prove square root of 2, square root of 3 is not rational

[transgalactic]

prove that the square root of 2 and the square root of 3 is not rational?

does always the sum of two not rational numbers is a not rational number?

 

[fasteddie65]

Re: a proof question..

I don't think they are asking for the sum of the two numbers; they would state that.

It involves setting ?2 = a/b, assuming that it is rational. 2 = a^2/b^2
2b^2 = a^2

Eventually you will find that a and b are not relatively prime, but I forget the details.

 

[pka]

Re: a proof question..

does always the sum of two not rational numbers is a not rational number?

\(\displaystyle \left( {\sqrt 2 } \right) + \left( { - \sqrt 2 } \right) = 0\)

 

[transgalactic]

i know the proof 2 = a^2/b^2
i separately proved that square root of 2 and square root of 3 are irrational

how two prove that the sum of two such numbers is irrational too?

 

[Unco]

how two prove that the sum of two such numbers is irrational too?

Pay attention, Trans; Pka answered that in the post above yours...

 

[transgalactic]

you assumed that (2)^0.5 and (3)^0.5 are rational numbers.

why do you say that (6)^0.5 is irrational?

(6)^0.5=(2)^0.5 * (3)^0.5

the multiplication of two rational numbers
is a rational number

??

 

[pka]

The simple fact is: For any non-square positive integer, its square root is irrational.
Therefore the square roots of both 2 & 8 are irrational.
However, the product of the square of 2 with the square of 8 equals 4 which is rational. Moreover the square of 2 plus the negative of the square of 2 in zero which is also rational. Thus the set of irrational numbers is not closed with respect to addition nor multiplication.

However, any rational number plus any irrational equals an irrational number.
Any rational number times any irrational equals an irrational number.

 

[transgalactic]

whats the formal equation proof?

 

[transgalactic]

ok i will try to prove by contradiction:
suppose
(2)^0.5 + (3)^0.5 is a rational number
(if we multiply a rational number by a rational number we will get a rational number "h")
5+2*(2)^0.5 * (3)^0.5=h
24=h^2 -10*h +25

h^2 -10*h +1=0
by this rational roots theorem
the possible roots is +1 and -1

not one of the represent the actual roots of h^2 -10*h +1=0

what is the next step in the prove?

 

[pka]

whats the formal equation proof?

I will use Roman letters for rational numbers and Greek letters for irrationals.

Theorem. The sum of a rational and an irrational is irrational.
Proof If \(\displaystyle r + \delta = s\) then \(\displaystyle \delta = s - r\).
But \(\displaystyle s-r\) is rational so that is a contradiction.

Theorem. The product of a nonzero rational and an irrational is irrational.
Proof If \(\displaystyle r ( \delta ) = s\) then \(\displaystyle \delta = \frac{s}{r}\).
But \(\displaystyle \frac{s}{r}\) is rational so that is a contradiction.