I'm having trouble understanding something in Hardy and Wright's 'An Introduction to the Theory of Numbers'.
The work in question can be viewed here on Google books.
I don't understand the part beginning with 'If we divide 1 by x...'.
As best as I can figure out, we are to divide 1 by x, and then subtract 1 (that is what I think 'taking the largest possible integral quotient' means). Since x is between 1/2 and 1, 1 divided by x must be between (but not including) 1 and 2; so subtracting 1 from 1/x would leave us with just the remainder (a proper fraction) of 1/x. But 1/x - 1 = x, right? But the book says the result is 1 - x = x^2, which is not equal to x.
Any help would be great.