How to prove the density of rational numbers on real numbers

Proofing 1: Let two rational numbers a,b,a>b,a-b=d, d is a rational number, d is not equal to 0, d 2 is not equal to 0, a-d 2 is a rational number, a>(a-d 2) >b

Proof-2: Let a, b r, exist z e, ax r, let c>0, then x+c>x

There is c1>0 so that x-like can be selected to c2, c3, ,..Such that } is included in e

Now let's prove that cn can be selected such that the limit of an=x+cn is x

Conversely, if any CN satisfies such that AN is greater than X and AN is monotonic (AN converges), then AN converges to A>X

However, you can choose A'>0 such that xAdditional Information:

A set of rational numbers is a set of integers.

of expansion. Within the set of rational numbers, addition, subtraction, multiplication, division (divisor.

Not zero) 4 kinds of operations are unimpeded.

For the mixed operation of addition, subtraction, multiplication and division of rational numbers, if there is no parentheses to indicate what operation to do first, it is carried out in the order of "multiplication and division first, then addition and subtraction", and if it is a sibling operation, it is calculated in order from left to right.

Subtracting a number from a rational number is equal to adding the opposite of the number.

That is, the subtraction of rational numbers uses the opposite of numbers to add for operation.

Division and multiplication of rational numbers are inverse operations.

1. Let e be a non-empty subset of r satisfies :

Let a, b r, there is z e, so that a considers x r, and let c>0, then x+c>x. So there is c1>0 so that x-like can be selected to c2, c3, ,..Such that } is included in e

Now let's prove that cn can be selected such that the limit of an=x+cn is x

Conversely, if any CN satisfies such that AN is greater than X and AN is monotonic (AN converges), then AN converges to A>X.

However, it is known that a can be selected'> 0 so that x is the gathering point of e, and the closure of e is r from the arbitrariness of x.

2. Let two rational numbers a and b.

a>b, a-b=d, d is a rational number, d is not equal to 0, d 2 is not equal to 0, a-d 2 is a rational number, a>(a-d 2)>b;

Solution: The proof makes use of the properties of rational numbers. Let the two rational numbers a, b, a>b.

a-b = d, d is a rational number, d is not equal to 0, d 2 is not equal to 0, a-d 2 is a rational number, a>(a-d 2) >b.

A binox in interval [a,b]: a (a+b) 2 b;

Two thirds in interval [a,b]: a (2a+b) 3 (a+2b) 3 b;

Three quartile points in interval [a,b]: a (3a+b) 4 (2a+2b) 4 (a+3b) 4 b;

Thus there is an infinite number of rational numbers between two rational numbers a,b(a b).

Definition of Denseness: If a set has elements in any of the openings of a space, then we call the set dense in that space.

Let e be a non-empty subset of r satisfying:

1.Let a, b r, there be z e, such that a0, then x+c>xSo there is c1>0 such that xx

But by 1I know that I can choose A'> 0 such that xb, a-b = d, d is a rational number, d is not equal to 0, d 2 is not equal to 0, a-d 2 is a rational number, a>(a-d 2) >b,

Because there are an infinite number of rational and irrational numbers between any two rational numbers. At the same time, there are also an infinite number of rational and irrational numbers between any two irrational numbers. So both rational and irrational numbers are dense.

Rational numbers are a collective term for integers (positive integers, 0s, negative integers) and fractions, and are a collection of integers and fractions.

Irrational numbers, also known as infinite non-cyclic decimals, cannot be written as a ratio of two integers.

Rational numbers are a collective term for integers (positive integers, 0s, negative integers) and fractions. Positive integers and positive fractions are collectively called positive rational numbers, and negative integers and negative fractions are collectively referred to as negative rational numbers. Therefore, the number of rational numbers in the set of rational numbers can be divided into positive rational numbers, negative rational numbers, and zeros.

Since any integer or fraction can be reduced to a decimal cyclic decimal, and conversely, every decimal cyclic decimal can also be reduced to an integer or fraction, therefore, a rational number can also be defined as a decimal cyclic decimal.

In mathematics, an irrational number is all real numbers that are not rational numbers, which are numbers made up of ratios (or fractions) of integers. Segments are also described as non-comparable when the ratio of length of two segments is irrational, meaning that they cannot be "measured", i.e. there is no length ("measure").

Given any two real numbers a, b assumes that a is easy to operate on the square of the fraction with a clear beard.

It is obviously wrong to mistake the whole part and the fraction part separately. The arithmetic square root of the sum is prone to appear.

Confusion with the sum of the square root of arithmetic and errors should be taken to avoid such errors.

Note for real numbers: the open square generally involves two aspects: one is that the opened square number is an integer, and the opened square number should be decomposed into a square number.

with a non-square product, square the square number; The second is that the square number is a fraction, to multiply the numerator and denominator by an appropriate number, turn the denominator into a square number, and then square the denominator.

Because there are an infinite number of rational and irrational numbers between any two rational numbers.

At the same time, there are also an infinite number of rational and irrational numbers between any two irrational numbers. So both rational and irrational numbers are dense.

Let a and b be any two rational numbers, and a divisor.

Not zero) 4 kinds of operations are unimpeded.

The order of the magnitude of the rational numbers a and b: if a-b are positive rational numbers, then when a is greater than b or b is less than a, it is denoted as the absolute value of a>b or b.

Add. 2. The two numbers of different signs are added, and if the absolute value is equal, they are opposite to each other.

The sum of two numbers is 0; If the absolute values are not equal, take the sign of the addition with the greater absolute value, and subtract the smaller absolute value from the larger absolute value.

3. Add two numbers that are opposite to each other to get 0.

4. Add a number to 0 and still get this number.

5. Two numbers that are opposite to each other can be added first.

6. Numbers with the same symbol can be added first.

7. Numbers with the same denominator can be added first.

8. If several numbers can be added to get an integer, they can be added first.

Subtracting a number is equivalent to adding the opposite of the number, that is, the subtraction of rational numbers uses the opposite number of numbers to add for operation.

1. The same sign is positive, the different sign is negative, and the absolute value is multiplied.

2. Multiplying any number by zero gives zero.

3. Multiply several numbers that are not equal to zero, and the sign of the product is determined by the number of negative factors, when there are odd numbers of negative factors, the product is negative, and when there are even numbers of negative factors, the product is positive.

4. When several numbers are multiplied, there is a factor that is zero, and the product is zero.

5. Multiply several numbers that are not equal to zero, first determine the sign of the product, and then multiply the absolute value.

1. Divide by a number that is not equal to zero, which is equal to the reciprocal of multiplying this number.

2. Divide the two numbers, the same sign is positive, the different sign is negative, and the absolute value is divided. Zero divided by any number that is not equal to zero gives zero.

Note: Zero cannot be a divisor and denominator.

Division and multiplication of rational numbers are inverse operations.

When doing division operations, the symbols are determined first according to the rule that the same sign is positive and the different sign is negative, and then the absolute values are divided. If there is a band fraction in the equation.

Generally, it is converted into a false score first.

Make the calculations. If it is not divisible, then division operations are converted to multiplication operations.

Both irrational and rational numbers are dense, that is, there are an infinite number of rational numbers and an infinite number of irrational numbers between any two unequal real numbers. There are many more irrational numbers than rational numbers. There are infinitely many rational numbers, as many as natural numbers, so they are called countable infinity.

There are as many irrational numbers as real numbers, and they are uncountable. On the interval [0,1], the measure of rational numbers is 0 and that of irrational numbers is 1.

Irrational numbers are all real numbers that are not rational numbers, which are numbers made up of ratios (or fractions) of integers. Segments are also described as incomparable when the ratio of the length of two segments is a corteck irrational number, meaning that they cannot be "measured", i.e. there is no length ("measure").

Obviously, take the two real numbers x1 and x2. (x1 then x0=(x1+x2) 2 (x1,x2) and then x1,x0, do the same change, the same is true.

It goes on infinitely, so the real number is dense.

Arbitrarily given two real numbers a, b

Suppose a then there is x = (a+b) 2

satisfying a so the real number is dense.