and the denominator is a natural number. There were such fractions, no matter how difficult it is to guess, in ancient Egypt. When calculating, all the shares tried to record in the form of such sums (for example, 1/2 + 1/4 + 1/8). Separate notation had only the fractions 2/3 and 3/4, the rest were broken into the components. There were special tables in which the shares of the number were presented in the form of the amount. The most ancient known references to such a system is found in the mathematical papyrus of Rinda, dating from the beginning of the second millennium BC. It includes a spreadsheet and mathematical tasks with solutions and answers presented in the form of fractions. The Egyptians were able to fold, share and multiply the number of numbers. The fraction in the Nile Valley was recorded using hieroglyphs. The representation of the share of the number in the form of the sum of the terms of the form 1 / N, characteristic of ancient Egypt, was used by mathematicians not only of this country. Up until the Middle Ages, Egyptian fractions were used in Greece and other states. Otherwise, mathematics looked in the Babylonian kingdom. The history of the fractions here is directly related to the characteristics of the number system that has given an ancient state into the inheritance from the predecessor, Sumero-Akkada civilization. The design technique in Babylon was more convenient and perfect than in Egypt. Mathematics in this country solved a much greater range of tasks. You can judge the achievements of Babylonian today in the preserved clay plates filled with clock. Thanks to the peculiarities of the material, they reached us in large quantities. According to some in Babylon, before Pythagora discovered the well-known theorem, which undoubtedly testifies to the development of science in this ancient state. The number system in Babylon was sixteen. Each new rank was different from the previous 60. Such a system was preserved in the modern world to denote the time and values \u200b\u200bof the corners. The fractions were also sixteen. Special icons used for recording. As in Egypt, examples with fractions contained separate characters to designate 1/2, 1/3 and 2/3. The Babylonian system did not disappear along with the state. Scrolls written in a 60-tiric system, used antique and Arabic astronomers and mathematics. The history of ordinary fractions has little enriched in ancient Greece. Eldlast residents believed that mathematics should operate only with integers. Therefore, expressions with fractions on the pages of ancient Greek treatises almost did not meet. However, a certain contribution to this section of mathematics was made by Pythagoreans. They understood the fraction as a relationship or proportion, and the unit was considered indivisible. Pythagoras with students built a general theory of fractions, learned to carry out all four arithmetic operations, as well as a comparison of fractions by bringing them to a common denominator. The Roman fraction system was associated with a weight measure called "Ass". She shared on 12 dollars. 1/12 ACCs was called oz. For the designation of fractions there were 18 titles. Here are some of them: semis - Half Ass; sextant - Sixth Share of ACCA; semiduction - half oz or 1/24 ACCA. The inconvenience of such a system was in the impossibility of presenting a number in the form of a fraction with a denominator 10 or 100. Roman mathematics overcame the difficulty with the use of interest. In antiquity, the fraction has already written familiar to us: one number over the other. However, there was one significant difference. The numerator was located under the denominator. For the first time, writing the fraci began in ancient India. Modern method for us began to use Arabs. But none of these nations applied a horizontal trait to separate the numerator and denominator. For the first time, it appears in the works of Leonardo Pisansky, better known as Fibonacci, in 1202. If the history of the occurrence of ordinary fractions began in Egypt, then the decimal first appeared in China. In the subway empire they began to use them from about the III century to our era. The history of decimal fractions began with Chinese mathematics Liu Huey, who offered to use them when removing square roots. In the third century of our era, decimal fractions in China began to be used when calculating weight and volume. Gradually, they began to penetrate into mathematics deeper. In Europe, however, decimal fractions began to be used much later. Regardless of the Chinese predecessors, decimal fractions opened Al-Kashi astronomer from the ancient city of Samarkand. He lived and worked in the XV century. The scientist outlined his theory in the treatise "The key to arithmetic", saw the light in 1427. Al-Kashi proposed to use a new shot of fractions. And the whole, and the fractional part now was written in the same line. For their separation, Samarkand astronomer did not use comma. He wrote an integer and fractional part with different colors using black and red ink. Sometimes for the separation of al-Kashi also used a vertical line. A new type of frains began to appear in the writings of European mathematicians from the XIII century. It should be noted that with the works of Al-Kashi, as in the invention, they were not familiar. Decimal fractions appeared in the works of Jordan Nemoraria. Then they used them already in the XVI century, the French scientist wrote a "mathematical canon", which contained trigonometric tables. In them, Viet used decimal fractions. For the separation of the whole and fractional part, the scientist applied a vertical feature, as well as a different font size. However, these were only private cases of scientific use. To solve everyday tasks, decimal fractions in Europe began to be applied slightly later. This happened due to the Dutch scientist Simon Stevin at the end of the XVI century. He issued the mathematical work "Tenth" in 1585. In it, the scientist outlined the theory of use of decimal fractions in arithmetic, in the monetary system and to determine measures and weights. Stevech also did not use the comma. He separated two parts of the fraction with zero, circled into a circle. For the first time, the comma divided two parts of the decimal fraction only in 1592. In England, however, instead of it began to apply a point. On the territory of the United States, the decimal fractions write in this way. One of the initiators of using both punctuation marks for the separation of the whole and fractional part was the Scottish mathematician John Necess. He expressed his proposal in 1616-1617. A German scientist enjoyed the comma In Russian land, the first mathematician, which settled the division of the whole to the part, was the Novgorod monk Kirik. In 1136, he wrote the work in which the method of the number of years have outlined. Kirik was engaged in issues of chronology and calendar. In his work, he led, including the division of an hour to Part: the fifth, twenty fifth, and so on, the shares. The division of the whole on the part was applied when calculating the amount of tax in the XV-XVII centuries. Operations of addition, subtraction, division and multiplication of fractional parts were used. The word "fraction" appeared in Russia in the VIII century. It happened from the verb to "rub, divide into parts". For the names of the fractions, our ancestors used special words. For example, 1/2 was designated as half or Poltina, 1/4 - Check, 1/8 - Hollow, 1/16 - half and so on. The complete theory of fractions, which differs little from modern, was set out in the first textbook on arithmetic, written in 1701 by Leonthius Filippovich Magnitsky. "Arithmetic" consisted of several parts. About the fractions in detail the author tells in the section "On the numbers of broken or with shames". Magnitsky leads operations with "broken" numbers, their different designations. Today, still among the most complex sections of mathematics are called fractions. The story of fractions was also not simple. Different peoples are sometimes independent of each other, and sometimes borrowing the experience of predecessors, they came to the need to introduce, mastering and using the number of numbers. Always the doctrine of fractions crossed from practical observations and thanks to pressing issues. It was necessary to share bread, place 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 system of numbering in the state and on the overall level of mathematics. Anyway, overcome not one thousand years, the section of algebras dedicated to the shares of numbers was formed, developed and successfully used today for various needs of both practical nature and theoretical. 1
/
5 Ordinary (or simple) Fraction - a recording of a rational number in the form ± M n (\\ displaystyle \\ pm (\\ FRAC (M) (N))) or ± m / n, (\\ displaystyle \\ pm m / n,) Where N ≠ 0. (\\ DisplayStyle N \\ NEQ 0.) The horizontal or oblique feature is a sign of division, resulting in a private one. Delimi called numerator fractiona, and divider - denominator. There are several types of recruitment of ordinary fractions in print form: Right It is called a fraction in which the numerator module is less than the denominator module. Fraction not correctly called wrong, and represents a rational number, the module is greater or equal to one. For example, the fraci 3 5 (\\ DisplayStyle (\\ FRAC (3) (5))), 7 8 (\\ DisplayStyle (\\ FRAC (7) (8))) and - the right fractions, while 8 3 (\\ displayStyle (\\ FRAC (8) (3))), 9 5 (\\ DisplayStyle (\\ FRAC (9) (5))), 2 1 (\\ DisplayStyle (\\ FRAC (2) (1))) and 1 1 (\\ displayStyle (\\ FRAC (1) (1))) - Incorrect fractions. Any non-zero integer can be represented as an irregularly ordinary fraction with denominator 1. The fraction recorded in the form of an integer and correct fraction is called mixed fraction And it is understood as the amount of this number and fraction. Any rational number can be written in the form of a mixed fraction. In contrast to the mixed fraction, the fraction containing only the numerator and the denominator is called plain. For example, 2 3 7 \u003d 2 + 3 7 \u003d 14 7 + 3 7 \u003d 17 7 (\\ DisplayStyle 2 (\\ FRAC (3) (7)) \u003d 2 + (\\ FRAC (3) (7)) \u003d (\\ FRAC (14 ) (7)) + (\\ FRAC (3) (7)) \u003d (\\ FRAC (17) (7))). In strict mathematical literature, this record is preferred not to use due to the similarity of the mixed fraction with the designation of the product of an integer on the fraction, as well as due to the more cumbersome record and less convenient computing. A multi-storey, or composite, fraquence is called an expression containing several horizontal (or less often - inclined) damn: A decimal fraction is called a positional entry of a fraction. It looks like this: Example: 3,141 5926 (\\ DisplayStyle 3 (,) 1415926). Part of the recording that stands to the positional semicol, is an integer part of the number (fracted), and after a semicolon - a fractional part. Any ordinary fraction can be converted to decimal, which in this case either has a finite number of semicolons, or is a periodic fraction. Generally speaking, not only a decimal number system can be used for the positional record of the number, but also other (including specific, such as Fibonacchiyev). The fraction is just a record of the number. The same number can correspond to different fractions, both ordinary and decimal. This section discusses actions on ordinary fractions. For actions over decimal fractions, see decimal fraction. For comparison, addition and subtraction of fractions should be converted ( lead) To the form with the same denominator. Let two fractions are given: A B (\\ DisplayStyle (\\ FRAC (A) (B))) and C D (\\ DisplayStyle (\\ FRAC (C) (D))). Procedure: After that, the denominators of both fractions coincide (equal M.). Instead of the smallest common multiple, you can take in simple cases as M. Any other common multiple, for example, the product of denominators. For example, see below in the Comparison section. To compare two ordinary fractions, you should bring them to a common denominator and compare the numerals of the collaboration. The fraction with a large numerator will be more. Example. Compare 3 4 (\\ DisplayStyle (\\ FRAC (3) (4))) and 4 5 (\\ DisplayStyle (\\ FRAC (4) (5))). NOK (4, 5) \u003d 20. We give the fractions to the denominator 20. Hence,
3
4
<
4
5
{\displaystyle {\frac {3}{4}}<{\frac {4}{5}}}
To fold two ordinary fractions, it should be brought to a common denominator. Then fold the numerals, and the denominator should be left unchanged: Nok denominators (here 2 and 3) is 6. We give a fraction 1 2 (\\ DisplayStyle (\\ FRAC (1) (2))) To the denominator 6, for this, the numerator and the denominator must be multiplied by 3. NOK denominators (here 2 and 4) is equal to 4. We give a fraction 1 2 (\\ DisplayStyle (\\ FRAC (1) (2))) to the denominator 4, for this it is necessary to multiply the numerator and denominator to 2. get 2 4 (\\ DisplayStyle (\\ FRAC (2) (4))). To multiply two ordinary fractions, you need to multiply their numerators and denominators: In particular, to multiply the fraction on the natural number, it is necessary to multiply the numeric number, and the denominator should be left the same: In general, the numerator and denominator of the resulting fraction may not be mutually simple, and it may be necessary to reduce the fraction, for example: To divide one ordinary fraction to another, you need to multiply the first to fraction, reverse the second: 1 2: 1 3 \u003d 1 2 ⋅ 3 1 \u003d 3 2. (\\ displayStyle (\\ FRAC (1) (2)): (\\ FRAC (1) (3)) \u003d (\\ FRAC (1) (2)) \\ CDOT (\\ FRAC (3) (1)) \u003d (\\ Conversion between different recording formats fractional Fractional numbers occur when one object (apple, watermelon, cake, loaf of bread, sheet of paper) or unit of measure (meter, hour, kilogram, degrees) divide into several. equal Parts. words like "half-bar", "Polbathone", Polkilogram, "half-liter", "quarter of an hour", "third paths", "one and a half meters", probably, you hear every day. Half, quarter, a third, one hundred and one and a half are examples of fractional numbers. Consider an example. For a birthday to you, 10 friends came to visit you. The festive cake was divided into 10 equal parts (Fig. 185). Then every guest got one tenth cake. Write: Cake (read: "One tenth cake"). Such a "two-story" record is used to designate and other fractional numbers. For example: Polkylogram - Kg (read: "one second kilogram"); quarter H (read: "one fourth hour"); Third ways - Ways (read: "One third path"). If two of your guests do not like sweet, then sweet tooth will get Cake (read: "three tenth of cake"; Fig. 186). Records of type Etc. Call ordinary fractions or shorter - fractions Ordinary fractions are written with two natural numbers and damage fractions.
The number recorded above the feature is called numerator of the shot. ; The number recorded under the line is called ranger Drobi. . The denominator of the fraci shows how much equal parts were divided by something, and the numerator - how many such parts took. So in Figure 187, the equilateral triangle ABC was divided into 4 equal parts - 4 equal triangles. Three of them are painted. We can say that the figure is painted, the area of \u200b\u200bwhich is ABC triangle square. Or say: painted Triangle ABC. In Figure 188, a single segment of the coordinate beam is divided into five equal parts. OB Cut is Single segment OA. Point B depicts the number Number Refer to the coordinate point b and write b ( ). Since the segment OC is Single segment OA, then the coordinate point C is equal Those. C ( Example 1
.
24 wood grow in the garden, of which 7 are apple trees. What part of all the trees make up an apple tree? Decision. Since 24 wood grows in the garden, then one apple tree is All trees, and 7 apple trees - All trees. . Example 2
.
24 wood grows in the garden, of which Make up cherries. How many cherry trees grows in the garden? Decision. Ranger Drobi. It shows that the number of all trees growing in the garden should be divided into 8 equal parts. Since 24 wood grows in the garden, then one part is 24: 8 \u003d 3 (wood). The crusher is crushed 3, then 8 * 3 \u003d 24 (wood) grows in the garden. Answer: 24 wood. Fraction In mathematics - a number consisting of one or more parts (fractions) of a unit. The fractions are part of the rational field field. By a method for recording, fractions are divided into 2 formats: ordinary species I. decimal . Ploba Numerator - A number indicating the number of shares taken (located at the top of the fraction - above the line). Ranger Drobi. - The number indicating how much fraction is divided (located under the line - at the bottom). In turn, are divided into: right and wrong, mixed and compound Closely related to units of measurement. 1 meter contains 100 cm in itself. Which means that 1 m is divided into 100 equal shares. Thus, 1 cm \u003d 1/100 m (one centimeter is equal to one hundred meters). or 3/5 (three fifths), here 3 - Numerator, 5 - denominator. If the numerator is less than the denominator, then the fraction less than the unit and is called right: If the numerator is equal to the denominator, the fraction is equal to one. If the numerator is greater than the denominator, the fraction more units. In both recent cases, the fraction is called wrong: To select the largest integer contained in incorrect fraction, you need to divide the numerator to the denominator. If division is performed without a balance, then the wrong fraction is equal to the private: If the division is performed with the residue, then (incomplete) private gives a suitable integer, the balance becomes the fractional part number; The valve of the fractional part remains the same. The number containing the whole and fractional part is called mixed. Fractional part mixed numbermaybe I. incorrect fraction. Then you can select the greatest integer from the fractional part and present a mixed number in this form so that the fractional part becomes the right fraction (or disappeared at all).Positive and negative fractions
Actions with fractions
What is a fraction?
Came from the depths of centuries
Ancient Egypt
Mathematics development in Babylon
DRISTI: The story of fractions in Babylon
Ancient Greece
Sacred Roman Empire
Writing ordinary fractions
China
Al-porridge from Samarkand
Decimal fractions in Europe
Point, dot, comma
Fruit in Russia
Encyclopedic YouTube.
Ordinary fractions designations
Right and incorrect fractions
Mixed fractions
Composite fractions
Decimal fractions
The value of the fraction and the main property of the fraction
Actions with fractions
Bringing to a common denominator
Comparison
Addition and subtraction
Happened 3 6 (\\ displayStyle (\\ FRAC (3) (6))). We bring a fraction 1 3 (\\ displayStyle (\\ FRAC (1) (3))) In addition, the denominator, for this, the numerator and the denominator must be multiplied by 2. Opened 2 6 (\\ DisplayStyle (\\ FRAC (2) (6))).
To get the difference of fractions, they should also be given to a common denominator, and then subtract numerators, denominator to leave unchanged: Multiplication and division
You know that, except for natural numbers and zero, there are other numbers -
