How to prove that a rational number is a minimum number field

Number of fieldsTo put it simply, a set of 0s and 1s is closed to the four operations (the result of the computation still belongs to this set).

Definition of rational numbers: Rational numbers can be written as two integer numbers that are proportional (so the result of division between rational numbers must still be numbers).

A set of rational numbers is not a field of numbers: it is clearly true. In fact, because it contains 0,1 and it is closed to four operations (the addition and subtraction of any two rational numbers is obviously a rational number, and the result of multiplication and division of a rational number is also a rational number).

A rational number field is the smallest number field:

1. Other number fields contain rational number fields. Because the set of rational numbers is the set of real numbers.

A true subset of complex sets.

Then the sets constructed on the basis of these sets (e.g., q(sqrt(2)), Gaussian.

number field, etc.), it must not run out of the largest set of numbers, the set of complex numbers, and it must also contain a field of rational numbers (because the set of integers is not a field).

2. The set of integers is not a field of numbers: the set of integers contains 0,1, which is closed to addition, subtraction, multiplication, but not to division. For example: 1 2=; 1,2 are integers, but the division result is not an integer. So, an integer set is not a field of numbers.

3. The set of integers smaller than the set of rational numbers is not a number field, but the set of rational numbers is a field of numbers and is a true subset of other number fields, so the field of rational numbers is the smallest number field.

That is, let x be a rational number, e.g. x=m n, m is an integer, n is a positive integer (m is the same as x), and m and n are coprime. From this, if m and n can be found, then x is a rational number; If a contradiction is introduced, x is an irrational number.

For example, the irrationality of 2 is proved in this way (squared on both sides,......found that m and n have common factors again), and we will all be.

For another example, like the irrationality of this number, we make its loop have n bits to derive contradictions.

However, this approach is not a panacea. For example, the irrationality of and e (the base of the natural logarithm) is almost impossible to prove in the above way. Therefore, their irrationality has its own proof.

For example, whether the number e (to the power of e) is an irrational number has been debated for a long time. It's because there's no way to prove or deny it.

In fact, there are far more irrational numbers than imagined, and there is no general proof to prove or deny whether all numbers are rational or not.

First of all, there must be a non-0 element s in the number fieldBy subtracting and dividing the pairs, we get x-x=0

and x x = 1 in the number field.

So 0,1 must be in the number field.

Since the number field is closed to addition.

So 1+1=2

All positive integers are in the number field.

It is then closed by pair subtraction, so 0-n=-n are all in the number field.

This gives all the integers in the number field.

Then by closing the division and dividing between the integers, all rational numbers can be obtained in the number field.

Therefore, a number field should contain at least rational numbers.

Introduction: An integer can also be seen as a fraction with a denominator of one. Real numbers that are not rational numbers are called irrational numbers, i.e., the fractional part of an irrational number is an infinite number that is not cyclical. It is one of the important contents in the field of "number and algebra", which has a wide range of applications in real life, and is the basis for continuing to learn real numbers, algebraic formulas, equations, inequalities, Cartesian coordinate systems, functions, statistics and other mathematical content and related subject knowledge.

The set of rational numbers can be represented by the uppercase black orthography symbol q. But q does not denote a rational number, and a set of rational numbers and a rational number are two different concepts. A rational number set is a set of elements that are all rational numbers, while a rational number is a set of all the elements in a rational number set.

It is enough to prove that any range of numbers contains a field of rational numbers.

Proof: Let any number field k, and know 0 1 k from the concept of number field.

So there are 1+1=2, 1+2=3, 1+3=4,..

Therefore, z k is known from the concept of number fields and rational numbers.

If the result of any two operations on any two numbers in the set p is still in p, then the set p is closed to the operation.

Definition of equivalence of number field: If a set of numbers p containing 0,1 is closed for addition, subtraction, multiplication and division (the divisor is not 0), then the set p is said to be a field of numbers.

Since 0 and 1 are in the field of numbers, by the closure of addition and subtraction, it is known that any a z is in the field of numbers, and by the closure of division, b|a is also in the number field, b, a z. It is known that any number field contains a field of rational numbers.

The smallest number field is a=

The smallest number field containing 5 is.

Since 0 and 1 are in the field of numbers, by the closure of addition and subtraction, it is known that any a z is in the field of numbers, and by the closure of division, b|a is also in the number field, b, a z. It is known that any number field contains a field of rational numbers.

Since 0 and 1 are in the field of numbers, by the closure of addition and subtraction, it is known that any a z is in the field of numbers, and by the closure of division, b|a is also in the number field, b, a z. It is known that any number field contains a field of rational numbers.

The smallest bumper number domain is the unreasonable laughter and noisy number domain. （）

a.That's right. b.Mistake.

Correct Answer: Happens to B

The so-called number field needs to meet two conditions, 1Contains 0 and 1; 2.For the four operations, the closure is satisfied (the divisor is not 0).

Since 1 belongs to the number domain, it can be seen from the addition closure that any positive integer n also belongs to the number domain, and because 0 belongs to the number domain, it can be seen from the subtraction closure that any negative integer -n=0-n also belongs to the number domain, so any integer belongs to the number domain, and then according to the division closure, we can know that the ratio of any two integers also belongs to the number domain, so any rational number belongs to the number domain. Therefore, a rational number field is the smallest number field, and any number field contains it.

Whoever says that, [2, 2] does not include the field of rational numbers.

Since 0 and 1 are in the field of numbers, by the closure of addition and subtraction, it is known that any a z is in the field of numbers, and by the closure of division, b|a is also in the number field, b, a z. It is known that any number field contains a field of rational numbers.