Can you give me an example of an irrational number between #5 and 6?#

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Answer 1 · 2018-04-24T06:14:19

#sqrt(26) = 5+1/(10+1/(10+1/(10+...))) ~~ 5.0990195#

Since:

#5^2 = 25 < 26 < 36 = 6^2#

we have:

#5 < sqrt(26) < 6#

To show that #sqrt(26)# is irrational, suppose:

#x = 5+1/(5+x)#

for some #x > 0#

Multiplying both sides by #5+x# this becomes:

#5x+x^2 = 25+5x+1#

Subtracting #5x# from both sides, and simplifying, this becomes:

#x^2 = 26#

So:

#x = sqrt(26)#

So we have shown:

#sqrt(26) = 5 + 1/(5+sqrt(26))#

#color(white)(sqrt(26)) = 5 + 1/(10+1/(10+1/(10+1/(10+...))))#

Since this continued fraction does not terminate, it is not expressible as a terminating fraction. So it is irrational.