rational number is 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 a real number is not rational, then it is irrational number. Decimal fractions expressing irrational numbers are infinite and not periodic. The set of irrational numbers is usually denoted by the capital Latin 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 transcendent.
Some properties:
The set of rational numbers is everywhere dense 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, one can establish a one-to-one correspondence between them (all elements of the set of rational numbers can be renumbered).
The set Q of rational numbers is closed under 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 real line: between any two numbers there is an irrational number (and hence an infinite set of irrational numbers).
The set of irrational numbers is uncountable.
When solving problems, it is convenient, together with the irrational number a + b√ c (where a, b are rational numbers, c is an integer that is not a square of a natural number), to consider the number “conjugate” with it a - b√ c: its sum and product with the original - rational numbers. So a + b√ c and a – b√ c are the roots of a quadratic equation with integer coefficients.
Problems with solutions
1. Prove that
a) number √ 7;
b) number lg 80;
c) number √ 2 + 3 √ 3;
is irrational.
a) Assume 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, then p 2, and hence p is divisible by 7. Then р = 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, so the number √ 7 is irrational.
b) Assume that the number lg 80 is rational. Then there are natural p and q such that lg 80 = p/q, or 10 p = 80 q , whence we get 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 to be natural.
So, the assumption is false, so the number lg 80 is irrational.
c) Let's denote this number by x.
Then (x - √ 2) 3 \u003d 3, or x 3 + 6x - 3 \u003d √ 2 (3x 2 + 2). After squaring this equation, we get that x must satisfy the equation
x 6 - 6x 4 - 6x 3 + 12x 2 - 36x + 1 = 0.
Its rational roots can only be the numbers 1 and -1. The check 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 numbers a – b and √ a –√ b , is rational because the quotient of two rational numbers is a rational number. Sum of two rational numbers
½ (√ a + √ b) + ½ (√ a - √ b) = √ a
is a rational number, their difference,
½ (√ a + √ b) - ½ (√ a - √ b) = √ b,
is also a rational number, which was to be proved.
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 is satisfied, and the numbers a, b, c, d are rational, then the equality is also satisfied:
(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: they don't 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 remaining cases of checking the triangle inequality are considered similarly, from which the conclusion follows.
6. Prove that the infinite decimal fraction 0.1234567891011121314... (all natural numbers are listed in order after the decimal point) is an irrational number.
As you know, rational numbers are expressed as decimal fractions, which have a period starting from a certain sign. Therefore, it suffices to prove that this fraction is not periodic with any sign. Suppose that this is not the case, and some sequence T, consisting of n digits, is the period of a fraction, starting from the mth decimal place. It is clear that there are nonzero digits after the mth digit, so there is a nonzero digit in the sequence of digits T. This means that starting from the m-th digit after the decimal point, among any n digits in a row there is a non-zero digit. However, in the decimal notation of this fraction, there must be a decimal notation for 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 contain more than n zeros in a row. Thus, we obtain a contradiction, which completes the proof.
7. Given an infinite decimal fraction 0,a 1 a 2 ... . Prove that the digits 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 from some 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 a periodic fraction, which can be obtained from the original permutation of the digits. 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 entry of the original fraction. The first class digits written will precede the period in the fractional part of the decimal. Next, we write down the numbers from the second class in some order once. We will declare this combination a period and repeat it an infinite number of times. Thus, we have written out the required periodic fraction expressing some rational number.
8. Prove that in each infinite decimal fraction there is a sequence of decimal digits of arbitrary length, which occurs infinitely many times in the expansion of the fraction.
Let m be an arbitrarily given natural number. Let's break this infinite decimal fraction into segments, each with m digits. There will be infinitely many such segments. On the other hand, there are only 10 m of 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 don't even know which digit is repeated infinitely many times in the infinite decimals that represent them, although each of these numbers can easily be shown to contain at least two distinct such digits.
9. Prove in an elementary way that the positive root of the equation
is irrational.
For x > 0, the left side of the equation increases with x, and it is easy to see that at x = 1.5 it is less than 10, and at x = 1.6 it is greater than 10. Therefore, the only positive root of the equation lies inside 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 \u003d 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 circumscribed circle are also rational.
c) Does there exist a sphere on which there is exactly one rational point? (A rational point is a point for which all three Cartesian coordinates are rational numbers.)
a) Yes, there are. Let C be the midpoint of 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 circumscribed circle are given by the system of equations:
(x - a 1) 2 + (y - b 1) 2 \u003d (x - a 2) 2 + (y - b 2) 2,
(x - a 1) 2 + (y - b 1) 2 \u003d (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 remaining points of the sphere are irrational. Let's prove it.
Assume 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 for x = 0 there is a unique solution (0; 0; 0), which we cannot 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, O(0; 0; 0) is the only rational point on the sphere under consideration.
Problems without solutions
1. Prove that the number
\[ \sqrt(10+\sqrt(24)+\sqrt(40)+\sqrt(60)) \]
is irrational.
2. For what 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 positive integer n the equation (x + y √ 3 ) 2n = 1 + √ 3 has no solutions in rational numbers (x; y).
With a segment of unit length, ancient mathematicians already knew: they knew, for example, the incommensurability of the diagonal and the side of the square, which is equivalent to the irrationality of the number.
Irrational are:
Irrationality Proof Examples
Root of 2
Assume the contrary: it is rational, that is, it is represented as an irreducible fraction, where and are integers. Let's square the supposed equality:
.From this it follows that even, therefore, even and . Let where the whole. Then
Therefore, even, therefore, even and . We have obtained that and are even, which contradicts the irreducibility of the fraction . Hence, the original assumption was wrong, and is an irrational number.
Binary logarithm of the number 3
Assume the contrary: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be taken positive. Then
But it's clear, it's odd. We get a contradiction.
e
Story
The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found that the square roots of some 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 lengths of the sides of a pentagram. In the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment. However, Hippasus argued 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 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.
- According to the Pythagorean theorem: a² = 2 b².
- Because a² even, a must be even (since the square of an odd number would be odd).
- Because the 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 alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe, which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometric objects are one and inseparable.
see also
Notes
| Numerical systems | |
|---|---|
| Counting sets |
Integers () |
The set of all natural numbers is denoted by the letter N. Natural numbers are the numbers that we use to count objects: 1,2,3,4, ... In some sources, the number 0 is also referred to natural numbers.
The set of all integers is denoted by the letter Z. Integers are all natural numbers, zero and negative numbers:
1,-2,-3, -4, …
Now let's add to the set of all integers the set of all ordinary fractions: 2/3, 18/17, -4/5, and so on. Then we get the set of all rational numbers.
Set of rational numbers
The set of all rational numbers is denoted by the letter Q. The set of all rational numbers (Q) is the set consisting of numbers of the form m/n, -m/n and the number 0. Any natural number can be used as n,m. It should be noted that all rational numbers can be represented as a finite or infinite PERIODIC decimal fraction. The converse is also true, that any finite or infinite periodic decimal fraction can be written as a rational number.
But what about, for example, the number 2.0100100010…? It is an infinitely NON-PERIODIC decimal. And it does not apply to rational numbers.
In the school course of algebra, only real (or real) numbers are studied. The set of all real numbers is denoted by the letter R. The set R consists of all rational and all irrational numbers.
The concept of irrational numbers
Irrational numbers are all infinite decimal non-periodic fractions. Irrational numbers have no special notation.
For example, all numbers obtained by extracting the square root of natural numbers that are not squares of natural numbers will be irrational. (√2, √3, √5, √6, etc.).
But do not think that irrational numbers are obtained only by extracting square roots. For example, the number "pi" is also irrational, and it is obtained by division. And no matter how hard you try, you can't get it by taking the square root of any natural number.
And they derived their roots from the Latin word "ratio", which means "reason". Based on the literal translation:
- A rational number is a "reasonable number".
- An irrational number, respectively, is an “unreasonable number”.
General concept of a rational number
A rational number is one that can be written as:
- Ordinary positive fraction.
- Negative ordinary fraction.
- Zero (0) as a number.
In other words, the following definitions will fit a rational number:
- Any natural number is inherently rational, since any natural number can be represented as an ordinary fraction.
- Any integer, including the number zero, since any integer can be written both as a positive ordinary fraction, as a negative ordinary fraction, and as the number zero.
- Any ordinary fraction, and it does not matter here whether it is positive or negative, also directly approaches the definition of a rational number.
- A mixed number, a finite decimal fraction or an infinite periodic fraction can also be included in the definition.
Rational Number Examples
Consider examples of rational numbers:
- Natural numbers - "4", "202", "200".
- Integers - "-36", "0", "42".
- Ordinary fractions.
From the above examples, it is clear that rational numbers can be both positive and negative. Naturally, the number 0 (zero), which is also a rational number, at the same time does not belong to the category of a positive or negative number.
Hence, I would like to recall the general education program using the following definition: “Rational numbers” are those numbers that can be written as a fraction x / y, where x (numerator) is an integer, and y (denominator) is a natural number.
General concept and definition of an irrational number
In addition to the "rational numbers" we also know the so-called "irrational numbers". Let's briefly try to define these numbers.
Even ancient mathematicians, wanting to calculate the diagonal of a square along its sides, learned about the existence of an irrational number.
Based on the definition of rational numbers, you can build a logical chain and define an irrational number.
So, in fact, those real numbers that are not rational are, elementarily, irrational numbers.
Decimal fractions, expressing irrational numbers, are not periodic and infinite. 
Examples of an irrational number
Consider for clarity a small example of an irrational number. As we have already understood, infinite decimal non-periodic fractions are called irrational, for example:
- The number "-5.020020002 ... (it is clearly seen that the twos are separated by a sequence of one, two, three, etc. zeros)
- The number "7.040044000444 ... (here it is clear that the number of fours and the number of zeros increases by one each time in a chain).
- Everyone knows the number Pi (3.1415 ...). Yes, yes - it is also irrational.
In general, all real numbers are both rational and irrational. In simple terms, an irrational number cannot be represented as an ordinary fraction x / y.
General conclusion and brief comparison between numbers
We considered each number separately, the difference between a rational number and an irrational one remains:
- An irrational number occurs when taking the square root, when dividing a circle by a diameter, and so on.
- A rational number represents an ordinary fraction.
We conclude our article with a few definitions:
- An arithmetic operation performed on a rational number, in addition to dividing by 0 (zero), will also lead to a rational number in the final result.
- The end result, when performing an arithmetic operation on an irrational number, can lead to both a rational and an irrational value.
- If both numbers take part in the arithmetic operation (except for division or multiplication by zero), then the result will give us an irrational number.
What numbers are irrational? irrational number is not a rational real number, i.e. it cannot be represented as a fraction (as a ratio of two integers), where m is an integer, n- natural number . irrational number can be represented as an infinite non-periodic decimal fraction.
irrational number cannot be exact. Only in the format 3.333333…. For example, the square root of two - is an irrational number.
What is the irrational number? Irrational number(unlike rational ones) is called an infinite decimal non-periodic fraction.
Many irrational numbers often denoted by a capital Latin letter in bold without shading. That.:
Those. the set of irrational numbers is the difference between the sets of real and rational numbers.
Properties of irrational numbers.
- The sum of 2 non-negative irrational numbers can be a rational number.
- Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class of which there is no largest number, and in the upper class there is no smaller number.
- Every real transcendental number is an irrational number.
- All irrational numbers are either algebraic or transcendent.
- The set of irrational numbers is everywhere dense on the number line: between each pair of numbers there is an irrational number.
- 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 infinite, is a set of the 2nd category.
- The result of every arithmetic operation on rational numbers (except division by 0) is a rational number. The result of arithmetic operations on irrational numbers can be either a rational or an irrational number.
- The sum of a rational and an irrational number will always be an irrational number.
- The sum of irrational numbers can be a rational number. For example, let x irrational, then y=x*(-1) also irrational; x+y=0, and the number 0 rational (if, for example, we add the root of any degree of 7 and minus the root of the same degree of seven, we get a rational number 0).
Irrational numbers, examples.
γ — ζ (3) — ρ — √2 — √3 — √5 — φ — δs — α — e — π — δ