Naihati Institute of Science & Culture: The Golden Ratio

the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

or, alternatively,

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

By definition, the first two numbers in the Fibonacci sequence are 0 and 1 (alternatively, 1 and 1), and each subsequent number is the sum of the previous two. The relationship of this sequence to the Golden Ratio lies not in the actual numbers of the sequence, but in the ratio of the consecutive numbers. Let's look at some of these ratios:

2 / 1 = 2.0

3 / 2 = 1.5

5 / 3 = 1.67

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1.615

34 / 21 = 1.619

55 / 34 = 1.618

89 / 55 = 1.618

144 / 89 = 1.618

Notice that I have rounded my ratios to the third decimal place. If we examine 55/34 and 89/55 more closely, we will see that their decimal values are actually not the same. But what do you think will happen if we continue to look at the ratios as the numbers in the sequence get larger and larger? That's right: the ratio will eventually become the same number, and that number is the Golden Ratio! The Golden Ratio is what we call an irrational number: it has an infinite number of decimal places and it never repeats itself!

we can say, 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.

The Golden Ratio is denoted by φ (phi) & its value is 1.618 approximately. It has an infinite number of decimal places and it never repeats itself.