What is a irrational number between 7.7 and 7.9

There is of course no one true answer, but an infinite number of them, all correct. Let's go with

Answer: 7.7 + π/100

Pi divided by 100 is. 031415926 ... so this is about 7.73141526.

Another answer, perhaps looking more like what you'd call a number, is the square root of 61, about 7.81.

Answer: √61