Verification: Root 2 is an irrational number (please prove it by using the opposite method or the reverse negative proposition)

Verification: Root 2 is an irrational number (please prove it by using the opposite method or the reverse negative proposition)

Pythagoras proved it
Suppose that the diagonal of a square with side length 1 can be written as the ratio of an integer to an integer (P: q) and PQ has no common divisor (when q = 1, P: q is an integer)
Pythagorean theorem: (P / Q) ^ 2 = 1 ^ 2 + 1 ^ 2
That is, P ^ 2 = 2q ^ 2
Because 2q ^ 2 is even, that is, P ^ 2 is even, so p is even (the square of any odd number is also odd)
Since PQ has no common divisor, q is odd and P is even. Let P = 2A, P ^ 2 = 4A ^ 2 = 2p ^ 2
Q^2=2a^2
That is, Q ^ 2 is an even number, q is an even number, q is an odd number, so it cannot be expressed by integer and integer ratio, so it is irrational
Write dead!

The monotonicity of function f = x + 1 is discussed

f'(x)=1-1/x^2
When x = 1, f '(x) = 0 has a maximum of 2
When x = - 1, f '(x) = 0 has a minimum of - 2
-1

Time addition and subtraction
For example, it's 17:00 on the 23rd to 5:00 on the 24th. How should it be calculated?

(24-17)+5
=7+5
=12
First, use the 24-hour system to calculate how many hours there are on the 23rd, plus the five hours on the 24th

Y = (1-x ^ 2) / (1 + x ^ 2) range derivative method!) the answer is (- 1,1]
Let f (x) = 1-x ^ 2, G (x) = 1 + x ^ 2, f '(x) = - 2x, G' (x) = 2x
y=f(x)/g(x)=(f'g-fg')/g^2=(-2x*(1+x^2)-(1-x^2)*2x)/(1+x^2)^2=(-4x)/(1+x^2)^2

You can use the derivative to get the maximum of this problem, according to the derivative you get
(1 + x ^ 2) ^ 2 is always greater than 0
So when you think about X

300 oral arithmetic questions in grade 3 with answers
Summer vacation is coming to an end, it must be 300,

640÷80= 815×5=75 23×3= 6912×2×5=120 480÷80= 616×5= 8027×3= 8190÷15= 648÷4= 12640÷16= 4039÷3= 1324×20= 48032×3= 9648÷16= 312×8= 9627×3= 8156÷14= 424÷8= 314×2= 2883－45= 38560÷80= 7...

If the solutions of the system x + 2Y = 6 and X-Y = 9-3a are a pair of identical numbers, what is the value of a?
have to do urgently

That is, x = y
Substituting into the second equation, 0 = 9-3a, a = 3, then x = y = 2

There are nine boxes in the following formula. Now fill in the nine numbers 1 to 9 in the following box to make the equation hold

6x29 = 174 = 58x3

The center of the hyperbola is at the origin, the distance between the vertices is 8, the asymptote equation is y = ± 3 / 2x, and the standard equation and focus coordinates of the hyperbola are obtained

If the focus is on the x-axis: the equation is X & # 178 / 16-y & # 178 / 36 = 1, the focus coordinates are (- 2 √ 13,0), (2 √ 13,0)
If the focus is on the y-axis, the equation is Y & # 178 / 16-x & # 178; / (64 / 9) = 1, and the focus coordinates are (0), (4 / 3) √ 13), (0, (4 / 3) √ 13)

A famous math teacher in the sixth grade of primary school
There are 5 buses and 10 minibuses, which are just full of 550 people. Each bus carries 20 more passengers than the minibus. How many passengers does each bus and minibus carry?

If each passenger car carries x people, the bus carries (x + 20) people
10X+5（X+20）=550
10X+5X+100=550
15x+100=550
15X=450
X=30
X+20=50
Answer: each small passenger carries 30 people, then the bus carries 50 people

In the quadratic function y = AX2 + BX + C, how to judge the sign of 2a-b according to the image?

Compare with - B / 2a and - 1 of parabola