Determine which real numbers exceed their reciprocals by exactly 1 .

Let n represent one such number. Then:

n - 1/n = 1 (the difference between the number n and its reciprocal is 1).

Solve for n: Multiply all three terms by n, to remove fractions:

n^2 - 1 = n, or n^2 - n - 1 = 0

Solving this for n:

-(-1) plus or minus √(1-4[1][-1])

n =

2

1 plus or minus √5

=

2

or

x = (1+√5)/2 or x = (1-√5)/2