There are more irrational numbers than rational ones. What are rational and irrational numbers

Integers

Natural numbers are defined as positive integers. Natural numbers are used for counting objects and for many other purposes. These are the numbers:

This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
how many natural numbers exists? There are an infinite number of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to indicate it, because there is an infinite number of natural numbers.

The sum of natural numbers is a natural number. So, the addition of natural numbers a and b:

The product of natural numbers is a natural number. So, the product of natural numbers a and b:

c is always a natural number.

Difference of natural numbers There is not always a natural number. If the subtracted is greater than the subtracted, then the difference of natural numbers is a natural number, otherwise it is not.

The quotient of natural numbers There is not always a natural number. If for natural numbers a and b

where c is a natural number, this means that a is divisible by b completely. In this example, a is the dividend, b is the divisor, c is the quotient.

The divisor of a natural number is a natural number by which the first number is evenly divisible.

Each natural number is divisible by one and by itself.

Prime natural numbers are divisible only by one and by themselves. Here it is meant to divide completely. Example, numbers 2; 3; 5; 7 are divisible only by one and by themselves. These are prime natural numbers.

The unit is not considered a prime number.

Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:

The unit is not considered a composite number.

The set of natural numbers is one, prime numbers and composite numbers.

The set of natural numbers is denoted by the Latin letter N.

Properties of addition and multiplication of natural numbers:

displacement property of addition

combination property additions

(a + b) + c = a + (b + c);

travel multiplication property

combination property of multiplication

(ab) c = a (bc);

distribution property of multiplication

A (b + c) = ab + ac;

Whole numbers

Integers are natural numbers, zero, and the opposite of natural numbers.

Numbers opposite to natural numbers are negative integers, for example:

1; -2; -3; -4;...

The set of integers is denoted by the Latin letter Z.

Rational numbers

Rational numbers are whole numbers and fractions.

Any rational number can be represented as a periodic fraction. Examples:

1,(0); 3,(6); 0,(0);...

The examples show that any integer is a periodic fraction with a period of zero.

Any rational number can be represented as a fraction m / n, where m is an integer number, n natural number. Let's represent in the form of such a fraction the number 3, (6) from the previous example.

Rational number- a number represented by an ordinary fraction m / n, where the numerator m is an integer and the denominator n is a natural number. Any rational number can be represented as a periodic infinite decimal fraction. The set of rational numbers is denoted by Q.

If the real number is not rational, then it irrational number ... Decimal fractions expressing irrational numbers are infinite and not periodic. The set of irrational numbers is usually denoted by the capital letter I.

The real number is called algebraic if it is a root of some polynomial (nonzero degree) with rational coefficients. Any non-algebraic number is called transcendental.

Some properties:

The set of rational numbers is densely located on the number axis: between any two different rational numbers, there is at least one rational number (and hence an infinite set of rational numbers). Nevertheless, it turns out that the set of rational numbers Q and the set of natural numbers N are equivalent, that is, a one-to-one correspondence can be established between them (all elements of the set of rational numbers can be renumbered).

The set Q of rational numbers is closed with respect to addition, subtraction, multiplication and division, that is, the sum, difference, product and quotient of two rational numbers are also rational numbers.

All rational numbers are algebraic (the converse is not true).

Every real transcendental number is irrational.

Every irrational number is either algebraic or transcendental.

The set of irrational numbers is everywhere dense on the number line: between any two numbers there is an irrational number (and therefore an infinite set of irrational numbers).

The set of irrational numbers is uncountable.

When solving problems, it is convenient together with ir rational number a + b√ c (where a, b are rational numbers, c is an integer that is not the square of a natural number) consider the "conjugate" number a - b√ c: its sum and product with the original are rational numbers. So a + b√ c and a - b√ c are roots quadratic equation with integer coefficients.

Problems with solutions

1. Prove that

a) number √ 7;

b) the number lg 80;

c) the number √ 2 + 3 √ 3;

is irrational.

a) Suppose that the number √ 7 is rational. Then, there are coprime p and q such that √ 7 = p / q, whence we obtain p 2 = 7q 2. Since p and q are coprime, p is 2, and hence p is divisible by 7. Then p = 7k, where k is some natural number. Hence q 2 = 7k 2 = pk, which contradicts the fact that p and q are coprime.

So, the assumption is false, which means that the number √ 7 is irrational.

b) Suppose that the number lg 80 is rational. Then there exist natural numbers p and q such that lg 80 = p / q, or 10 p = 80 q, whence we obtain 2 p – 4q = 5 q – p. Taking into account that the numbers 2 and 5 are coprime, we get that the last equality is possible only for p – 4q = 0 and q – p = 0. Whence p = q = 0, which is impossible, since p and q are chosen natural.

So, the assumption is false, which means that the number lg 80 is irrational.

c) We denote this number by x.

Then (x - √ 2) 3 = 3, or x 3 + 6x - 3 = √ 2 (3x 2 + 2). After squaring this equation, we find that x must satisfy the equation

x 6 - 6x 4 - 6x 3 + 12x 2 - 36x + 1 = 0.

Only numbers 1 and –1 can be its rational roots. Verification shows that 1 and –1 are not roots.

So, the given number √ 2 + 3 √ 3 ​​is irrational.

2. It is known that the numbers a, b, √ a –√ b,- rational. Prove that √ a and √ b Are also rational numbers.

Consider the product

(√ a - √ b) (√ a + √ b) = a - b.

Number √ a + √ b, which is equal to the ratio of the numbers a - b and √ a –√ b, is rational, since the quotient of dividing two rational numbers is a rational number. The sum of two rational numbers

½ (√ a + √ b) + ½ (√ a - √ b) = √ a

- rational number, their difference,

½ (√ a + √ b) - ½ (√ a - √ b) = √ b,

is also a rational number, as required.

3. Prove that there are positive irrational numbers a and b for which the number a b is natural.

4. Are there rational numbers a, b, c, d satisfying the equality

(a + b √ 2) 2n + (c + d√ 2) 2n = 5 + 4√ 2,

where n is a natural number?

If the equality given in the condition holds, and the numbers a, b, c, d are rational, then the equality holds:

(a - b √ 2) 2n + (c - d√ 2) 2n = 5 - 4√ 2.

But 5 - 4√ 2 (a - b√ 2) 2n + (c - d√ 2) 2n> 0. The resulting contradiction proves that the original equality is impossible.

Answer: do not exist.

5. If segments with lengths a, b, c form a triangle, then for all n = 2, 3, 4,. ... ... segments with lengths n √ a, n √ b, n √ c also form a triangle. Prove it.

If segments with lengths a, b, c form a triangle, then the triangle inequality gives

Therefore we have

(n √ a + n √ b) n> a + b> c = (n √ c) n,

N √ a + n √ b> n √ c.

The rest of the cases of checking the triangle inequality are considered in a similar way, whence the conclusion follows.

6. Prove that the infinite decimal fraction 0.1234567891011121314 ... (after the decimal point in a row are written out all natural numbers in order) is an irrational number.

As you know, rational numbers are expressed in decimal fractions, which have a period starting from a certain sign. Therefore, it suffices to prove that the given fraction is not periodic from any sign. Suppose that this is not the case, and some sequence T, consisting of n digits, is a period of the fraction, starting from the mth decimal place. It is clear that among the digits after the m-th character there are nonzero ones, therefore there is a nonzero digit in the sequence of digits T. This means that starting from the m-th digit after the decimal point, there is a nonzero digit among any n digits in a row. However, in the decimal notation of this fraction, there must be a decimal notation of the number 100 ... 0 = 10 k, where k> m and k> n. It is clear that this entry will occur to the right of the m-th digit and contains more than n zeros in a row. Thus, we obtain a contradiction, which completes the proof.

7. You are given an infinite decimal fraction 0, a 1 a 2 .... Prove that the numbers in its decimal notation can be rearranged so that the resulting fraction expresses a rational number.

Recall that a fraction expresses a rational number if and only if it is periodic, starting with a certain sign. We divide the numbers from 0 to 9 into two classes: in the first class we include those numbers that occur in the original fraction a finite number of times, in the second class - those that occur in the original fraction an infinite number of times. Let's start writing out the periodic fraction, which can be obtained from the original permutation of the numbers. First, after zero and a comma, we write in random order all the numbers from the first class - each as many times as it occurs in the initial fraction. The first-class digits recorded will precede the period in the fractional part of the decimal fraction. Next, we write down in some order, one at a time, the numbers from the second class. We will declare this combination as a period and will repeat it an infinite number of times. Thus, we have written out the required periodic fraction, which expresses some rational number.

8. Prove that in each infinite decimal fraction there is a sequence of decimal places of arbitrary length, which occurs infinitely many times in the expansion of the fraction.

Let m be an arbitrary natural number. Let's split the given infinite decimal fraction into segments, with m digits in each. There will be infinitely many such segments. On the other hand, there are only 10 m different systems consisting of m digits, that is, a finite number. Consequently, at least one of these systems must be repeated here infinitely many times.

Comment. For irrational numbers √ 2, π or e we do not even know which digit is repeated infinitely many times in the infinite decimal fractions representing them, although each of these numbers, as can easily be proved, contains at least two different such digits.

9. Prove in an elementary way that positive root equations

is irrational.

For x> 0, the left side of the equation increases with increasing x, and it is easy to see that for x = 1.5 it is less than 10, and for x = 1.6 it is more than 10. Therefore, the only positive root of the equation lies within the interval (1.5 ; 1.6).

We write the root as an irreducible fraction p / q, where p and q are some coprime natural numbers. Then, for x = p / q, the equation will take the following form:

p 5 + pq 4 = 10q 5,

whence it follows that p is a divisor of 10, therefore, p is equal to one of the numbers 1, 2, 5, 10. However, writing out fractions with numerators 1, 2, 5, 10, we immediately notice that none of them falls inside the interval (1.5; 1.6).

So, the positive root of the original equation cannot be represented as an ordinary fraction, which means it is an irrational number.

10. a) Are there three points A, B and C on the plane such that for any point X the length of at least one of the segments XA, XB and XC is irrational?

b) The coordinates of the vertices of the triangle are rational. Prove that the coordinates of the center of its circumcircle are also rational.

c) Is there such a sphere on which there is exactly one rational point? (A rational point is a point at which all three Cartesian coordinates are rational numbers.)

a) Yes, they do. Let C be the midpoint of the segment AB. Then XC 2 = (2XA 2 + 2XB 2 - AB 2) / 2. If the number AB 2 is irrational, then the numbers XA, XB and XC cannot be rational at the same time.

b) Let (a 1; b 1), (a 2; b 2) and (a 3; b 3) be the coordinates of the vertices of the triangle. The coordinates of the center of its circumcircle are given by a system of equations:

(x - a 1) 2 + (y - b 1) 2 = (x - a 2) 2 + (y - b 2) 2,

(x - a 1) 2 + (y - b 1) 2 = (x - a 3) 2 + (y - b 3) 2.

It is easy to check that these equations are linear, which means that the solution of the considered system of equations is rational.

c) Such a sphere exists. For example, a sphere with the equation

(x - √ 2) 2 + y 2 + z 2 = 2.

Point O with coordinates (0; 0; 0) is a rational point lying on this sphere. The rest of the points of the sphere are irrational. Let's prove it.

Suppose the opposite: let (x; y; z) be a rational point of the sphere, different from the point O. It is clear that x is different from 0, since at x = 0 there is a unique solution (0; 0; 0), which we are not now interested. Let's expand the brackets and express √ 2:

x 2 - 2√ 2 x + 2 + y 2 + z 2 = 2

√ 2 = (x 2 + y 2 + z 2) / (2x),

which cannot be for rational x, y, z and irrational √ 2. So, О (0; 0; 0) is the only rational point on the considered sphere.

Tasks without solutions

1. Prove that the number

\ [\ sqrt (10+ \ sqrt (24) + \ sqrt (40) + \ sqrt (60)) \]

is irrational.

2. For which integers m and n does the equality (5 + 3√ 2) m = (3 + 5√ 2) n hold?

3. Is there a number a such that the numbers a - √ 3 and 1 / a + √ 3 are integers?

4. Can the numbers 1, √ 2, 4 be members (not necessarily adjacent) of an arithmetic progression?

5. Prove that for any natural number n the equation (x + y√ 3) 2n = 1 + √ 3 has no solutions in rational numbers (x; y).

What are irrational numbers? Why are they called that? Where are they used and what are they? Few can answer these questions without hesitation. But in fact, the answers to them are quite simple, although not everyone needs them and in very rare situations.

Essence and designation

Irrational numbers are infinite non-periodic The need to introduce this concept is due to the fact that the previously existing concepts of real or real, integer, natural and rational numbers were already insufficient for solving new emerging problems. For example, in order to figure out how square 2 is, you need to use non-periodic infinite decimal fractions. In addition, many of the simplest equations also do not have a solution without introducing the concept of an irrational number.

This set is denoted as I. And, as it is already clear, these values ​​cannot be represented as a simple fraction, in the numerator of which there will be an integer, and in the denominator -

For the first time, one way or another, Indian mathematicians faced this phenomenon in the 7th century when it was discovered that square roots some of the quantities cannot be indicated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this in the process of studying the isosceles right triangle... Some scientists who lived before our era made a serious contribution to the study of this set. The introduction of the concept of irrational numbers entailed a revision of the existing mathematical system which is why they are so important.

origin of name

If ratio in Latin is "fraction", "ratio", then the prefix "ir"
gives this word the opposite meaning. Thus, the name of the set of these numbers means that they cannot be correlated with whole or fractional numbers, they have a separate place. This follows from their essence.

Place in the general classification

Irrational numbers, along with rational numbers, belong to the group of real or real numbers, which in turn are complex. There are no subsets, however, there are algebraic and transcendental varieties, which will be discussed below.

Properties

Since irrational numbers are part of the set of real numbers, then all their properties that are studied in arithmetic (they are also called basic algebraic laws) are applicable to them.

a + b = b + a (commutativity);

(a + b) + c = a + (b + c) (associativity);

a + (-a) = 0 (existence of the opposite number);

ab = ba (displacement law);

(ab) c = a (bc) (distributivity);

a (b + c) = ab + ac (distribution law);

a x 1 / a = 1 (existence of a reciprocal);

The comparison is also carried out in accordance with general laws and principles:

If a> b and b> c, then a> c (the transitivity of the relation) and. etc.

Of course, all irrational numbers can be converted using basic arithmetic. There are no special rules for this.

In addition, the action of the Archimedes axiom extends to irrational numbers. It says that for any two quantities a and b, the statement is true that by taking a as a term a sufficient number of times, you can exceed b.

Usage

Despite the fact that in ordinary life not so often you have to deal with them, irrational numbers do not lend themselves to counting. There are a lot of them, but they are almost invisible. We are surrounded by irrational numbers everywhere. Examples familiar to everyone are pi, equal to 3.1415926 ..., or e, which is essentially the base natural logarithm, 2.718281828 ... In algebra, trigonometry and geometry, they have to be used constantly. By the way, the famous meaning of the "golden ratio", that is, the ratio of both the greater part to the lesser, and vice versa, is also

refers to this set. The less well-known "silver" - too.

On the number line, they are located very densely, so that between any two quantities referred to the set of rational ones, an irrational one is necessarily encountered.

Until now, there are a lot of unsolved problems associated with this set. There are criteria such as the measure of irrationality and the normality of a number. Mathematicians continue to examine the most significant examples for belonging to one group or another. For example, it is believed that e is a normal number, that is, the probability of different digits appearing in its record is the same. As for pi, research is still underway on it. The measure of irrationality is a quantity that shows how well a particular number can be approximated by rational numbers.

Algebraic and transcendental

As already mentioned, irrational numbers are conventionally divided into algebraic and transcendental. Conditionally, since, strictly speaking, this classification is used to divide the set C.

This designation hides complex numbers that include real or real.

So, algebraic is a value that is a root of a polynomial that is not identically zero. For example, the square root of 2 would be in this category because it is the solution to the equation x 2 - 2 = 0.

All other real numbers that do not satisfy this condition are called transcendental. This variety includes the most famous and already mentioned examples - the number pi and the base of the natural logarithm e.

Interestingly, neither one nor the second was originally deduced by mathematicians in this capacity, their irrationality and transcendence were proved many years after their discovery. For pi, the proof was presented in 1882 and simplified in 1894, ending the 2,500 year controversy over the problem of squaring the circle. It is still not fully understood, so modern mathematicians have something to work on. By the way, the first sufficiently accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too rough.

For e (Euler's or Napier's number), evidence of its transcendence was found in 1873. It is used in solving logarithmic equations.

Other examples include sine, cosine, and tangent values ​​for any algebraic nonzero values.

Irrational number- this is real number, which is not rational, that is, it cannot be represented as a fraction, where are integers,. An irrational number can be represented as an infinite non-periodic decimal fraction.

A lot of irrational numbers are usually indicated by an uppercase Latin letter in bold, no fill. Thus: i.e. the set of irrational numbers is the difference between the sets of real and rational numbers.

On the existence of irrational numbers, more precisely segments incommensurable with a segment of unit length were already known to ancient mathematicians: they knew, for example, the incommensurability of the diagonal and the side of a square, which is tantamount to the irrationality of a number.

Properties

• Any real number can be written in the form of an infinite decimal fraction, while irrational numbers and only they are written in non-periodic infinite decimal fractions.
• Irrational numbers define Dedekind sections in the set of rational numbers, which do not have the largest number in the lower class and do not have the smallest number in the upper class.
• Every real transcendental number is irrational.
• Every irrational number is either algebraic or transcendental.
• The set of irrational numbers is everywhere dense on the number line: there is an irrational number between any two numbers.
• The order on the set of irrational numbers is isomorphic to the order on the set of real transcendental numbers.
• The set of irrational numbers is uncountable, it is a set of the second category.

Examples of

Irrational numbers - ζ (3) - √2 - √3 - √5 - - - - -

Irrational are:

Examples of proof of irrationality

Root of 2

Suppose the opposite: rational, that is, it is represented as an irreducible fraction, where is an integer and is a natural number. Let's square the assumed equality:

.

Hence it follows that even means even and. Let it be, where is the whole. Then

Therefore, even means even and. We got that and are even, which contradicts the irreducibility of the fraction. This means that the initial assumption was wrong, and - an irrational number.

Binary logarithm of 3

Suppose the opposite: rational, that is, represented as a fraction, where and are integers. Since, and can be chosen as positive. Then

But even and odd. We get a contradiction.

e

History

The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manava (c. 750 BC - c. 690 BC) figured out that the square roots of certain natural numbers, such as 2 and 61 cannot be explicitly expressed.

The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontus (c. 500 BC), a Pythagorean who found this proof by studying the side lengths of the pentagram. At the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which enters any segment an integer number of times. However, Hippasus proved that there is no single unit of length, since the assumption of its existence leads to a contradiction. He showed that if the hypotenuse of an isosceles right-angled triangle contains an integer number of unit segments, then this number must be both even and odd at the same time. The proof looked like this:

• The ratio of the length of the hypotenuse to the length of the leg of an isosceles right triangle can be expressed as a:b, where a and b selected as the smallest possible.
• By the Pythagorean theorem: a² = 2 b².
• Because a² even, a must be even (since the square of an odd number would be odd).
• Insofar as a:b irreducible, b must be odd.
• Because a even, denote a = 2y.
• Then a² = 4 y² = 2 b².
• b² = 2 y², therefore b Is even, then b even.
• However, it has been proven that b odd. Contradiction.

Greek mathematicians called this ratio of incommensurable quantities aalogos(ineffable), however, according to the legends, they did not give Hippas the respect he deserved. Legend has it that Hippasus made a discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe that denies the doctrine that all entities in the universe can be reduced to whole numbers and their relationships." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometrical objects are one and indivisible.

We have shown earlier that $ 1 \ frac25 $ is close to $ \ sqrt2 $. If it were exactly $ \ sqrt2 $,. Then the ratio - $ \ frac (1 \ frac25) (1) $, which can be turned into the ratio of integers $ \ frac75 $, multiplying the upper and lower parts of the fraction by 5, and would be the desired value.

Unfortunately, $ 1 \ frac25 $ is not an exact value for $ \ sqrt2 $. A more accurate answer, $ 1 \ frac (41) (100) $, gives us the relation $ \ frac (141) (100) $. We achieve even greater precision when we equate $ \ sqrt2 $ with $ 1 \ frac (207) (500) $. In this case, the ratio in integers will be $ \ frac (707) (500) $. But also $ 1 \ frac (207) (500) $ is not the exact value of the square root of 2. Greek mathematicians spent a lot of time and effort to calculate exact value$ \ sqrt2 $, but they never succeeded. They were unable to represent the ratio $ \ frac (\ sqrt2) (1) $ as a ratio of integers.

Finally, the great Greek mathematician Euclid proved that no matter how the accuracy of the calculations increases, it is impossible to get the exact value of $ \ sqrt2 $. There is no fraction that, when squared, will result in 2. They say that Pythagoras was the first to come to this conclusion, but this inexplicable fact so amazed the scientist that he himself swore and took an oath from his students to keep this discovery a secret. ... However, it is possible that this information does not correspond to reality.

But if the number $ \ frac (\ sqrt2) (1) $ cannot be represented as a ratio of integers, then none containing $ \ sqrt2 $, for example $ \ frac (\ sqrt2) (2) $ or $ \ frac (4) (\ sqrt2) $ also cannot be represented as a ratio of integers, since all such fractions can be converted to $ \ frac (\ sqrt2) (1) $ multiplied by some number. So $ \ frac (\ sqrt2) (2) = \ frac (\ sqrt2) (1) \ times \ frac12 $. Or $ \ frac (\ sqrt2) (1) \ times 2 = 2 \ frac (\ sqrt2) (1) $, which can be transformed by multiplying the top and bottom by $ \ sqrt2 $ to get $ \ frac (4) (\ sqrt2) $. (Remember that no matter what the number $ \ sqrt2 $ is, if we multiply it by $ \ sqrt2 $, we get 2.)

Since the number $ \ sqrt2 $ cannot be represented as a ratio of integers, it is called irrational number... On the other hand, all numbers that can be represented as a ratio of integers are called rational.

All integers and fractional numbers, both positive and negative, are rational.

As it turns out, most square roots are irrational numbers. Only numbers in a series of square numbers have rational square roots. These numbers are also called perfect squares. Rational numbers are also fractions made up of these perfect squares. For example, $ \ sqrt (1 \ frac79) $ is a rational number because $ \ sqrt (1 \ frac79) = \ frac (\ sqrt16) (\ sqrt9) = \ frac43 $ or $ 1 \ frac13 $ (4 is the root square of 16, and 3 is the square root of 9).