How to prove that any point on the number axis is either rational or irrational If there is any reference, it would be better

How to prove that any point on the number axis is either rational or irrational If there is any reference, it would be better

Yes, real number set is the union of rational number set and irrational number set, but this "definition" can't solve the problem
The real definition of real number set R is: all the limit points of all convergent rational number sequences
The set of irrational numbers is defined: real numbers that are not rational numbers are called irrational numbers
So the question is: why do the points on the number axis correspond to the real numbers one by one?
When the origin o on the number axis is determined, for any point m on the right side of O, the length of line segment OM is the coordinate X of M. since the length of line segment om | om | (whether it is rational or not) can be infinitely approximated by a series of rational numbers (R n), that is, x = LIM (R n), it is known that x is a real number by definition. Therefore, any point m on the right side of O corresponds to a real number X
In the same way, it can be shown that any point m to the left of O also corresponds to a real number
On the other hand, given a real number x, if x > 0, then it corresponds to a point m with distance x to the right of O. if x

It is known that the equation (AX + 1) / (x-1) - 1 = 0 has no solution, and the value of a is obtained
Two solutions are required, and the solution when x = 1 has been made,

（ax+1）/(x-1)-1=0,
（ax+1）/(x-1)=1,
（a-1）x=-2,
When a = 1, the equation has no solution

What's the difference between number of and the number of? And who follows the singular predicate and the plural predicate?

A number of is a large number of equal to a lot of, followed by the plural noun
The number of is followed by the plural
For example: a number of apples are red
Many apples are red
The number of students is 2000
The number of students is 2000

As shown in the figure, in △ ABC, D is on AC, point E is on BC, and de ‖ AB, rotate △ CDE clockwise around point C to get △ cd'e '(make ∠ BCE' < 180 °), and connect ad ', be'. Let the line be 'and AC, ad' intersect at points o and f respectively
(1) Ask if there is an internal angle in △ Dec equal to ∠ AFO? If so, please write and prove it. If not, please explain the reason
(2) If ∠ ACB = 60 ° in △ ABC, AC = root 3, BC = root 2. E is the midpoint of BC, is there a maximum area of △ ACD '

(1) the proof of the "AFO = - ACB: because of the fact that de ‖ AB (CE / BC) = (CD / AC) because △ CDE is similar to △ CD \35; 39; e # 39; e # 39;; (CE \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8743; 2 ∧ 4 =

Find the synonyms of some English phrases or words,
Find some synonyms of English phrases or words, that is, when writing a composition, which of the following common words or phrases can be used instead
1.people
2.because
3.first,second...
4.more and more
5.After all,in the end
6.become
7.in the future
(of course, if anyone puts forward more suggestions, I will be rewarded.)

Individuals, characters, folds replace (people, persons) 2: position, favorable, Rose (beautiful), promising (promising), perfect, pleasant, outstanding, outstanding, superior replace good

（-90）+(-89）+（-88）+… +（+99）+（+100）=?

It can be found that many of them are 0 in pairs
For example, 90 and - 90
-89 and 89
So the original formula is 0 + 0 + 0 + 0. + 91 + 92 + 93 +.. 100 = 955

Give out give away hand out
Which friend can explain the difference between give out, give away and hand out for me. I only know that they all have the meaning of distribution. I don't know their usage. It's better to give examples

The teacher played out the examination papers. I have to make some copies to hand out

When the numerator and denominator of 5 out of 9 are added at the same time, the result is equal to 2 out of 3

Let the number added to the solution be x (5 + x) / 9 + x = 2 / 3, which can be regarded as a proportional solution. 3 × (5 + x) = 2 × (9 + x) 15 + 3x = 18 + 2x 3x-2x = 18-15, and the number added to X = 3 is 3

Is one apple and a half plural or singular?

It's plural

Is y a function of X in the following formula? Why
Y = 3x-5, y = X-2 / X-1, y = X-1 under radical

Where y varies with X, X is the independent variable, and K is the proportional coefficient (k is not equal to 0)
Y = 3x-5, is a linear function of the form y = KX + B
Y = X-2 / X-1 is a quadratic function
X-1 is a power function under y = root