Question

prove that
$4 - 2 \sqrt{5}$
is an irrational number​

Answer 1

Answer:

What I will actually do (because it's easier) is to show that it can't be rational.

Suppose it were true that  4+25–√=p  for some rational  p .

Then, rearranging a little,

5–√=p−42  

The difference of two rational numbers is rational, and so is the quotient of two rational numbers (as long as we're not dividing by  0 , and we aren't). The right-hand side is therefore rational, so the left-hand side must be rational.

All right,  5–√=rs  

for integers  r,s  (and  s  isn't zero: in this case we can see that  r  isn't zero either).

Without loss of generality, we can assume that  rs  is in its lowest terms — that is, the g.c.d.  (r,s)=1 .

Multiply by  s  and square:

5s2=r2

The left-hand side divides by  5 , so the right-hand side must. However, if  5  doesn't divide r , then it doesn't divide  r2  either (because  5  is a prime number). So in fact  5  divides  r , and  52  divides  r2 . Now we have

5s2=r2=52t2  for  5t=r .

We can take out the common factor  5  (dividing by  5 , because it's not  0 ).

s2=5t2  

Exactly the same argument as before, for  r  and  s , now applies to  s  and  t . This time,  5  must divide  s . However, that means that  5  divides both  r  and  s , and the greatest common divisor  (r,s)  must be a multiple of  5 , contradicting our earlier requirement that it be  1 .

Step-by-step explanation: