The fact that many square roots are irrational numbers, does not detract from their significance, in particular, the number $\sqrt2$ is very often used in various engineering and scientific calculations. This number can be calculated with the accuracy that is necessary in each specific case. You can get this number with as many decimal places as you have the patience for.
For example, the number $\sqrt2$ can be determined to six decimal places: $\sqrt2=1.414214$. This value is not very different from true value, because $1.414214 \times 1.414214=2.000001237796$. This answer differs from 2 by just over one millionth. Therefore, the value of $\sqrt2$, equal to $1.414214$, is considered quite acceptable for the solution of the majority practical tasks. In the case when greater accuracy is required, it is not difficult to obtain as many significant figures after the decimal point, as needed in this case.
However, if you show rare stubbornness and try to extract Square root from the number $\sqrt2$ until you achieve the exact result, you will never finish your work. It's an endless process. No matter how many decimal places you get, there will always be a few more.
This fact can amaze you as much as turning $\frac13$ into an infinite decimal $0.333333333…$ and so on infinitely or turning $\frac17$ into $0.142857142857142857…$ and so on infinitely. At first glance, it may seem that these infinite and irrational square roots are phenomena of the same order, but this is not at all the case. After all, these infinite fractions have a fractional equivalent, while $\sqrt2$ has no such equivalent. And why, exactly? The point is that the decimal equivalent of $\frac13$ and $\frac17$, as well as an infinite number other fractions are periodic finite fractions.
At the same time, the decimal equivalent of $\sqrt2$ is a non-periodic fraction. This statement is also true for any ir rational number.
The problem is that any decimal that is an approximation of the square root of 2 is not periodic fraction . No matter how far we advance in the calculations, any fraction we get will be non-periodic.
Imagine a fraction huge amount non-periodic digits after the decimal point. If suddenly after the millionth digit the whole sequence of decimal places is repeated, then decimal- periodic and for it there is an equivalent in the form of a ratio of integers. If a fraction with a huge number (billions or millions) of non-periodic decimal places at some point has an endless series of repeating digits, for example $…55555555555…$, this also means that this fraction is periodic and there is an equivalent for it in the form of a ratio of integers numbers.
However, in the case of their decimal equivalents are completely non-periodic and cannot become periodic.
Of course, you can ask next question: “And who can know and say for sure what happens to a fraction, say, after a trillion sign? Who can guarantee that the fraction will not become periodic? There are ways to irrefutably prove that irrational numbers are non-periodic, but such proofs require complex mathematical apparatus. But if it suddenly turned out that irrational number becomes periodic fraction, this would mean a complete collapse of the foundations mathematical sciences. And in fact, this is hardly possible. This is not just for you to throw on the knuckles from side to side, there is a complex mathematical theory here.
That if they know the theory of series, then without it, no metamatic concepts can be introduced. Moreover, these people believe that one who does not use it everywhere is ignorant. Let us leave the views of these people to their conscience. Let's better understand what an infinite periodic fraction is and how to deal with it for us, uneducated people who know no limits.
Divide 237 by 5. No, you don't need to run the Calculator. Let's better remember the middle (or even elementary?) school and just divide the column:
Well, do you remember? Then you can get down to business.
The concept of "fraction" in mathematics has two meanings:
- Non-integer.
- Notation form of a non-integer number.
- Simple (or vertical) fractions like 1/2 or 237/5.
- Decimals, such as 0.5 or 47.4.
In mathematics, in general, from time immemorial, a decimal account has been accepted, and therefore decimal fractions are more convenient than simple ones, that is, a fraction with decimal denominator(Vladimir Dal. Dictionary alive Great Russian language. "Ten").And if so, then I want to make any vertical fraction decimal (“horizontal”). And for this you just need to divide the numerator by the denominator. Take, for example, the fraction 1/3 and try to make it a decimal.
Even a completely uneducated person will notice: no matter how long it takes, they won’t split up: this is how triples will appear indefinitely. So let's write it down: 0.33... We mean "the number that is obtained when you divide 1 by 3", or, in short, "one third". Naturally, one third is a fraction in the first sense of the word, and "1/3" and "0.33 ..." are fractions in the second sense of the word, that is record forms a number that is on the number line at such a distance from zero that if you postpone it three times, you get one.
Now let's try to divide 5 by 6:
Let's write it down again: 0.833 ... We mean "the number that is obtained when you divide 5 by 6", or, in short, "five-sixths." However, confusion arises here: does it mean 0.83333 (and then the triples are repeated), or 0.833833 (and then 833 is repeated). Therefore, the record with ellipsis does not suit us: it is not clear where the repeating part starts from (it is called the “period”). Therefore, we will take the period in brackets, like this: 0, (3); 0.8(3).
0,(3) not just equals one third is there is one-third, because we specifically came up with this notation to represent this number in the form decimal fraction.
This entry is called an infinite periodic fraction, or just a periodic fraction.
Whenever we divide one number by another, if we don’t get a finite fraction, then we get an infinite periodic fraction, that is, sometime the sequences of numbers will begin to repeat. Why this is so can be understood purely speculatively, looking carefully at the division algorithm by a column:
In places marked with checkmarks, they cannot be obtained all the time different couples numbers (because there are, in principle, a finite set of such pairs). And as soon as such a pair appears there, which already existed, the difference will also be the same - and then the whole process will begin to repeat itself. There is no need to check this, because it is quite obvious that when the same actions are repeated, the results will be the same.
Now that we understand well essence periodic fraction, let's try multiplying one third by three. Yes, it will turn out, of course, one, but let's write this fraction in decimal form and multiply by a column (the ambiguity due to the ellipsis does not arise here, since all the numbers after the decimal point are the same):
And again we notice that nines, nines and nines will appear after the decimal point all the time. That is, using, inversely, bracket notation, we get 0, (9). Since we know that the product of one third and three is a unit, then 0, (9) is such a bizarre form of writing a unit. However, it is not advisable to use this form of notation, because the unit is perfectly written without using a period, like this: 1.
As you can see, 0,(9) is one of those cases where an integer is written as a fraction, like 3/3 or 7.0. That is, 0, (9) is a fraction only in the second sense of the word, but not in the first.
So, without any limits and rows, we figured out what 0, (9) is and how to deal with it.
But still remember that in fact we are smart and studied analysis. Indeed, it is hard to deny that:
But, perhaps, no one will argue with the fact that:
All this is, of course, true. Indeed, 0,(9) is both the sum of the reduced series and the doubled sine of the specified angle, and natural logarithm Euler numbers.
But neither one, nor the other, nor the third is a definition.
To say that 0,(9) is the sum of the infinite series 9/(10 n), when n is greater than one, is the same as to say that the sine is the sum of the infinite Taylor series:
This is quite right, and this is important fact for computational mathematics, but this is not a definition, and, most importantly, it does not bring a person closer to understanding essence sinus. The essence of the sine of a certain angle is that it is just attitude opposite corner catheter to the hypotenuse.
Well, the periodic fraction is just decimal fraction that results when when dividing by a column the same set of numbers will be repeated. There is no analysis here at all.
And here the question arises: where generally we took the number 0,(9)? What do we divide by a column to get it? Indeed, there are no such numbers, when dividing by each other in a column, we would have infinitely appearing nines. But we managed to get this number by multiplying the column 0, (3) by 3? Not really. After all, you need to multiply from right to left in order to correctly take into account transfers of digits, and we did this from left to right, cleverly taking advantage of the fact that transfers do not occur anywhere anyway. Therefore, the legitimacy of writing 0,(9) depends on whether we recognize the legitimacy of such multiplication by a column or not.
Therefore, one can generally say that the notation 0,(9) is incorrect - and to a certain extent be right. However, since the notation a ,(b ) is accepted, it's just ugly to drop it when b = 9; it is better to decide what such a record means. So, if we accept the notation 0,(9) at all, then this notation, of course, means the number one.
It remains only to add that if we used, say, a ternary number system, then when dividing a unit column (1 3) by a triple (10 3), we would get 0.1 3 (it reads “zero point one third”), and when dividing 1 by 2 would be 0,(1) 3 .
So the periodicity of the fraction-record is not some objective characteristic of the fraction-number, but only by-effect using one or another number system.
As is known, the set of rational numbers (Q) includes the sets of integers (Z), which in turn includes the set of natural numbers (N). In addition to integers, rational numbers include fractions.
Why, then, is the whole set of rational numbers sometimes considered as infinite decimal periodic fractions? After all, in addition to fractions, they include whole numbers, as well as non-periodic fractions.
The fact is that all integers, as well as any fraction, can be represented as an infinite periodic decimal fraction. That is, for all rational numbers, you can use the same notation.
How is an infinite periodic decimal represented? In it, a repeating group of numbers after the decimal point is taken in brackets. For example, 1.56(12) is a fraction in which the group of digits 12 is repeated, i.e. the fraction has a value of 1.561212121212... and so on without end. A repeating group of digits is called a period.
However, in this form, we can represent any number if we consider the number 0 as its period, which also repeats without end. For example, the number 2 is the same as 2.00000.... Therefore, it can be written as an infinite periodic fraction, i.e. 2,(0).
The same can be done with any finite fraction. For example:
0,125 = 0,1250000... = 0,125(0)
However, in practice, the transformation of a finite fraction into an infinite periodic fraction is not used. Therefore, finite fractions and infinite periodic fractions are separated. Thus, it is more correct to say that the rational numbers include
- all integers,
- final fractions,
- infinite periodic fractions.
At the same time, they simply remember that integers and finite fractions can be represented in theory as infinite periodic fractions.
On the other hand, the concepts of finite and infinite fractions are applicable to decimal fractions. If we talk about ordinary fractions, then both finite and infinite decimal fractions can be uniquely represented as an ordinary fraction. So, from the point of view of ordinary fractions, periodic and finite fractions are one and the same. In addition, whole numbers can also be represented as a common fraction if we imagine that we divide this number by 1.
How to represent a decimal infinite periodic fraction in the form of an ordinary? The most commonly used algorithm is:
- They bring the fraction to the form so that after the decimal point there is only a period.
- Multiply an infinite periodic fraction by 10 or 100 or ... so that the comma moves to the right by one period (that is, one period is in the integer part).
- The original fraction (a) is equated with the variable x, and the fraction (b) obtained by multiplying by the number N is equal to Nx.
- Subtract x from Nx. Subtract a from b. That is, they make up the equation Nx - x \u003d b - a.
- When solving the equation, it turns out common fraction.
An example of converting an infinite periodic decimal fraction to an ordinary fraction:
x = 1.13333...
10x = 11.3333...
10x * 10 = 11.33333... * 10
100x = 113.3333...
100x – 10x = 113.3333... – 11.3333...
90x=102
x=
There is another representation of the rational number 1/2, different from representations of the form 2/4, 3/6, 4/8, etc. We mean the representation as a decimal fraction of 0.5. Some fractions have finite decimal representations, for example,
while the decimal representations of other fractions are infinite:
These infinite decimals can be obtained from the corresponding rational fractions by dividing the numerator by the denominator. For example, in the case of the fraction 5/11, dividing 5.000... by 11 gives 0.454545...
What rational fractions have finite decimal representations? Before answering this question in the general case, consider specific example. Take, say, the final decimal fraction 0.8625. We know that
and that any finite decimal can be written as a rational decimal with a denominator equal to 10, 100, 1000, or some other power of 10.
Reducing the fraction on the right to an irreducible fraction, we get
The denominator 80 is obtained by dividing 10,000 by 125 - the largest common divisor 10 000 and 8625. Therefore, in the expansion into prime factors the numbers 80, like the numbers 10,000, include only two prime factors: 2 and 5. If we started not with 0.8625, but with any other finite decimal fraction, then the resulting irreducible rational fraction would also have this property. In other words, the decomposition of the denominator b into prime factors could only include prime numbers 2 and 5, since b is a divisor of some power of 10, and . This circumstance turns out to be decisive, namely, the following general statement holds:
An irreducible rational fraction has a finite decimal representation if and only if the number b does not have prime divisors, personal from 2 and 5.
Note that in this case b does not have to have both 2 and 5 among its prime divisors: it can be divisible by only one of them or not divisible by them at all. For example,
here b is equal to 25, 16, and 1, respectively. The essential thing is that b has no other divisors other than 2 and 5.
The above sentence contains an expression if and only if. So far, we have only proved the part that applies to turnover only then. It was we who showed that the expansion of a rational number into a decimal fraction will be finite only if b has no prime divisors other than 2 and 5.
(In other words, if b is divisible by a prime number other than 2 and 5, then irreducible fraction does not have a trailing decimal expression.)
The part of the sentence that refers to the word then states that if the integer b has no other prime divisors f other than 2 and 5, then an irreducible rational fraction can be represented by a finite decimal fraction. To prove this, we must take an arbitrary irreducible rational fraction, for which b has no other prime divisors except 2 and 5, and make sure that the corresponding decimal fraction is finite. Let's consider an example first. Let be
To obtain a decimal expansion, we convert this fraction into a fraction whose denominator is an integer power of ten. This can be achieved by multiplying the numerator and denominator by:
The above discussion can be extended to general case in the following way. Suppose b is of the form , where the type is non-negative integers (i.e., positive numbers or zero). Two cases are possible: either less than or equal (this condition is written ), or greater (which is written ). When we multiply the numerator and denominator of the fraction by
Already in primary school students are dealing with fractions. And then they appear in every topic. It is impossible to forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.
Why are fractions needed?
The world around us consists of whole objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.
For example, chocolate consists of several slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of slices of chocolate.
By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.
What is a "fraction"?
This is a number consisting of parts of one. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on the top (left) is called the numerator. The one on the bottom (right) is the denominator.
In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called a dividend, and the denominator can be called a divisor.
What are the fractions?
In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren are first introduced to primary school, calling them simply "fractions". The second learn in the 5th grade. That's when these names appear.
Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear that the two examples given are completely different numbers.
Every simple fraction can be written as a decimal. This statement is almost always true in reverse direction. There are rules that allow you to write a decimal fraction as an ordinary fraction.
What subspecies do these types of fractions have?
Better start at chronological order as they are being studied. Common fractions come first. Among them, 5 subspecies can be distinguished.
Correct. Its numerator is always less than the denominator.
Wrong. Its numerator is greater than or equal to the denominator.
Reducible / irreducible. It can be either right or wrong. Another thing is important, whether the numerator and denominator have common factors. If there are, then they are supposed to divide both parts of the fraction, that is, to reduce it.
Mixed. An integer is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.
Composite. It is formed from two fractions divided into each other. That is, it has three fractional features at once.
Decimals have only two subspecies:
final, that is, one in which the fractional part is limited (has an end);
infinite - a number whose digits after the decimal point do not end (they can be written endlessly).
How to convert decimal to ordinary?
If this is a finite number, then an association based on the rule is applied - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional line.
As a hint about the required denominator, remember that it is always a one and a few zeros. The latter need to be written as many as the digits in the fractional part of the number in question.
How to convert decimal fractions to ordinary ones if their whole part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.
How to make an ordinary fraction from a decimal if its integer part is different from zero? For example, 5.23 or 13.00108. Both examples read the integer part and write its value. In the first case, this is 5, in the second, 13. Then you need to move on to the fractional part. With them it is necessary to carry out the same operation. The first number has 23/100, the second has 108/100000. The second value needs to be reduced again. The response is like this mixed fractions: 5 23/100 and 13 27/25000.
How to convert an infinite decimal to a common fraction?
If it is non-periodic, then such an operation cannot be carried out. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.
The only thing that is allowed to be done with such a fraction is to round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal - will never give the initial value. That is, infinite non-periodic fractions are not translated into ordinary fractions. This must be remembered.
How to write an infinite periodic fraction in the form of an ordinary?
In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as they can be converted into ordinary fractions.
Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with any numbers, and then the repetition begins.
The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for these two types of numbers. It is quite easy to write pure periodic fractions as ordinary fractions. As with the final ones, they need to be converted: write the period into the numerator, and the number 9 will be the denominator, repeating as many times as there are digits in the period.
For example, 0,(5). The number does not have an integer part, so you need to immediately proceed to the fractional part. Write 5 in the numerator, and write 9 in the denominator. That is, the answer will be the fraction 5/9.
A rule on how to write a common decimal fraction that is a mixed fraction.
Look at the length of the period. So much 9 will have a denominator.
Write down the denominator: first nines, then zeros.
To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - it is without a period.
For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period is one digit. So zero will be one. There is also only one digit in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.
To determine the numerator from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.
How are common fractions converted to decimals?
by the most simple option it turns out the number in the denominator of which is the number 10, 100 and so on. Then the denominator is simply discarded, and between the fractional and whole parts a comma is placed.
There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. Only it is necessary to multiply not only the denominator, but also the numerator by the same number.
For all other cases, a simple rule will come in handy: divide the numerator by the denominator. In this case, you can get two answers: a final or a periodic decimal fraction.
Operations with common fractions
Addition and subtraction
Students get to know them earlier than others. And first with fractions same denominators and then different. General rules can be reduced to such a plan.
Find the least common multiple of the denominators.
burn additional multipliers to all ordinary fractions.
Multiply the numerators and denominators by the factors defined for them.
Add (subtract) the numerators of fractions, and leave the common denominator unchanged.
If the numerator of the minuend is less than the subtrahend, then you need to find out whether we have a mixed number or a proper fraction.
In the first case, the integer part needs to take one. Add a denominator to the numerator of a fraction. And then do the subtraction.
In the second - it is necessary to apply the rule of subtraction from a smaller number to a larger one. That is, subtract the modulus of the minuend from the modulus of the subtrahend, and put the “-” sign in response.
Look carefully at the result of addition (subtraction). If you get an improper fraction, then it is supposed to select the whole part. That is, divide the numerator by the denominator.
Multiplication and division
For their implementation, fractions do not need to be reduced to common denominator. This makes it easier to take action. But they still have to follow the rules.
When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have common factor, then they can be reduced.
Multiply numerators.
Multiply the denominators.
If you get a reducible fraction, then it is supposed to be simplified again.
When dividing, you must first replace division with multiplication, and the divisor (second fraction) with reciprocal(swap the numerator and denominator).
Then proceed as in multiplication (starting from point 1).
In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written in the form improper fraction. That is, with a denominator of 1. Then proceed as described above.
Operations with decimals
Addition and subtraction
Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.
Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Assign the missing number of zeros in it.
Write fractions so that the comma is under the comma.
Add (subtract) like natural numbers.
Remove the comma.
Multiplication and division
It is important that you do not need to append zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.
For multiplication, you need to write fractions one under the other, not paying attention to commas.
Multiply like natural numbers.
Put a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.
To divide, you must first convert the divisor: make it natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.
Multiply the dividend by the same number.
Divide a decimal by a natural number.
Put a comma in the answer at the moment when the division of the whole part ends.
What if there are both types of fractions in one example?
Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. There are two possible solutions to these problems. You need to objectively weigh the numbers and choose the best one.
First way: represent ordinary decimals
It is suitable if, when dividing or converting, final fractions are obtained. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.
The second way: write decimal fractions as ordinary
This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, you can get a very large ordinary fraction and decimal entries will allow you to calculate the task faster and easier. Therefore, it is always necessary to soberly evaluate the task and choose the simplest solution method.
