Ancient mathematicians already knew with a segment of unit length: they knew, for example, the incommensurability of the diagonal and the side of a square, which is tantamount to the irrationality of a number.
Irrational are:
Examples of proof of irrationality
Root of 2
Suppose the opposite: rational, that is, represented as an irreducible fraction, where and are integers. Let's square the assumed equality:
.Hence it follows that even means even and. Let 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 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 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 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.
- As 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.
- As 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(unspeakable), 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 integers and their relationships." 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 indivisible.
see also
Notes (edit)
| Number systems | |
|---|---|
| Counting multitudes |
Integers () |
Integers
The numbers used in counting are called natural numbers. For example, $ 1,2,3, etc. Natural numbers form the set of natural numbers, which denote $ N $. This notation comes from the Latin word naturalis- natural.
Opposite numbers
Definition 1
If two numbers differ only in signs, they are called in mathematics opposite numbers.
For example, the numbers $ 5 $ and $ -5 $ are opposite numbers, since differ only in signs.
Remark 1
For any number, there is an opposite number, and moreover, only one.
Remark 2
The number zero is the opposite of itself.
Whole numbers
Definition 2
Whole numbers are natural, opposite numbers and zero.
The set of integers includes many naturals and their opposite.
Denote integers $ Z. $
Fractional numbers
Numbers like $ \ frac (m) (n) $ are called fractions or fractional numbers. Fractional numbers can also be written in decimal notation, i.e. as decimal fractions.
For example: $ \ \ frac (3) (5) $, $ 0.08 $, etc.
Just like integers, fractional numbers can be either positive or negative.
Rational numbers
Definition 3
Rational numbers is called a set of numbers containing a set of integers and fractional numbers.
Any rational number, both integer and fractional, can be represented as a fraction $ \ frac (a) (b) $, where $ a $ is an integer and $ b $ is a natural number.
Thus, the same rational number can be written in different ways.
For example,
Hence it is clear that any rational number can be represented in the form of a finite decimal fraction or an infinite decimal periodic fraction.
The set of rational numbers is denoted by $ Q $.
As a result of performing any arithmetic operation on rational numbers, the resulting answer will be a rational number. This is easy to prove, due to the fact that when adding, subtracting, multiplying and dividing ordinary fractions, you get an ordinary fraction
Irrational numbers
In the course of studying a course in mathematics, you often have to deal with numbers that are not rational.
For example, to make sure that there is a set of non-rational numbers, solve the equation $ x ^ 2 = 6 $. The roots of this equation will be the numbers $ \ surd 6 $ and - $ \ surd 6 $. These numbers will not be rational.
Also, when finding the diagonal of a square with a side of $ 3 $, we apply the Pythagorean theorem and obtain that the diagonal will be equal to $ \ surd 18 $. This number is also not rational.
Such numbers are called irrational.
So, an irrational number is called an infinite decimal non-periodic fraction.
One of the most common irrational numbers is the number $ \ pi $
When performing arithmetic operations with irrational numbers, the result can be both rational and irrational.
Let us prove this by the example of finding the product of irrational numbers. Let's find:
$ \ \ sqrt (6) \ cdot \ sqrt (6) $
$ \ \ sqrt (2) \ cdot \ sqrt (3) $
Decision
$ \ \ sqrt (6) \ cdot \ sqrt (6) = 6 $
$ \ sqrt (2) \ cdot \ sqrt (3) = \ sqrt (6) $
This example shows that the result can be both a rational and an irrational number.
If rational and irrational numbers participate in arithmetic operations at the same time, then the result will be an irrational number (except, of course, multiplication by $ 0 $).
Real numbers
A set of real numbers is a set containing a set of rational and irrational numbers.
The set of real numbers $ R $ is denoted. The set of real numbers can be symbolically denoted $ (-?; +?). $
We said earlier that an irrational number is called an infinite decimal non-periodic fraction, and any rational number can be represented as a finite decimal fraction or an infinite decimal periodic fraction, therefore any finite and infinite decimal fraction will be a real number.
When performing algebraic actions, the following rules will be fulfilled
- when multiplying and dividing positive numbers, the resulting number will be positive
- when multiplying and dividing negative numbers, the resulting number will be positive
- when multiplying and dividing negative and positive numbers, the resulting number will be negative
Also, real numbers can be compared with each other.
Not all actions considered in algebra are feasible in the field of rational numbers. An example is the square root operation. So, if equality holds for values, then equality does not hold for any rational value. Let us prove this. First, we note that an integer cannot have a square equal to 2: for, we have a for certainly greater than 2. Suppose now that the fractional: (the fraction is considered irreducible) and
Hence, we have to be an even number (otherwise the square would not be even). Let's put.
Now it turns out that and is even, which contradicts the assumption that the fraction is irreducible
This shows that in the area of rational numbers, the square root cannot be extracted from the number 2, the symbol has no meaning in the area of rational numbers. Meanwhile, the task: "to find the side of a square, knowing that its area is S" - is just as natural with as with. The way out of this and other similar difficulties is to further expand the concept of number, in the introduction of a new type of numbers - irrational numbers.
Let us show how irrational numbers are introduced using the example of the problem of extracting the square root of the number 2; for simplicity, we restrict ourselves to a positive value of the root.
For every positive rational number, one of the inequalities or Obviously will take place. We then consider the numbers and find two neighboring numbers with the property that the first has a square less than two, and the second has a greater than two. Namely, Similarly, continuing this process, we obtain a number of inequalities (to obtain the decimal fractions written here, you can also use the well-known approximate square root extraction algorithm, p. 13):
Comparing first the whole parts, and then the first, second, third, etc. digits after the decimal point for rational numbers, between the squares of which there is 2, we can write out these decimal places in sequence:
The process of finding pairs of rational numbers (expressed in finite decimal fractions) that differ from each other by more and more m can be continued indefinitely. Therefore, we can consider the fraction (6.1) as an infinite decimal fraction (non-periodic, since in the case of periodicity it would represent a rational number).
This infinite non-periodic fraction, any number of decimal places of which we can write out, but for which it is impossible to record all the signs at the same time, and is taken as a number equal to (i.e., for a number whose square is 2).
We represent the negative value of the square root of two in the form
or, using an artificial form of writing numbers, in the form
We now introduce the following definition: an irrational number is any infinite non-periodic decimal fraction
where a - making part of the number (it can be positive, equal to zero or negative), and - decimal places (digits) of its fractional part.
An irrational number given by an infinite non-periodic fraction determines two sequences of finite decimal fractions, called decimal approximations for a deficiency and an excess:
For example, for we write
etc. Here, for example, 1.41 is a decimal approximation with an accuracy of 0.01 for a deficiency, and 1.42 for an excess.
Writing inequalities between an irrational number and its decimal approximations is included in the very definition of the concept of an irrational number and can be used as the basis for determining the ratio "more" and "less" for irrational numbers.
The possibility of representing irrational numbers by their more and more accurate decimal approximations also underlies the definition of arithmetic operations on irrational numbers, which are actually performed on their irrational approximations by deficiency or excess.
Many actions lead to irrational numbers, such as the action of extracting the root of a power from a rational number (if it is not a power of another rational number), taking the logarithm, etc. An irrational number is a number equal to the ratio of the circumference of a circle to its diameter (p. 229).
All rational and irrational numbers together form a set of real (or real) numbers. Thus, any decimal fraction, finite or infinite (periodic or non-periodic), always determines a real number.
Any nonzero real number is either positive or negative.
In this connection, we recall the following definition. The absolute value or modulus of a real number a is the number determined by the equalities a, if
Thus, the modulus of a non-negative number is equal to this number itself (the top line of the equality); the absolute value of a negative number is equal to this number, taken with the opposite sign (bottom line). For example,
It follows from the definition of the modulus that the modulus of any number is a non-negative number; if the modulus of the number is zero, then the number itself is equal to zero, in other cases the modulus is positive.
Real numbers form a numerical field - the field of real numbers: the result of rational actions on real numbers is again expressed by a real number. Note that irrational numbers taken separately do not form a field or even a ring: for example, the sum of two irrational numbers is equal to the rational number 3.
Our short essay on the development of the concept of number, built according to the scheme
we conclude by indicating the most important properties of the collection of real numbers.
1. Real numbers form a field.
2. Actions on real numbers are subject to the usual laws (for example, addition and multiplication - the laws of commutativity, associativity, distributivity, item 1).
3. For any two real numbers a and b, one and only one of the three relations holds: a is greater than b (a> b), a is less, and equal. Therefore, they say that the set of real numbers is ordered.
4. It is customary, finally, to say that the set of real numbers has the property of continuity. The meaning given to this expression is explained in Section 8. It is this property that essentially distinguishes the field of real numbers from the field of rational numbers.
