Real number theory (1) deduction of the basic properties of proportional numbers

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Question 1: Why A+b=b+a
Question 2.1/3 Why is a positive number, 1/3 why greater than 0
Question 3: Why a third proportion must exist between any two scale numbers
Question 4: How to prove the Archimedes nature of the proportional number
Question 5: Why the square root 2 is not a proportional number
Question 6: Why do you define Cauchy columns

From the simplest natural number, we know that the natural number for addition and multiplication is closed, by introducing the concept of subtraction in the natural number, you can construct an integer set, the integer set is not only for addition, multiplication operation closed, but also for subtraction is closed, by the integer set introduced division operation, You can construct a new set, a scale number, or a rational number.
In other words, by two operations, subtraction and unless, it is possible to construct integers and proportional numbers. However, for real collections, it is not possible to do so, since the scale is over to the real number, which is a discrete to continuous process. This requires a heart operation, a limit.
Once the real number is constructed, it is immediately proved that the most basic theorem, the exact boundary principle. With him, the whole calculus has a theoretical basis.
Of course you need to be aware of all the properties of the scale, and the premise here is that you have an understanding of all the basic properties of the whole number.

Why A+b=b+a
In life, if you ask others why 1/3+1/2=1/2+1/3. Someone will say two words to you: stupid.
Is dripping, these things are too intuitive, intuitive to never need to doubt his correctness. Then the intuition is not always correct.
In fact, to answer this definition, we need to give a strict definition of the proportional number. Of course, the strict definition of the proportional number must use some of the properties of integers, of course, we recognize some of the properties of integers: such as the addition of Exchange law, binding law.
After giving the definition of the proportional number, we need to give the definition of the operation of the proportional number, and then we can use the property of the integer to prove the operation nature of the proportional number.

Definition: Number of Scales
A//b is a proportional number, c//d is also a proportional number, a,b,c,d is an integer, a//b=c//d when and only if a*d=b*c.
The definition of the scale number here only uses the multiplication of integers and integers. Of course it is necessary to prove that this definition is successful, for example, you need to prove that he satisfies reflexive, symmetrical, transitive.
A proof of symmetry is given below
Proposition: if a//b=c//d so c//d=a//b
Prove:
A//B=C//D =>a*d=b*c (scale number definition)
=>d*a=c*b (integer multiplication interchange)
=>c*b=d*a (symmetry of integer equality)
=>c//d=a//b (scale number definition)

Proportional number addition, multiplication, definition of negative operation
Addition definition: a//b+c//d= (a*d+c*b)//b*d

Proposition: a//b+c//d=c//d+a//b
Prove
A//b+c//d=> (a*d+c*b)//b*d
C//d+a//b=> (Cb+ad)//db
Only need to prove (a*d+c*b)//b*d= (Cb+ad)//db.
Only need to prove (AD+CB) *db= (BD) * (CB+AD)
Just need to prove ad*db+cb*db=bd*cb+bd*ad
The commutative law of an integer multiplication and the Exchange law of addition are established above the equation.

It is interesting to prove this proposition by some of the simplest properties. Other algorithms can be proven in a similar way.

Proposition: Any proportional number C, exists only an integer N makes n<=c<n+1.
For any number of proportional cases m, there are positive natural numbers a and B, which makes m=a/b. According to the theorem of the divide division of the natural number, we know that there is only Q and r,0<=r<b that make a=bq*+r.
A/b=q+r/b
where 0<=r/b<1 (why)
Q<=q+r/b<q+1
For any negative proportional number m, a similar proof can be used.

Proposition: Any two positive proportional number c,d, there is a positive integer n, making C<nd
C/D are proportional to the number of cases, and there is a positive integer M that makes c/d<=m<m+1. Another n=m+1. Get c/d<n.
Multiply by D on both sides to get c<dn.


Definition: Proportional Case number
A proportional number c is positive when and only if he can write a ratio of two positive integers. A proportional number c is negative when and only if there is a positive proportional number z, which makes the c=-z;

Proof 1/3 is proportional to the number of cases
1/3 is positive, because there are two positive integers 1 and 3, which makes the proportional number 1/3=1/3.

Proof-1/3 is negative.
-1/3 is negative, because a positive ratio of 1/3, makes -1/3=-(1/3)


Proposition: A is a proportional number, then the following three have just one set up
1) A is 0
2) A is proportional to the number of cases
3) A is a negative proportional number
We need to prove that a can only be 1), 2), 3), but also to prove that a can not meet any of the above two.

Certificate
Suppose a=m/n, where M,n is an integer. By the three-qi of integers know that m,n is 0, or positive, or negative.
Now classified discussion.
1. M,n. If the m,n is in the same position, a is proportional to the number of cases, if the m,n is negative, a is a negative scale. Because there is a proportional example number z=-m/-n, so a=-z.
2. M,n. M is a positive number, n is negative, m/n is negative, because there is a proportional case number z=m/-n make, a=-z. The same is true if M is negative and N is a positive number, A is also a negative scale.
3. M is 0 o'clock, a is 0.
We have proved that no matter what the value of M,n (except when n=0), m/n can only be 0, or negative scale number, or proportional to the number of cases, it is not possible to be other additional things.
The following proves that m/n cannot satisfy 1), 2), 3) any two.
1. If m/n is 0, then M must be 0. At this time m/n cannot be 2), 3).
2. If the m/n is a proportional example, there is a proportional example number z=x/y, where x is positive and y is a positive number. If the m/n is also negative, there is a proportional example of r=e/f. A=-r. At this time-e/f=x/y. Because the definition of the scale number is known,-EY=FX. Y,f,x are positive. We have introduced a contradiction, because by the nature of positive numbers, a number cannot be both positive and negative.

It's over. This seemingly very, very simple proposition is proving to be a strength.

For convenience, we call proportional examples a positive number, and negative proportions are called negative numbers.

The definition of positive and negative numbers helps us define a more useful thing, preface!

Definition of the order:
A, B is a proportional number, a>b when and only if a-B is positive, a<b, and only if the-a is negative, a=b when and only if the-B is 0.

The definition order needs to use positive and negative numbers, but also the concept of subtraction, well, without them, a>b this kind of thing can not be defined.
(Of course, a>=b,a<=b is not yet defined, but it is easy to define the definition, for example, that A>=b is actually saying A-B is non-negative.)

There are several very, very important properties about the order
1. Any two scale number A, B, just a>b,a=b,a<b.
2.a>b equivalent to B<a
3.a>b,b>c, then A>c
3.a,b,c is the proportional number, if a>b, then A+c>b+c
4.a,b,c is the proportional number, and a>b>0, if c>0, then ac>bc>0

Proof 1.
A-B is either positive or negative, or 0, and just one is set up, so A>b,a<b,a=b has a set up.
Proof 2.
A>b =>a-b is positive. (Definition of the order)
=>-(A-B) is negative (integer negative definition)
=>-a+b is negative (the allocation rate of positive multiplication)
=>b-a is negative (the exchange rate of positive addition)
=>b<a (definition of the order)
Proof 3.
A>b = A-B is positive
B>c = B-c is positive

A-b+b-c is positive (why, why is a proportional number plus a proportional example number is positive)
A-c is positive.
A>c

Attach the previous proposition: A, b are positive numbers, and a+b are positive
Because A, B is proportional to the number of cases, there is a positive x,y,m,n make a=x/y,b=m/n.
a+b= (xn+my)/yn. Xn,my,xn+my,yn are all positive integers, so a+b is a positive proportional number.

Proof 4.
A>b=>a-b is positive.
=>a-b+0
=>a-b+ (C-C) is positive (because any integer c, there is c=c+0.)
=>a+c-b-c is positive.
=>a+c-(B+C) is positive.
=>a+c>b+c
A/b if the same number is positive, then A/b is proportional to the number of cases. If AB is negative, then a/b=-a/-B, where-A is positive, and-B is positive, so A/b is positive.
b, a similar can prove that A/b is negative.

Proof 5.
A>b>0=>a-b is positive.
= = (a) *c is positive (c is positive)
=>a*c-b*c is positive.
=>a*c>b*c>0


Proposition 1/2>1/3
Prove:
1/2-1/3= (1*3-1*2)/(2*3)
=1/6
And 1/6 is positive, so 1/2>1/3

Defines the algebraic operation of the scale number, also defines the order, if the absolute value is not defined is a pity. The absolute value is a very very wonderful thing, in all the definitions of the scale number, I like the absolute value of this thing, his wonderful behind can be seen.

Defining absolute values

Definition: |a|=a, when and only if a is a positive number. | A|=-a, when and only if a is negative, |a|=0, if and only if a=0. We'll call |a| the absolute value of a.

Set A,B,C is the proportional number, then the absolute value has the following properties
1) |a|>=0
2) If |a|<=b, then-b<=a<=b
3) |a+b|<=|a|+|b|,
4) |ab|=|a|*|b|

Proof 2)
Category discussion A.
If a is proportional to the number of cases, then |a|<=b contains A<=b, because A>0, and b>=a>0, so b!=0,b is proportional to the number of cases,-B is a negative scale.
So there are-b<a<b.

If a is negative proportional number, then |a|<=b contains-a<=b, similarly also can get-b<-a<b,-b<a<b.
If a is 0, it is also established.

Can prove that if B equals 0, then A is also equal to 0.

Proof 3.
Ideas to prove |a+b|<=|a|+|b|, according to Nature 2) only need proof-(|a|+|b|) <=a+b<=|a|+|b|
-|a|<=a<=|a| (why)
-|b|<=b<=|b| (why;)
-(|a|+|b|) <=a+b<=|a|+|b|

Auxiliary proposition: A is the proportional number, has-|a|<=a<=|a|
When a is proportional to the number of cases, the |a| is proportional to the number of cases and |a|=a,-|a|=-a. At this time-a<=a<a established, so-|a|<=a<=|a| set up
A is 0, established.
A is a negative proportional number, |a|=-a,-|a|=-(-a) =a. There are a<=a<=-a at this time. So-|a|<=a<=|a| was established.


If you define an absolute value, it is a pity to not define the concept of distance. The concept of distance is too important in the theory of real numbers.
Define distance : The distance between A and B is recorded as P (A, B) =|a-b|.
By the definition of absolute value can easily get the nature
1.P (A, B) >=0, when and only when a=b, the equals sign is established
2.P (A, B) =p (b,a)
3.P (A,c) <=p (A, B) +p (B,C)

Real number theory (1) deduction of the basic properties of proportional numbers

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