Advanced Pure Gold

[MAS1]

What positive number’s reciprocal is equal to one less than the positive number?

[MAS1]

Originally Posted by MAS1

What positive number’s reciprocal is equal to one less than the positive number?

No one wants to try this one?

[noreply]

Reciprocal is placing the number as 1/. So, let's say the positive number is x.

x = 1/x + 1
x^2 = 1 + x
x^2 - x - 1 = 0

x = (--1+/- sqrt(-1^2-4*1*-1))/2*1
x = (1 +/- sqrt(1--4))/2
x = (1 +/- sqrt(5))/2

Since x is positive, then it is (1 + sqrt(5))/2.

Checking:
Calculator says that (1 + sqrt(5))/2 is equal to 1.61803...
The reciprocal is 2/(1+sqrt(5)), which is equal to 0.61803...

Thus, I THINK that this is the answer.

[MAS1]

Originally Posted by noreply

Reciprocal is placing the number as 1/. So, let's say the positive number is x.

x = 1/x + 1
x^2 = 1 + x
x^2 - x - 1 = 0

x = (--1+/- sqrt(-1^2-4*1*-1))/2*1
x = (1 +/- sqrt(1--4))/2
x = (1 +/- sqrt(5))/2

Since x is positive, then it is (1 + sqrt(5))/2.

Checking:
Calculator says that (1 + sqrt(5))/2 is equal to 1.61803...
The reciprocal is 2/(1+sqrt(5)), which is equal to 0.61803...

Thus, I THINK that this is the answer.

Correct! This is the golden ratio also known as phi. Two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

If a > b and (a + b)/a = a/b then those ratios are equal to phi.

From Wikipedia:

"Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics."