# Is #4+sqrt7# rational?

##### 2 Answers

Therefore

#### Explanation:

The answer will be irrational.

If you use an irrational number in an operation, the answer will be irrational.

Note that

No

#### Explanation:

If

To see that

Suppose

#x = 2+1/(1+1/(1+1/(1+1/(2+x))))#

Then:

#x = 2+1/(1+1/(1+1/(1+1/(2+x))))#

#color(white)(x) = 2+1/(1+1/(1+(2+x)/(3+x)))#

#color(white)(x) = 2+1/(1+(3+x)/(5+2x))#

#color(white)(x) = 2+(5+2x)/(8+3x)#

#color(white)(x) = (21+8x)/(8+3x)#

Multiplying both ends by

#3x^2+8x = 21+8x#

Subtracting

#3x^2=21#

Hence:

#x^2 = 7#

So:

#x = sqrt(7)#

We have found:

#sqrt(7) = 2+1/(1+1/(1+1/(1+1/(2+sqrt(7)))))#

#color(white)(sqrt(7)) = 2+1/(1+1/(1+1/(1+1/(4+1/(1+1/(1+1/(1+1/(4+...))))))))#

Since this continued fraction does not terminate, it does not represent a rational number.

So