Welcome to Mr. Narain's Golden Ratio Page!

This page is meant to be a basic introduction to one of the most amazing discoveries in mathematics: the Golden Ratio. Please use the navigation bar on the left at any time to take you where you want to go. You can always come back to this page by clicking on the "Home" button. Please read the Introduction below and then go to the Activities page. The activities are meant to be done in sequence. After you complete all of them, you should go to the Assessment page to find out how much you have learned. Finally, you may use the Feedback button to communicate with Mr. Narain and tell him what you liked/disliked about this website. If you are a teacher, please visit the Teacher's Page to learn more about instruction for this website. Finally, the Links page will take you to a list of other Golden Ratio websites, many of them far more involving than this one.

Introduction

What is the Golden Ratio?

Well, before we answer that question let's examine an interesting sequence (or list) of numbers. We'll start with the numbers 1 and 1. To get the next number we add the previous two numbers together. So now our sequence becomes 1, 1, 2. The next number will be 3. What do you think the next number in the sequence will be? If you said 4, then unfortunately you are incorrect. Remember, we add the previous two numbers to get the next. So the next number should be 2+3, or 5. Here is what our sequence should look like if we continue on in this fashion for a while:

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

Now, I know what you might be thinking: "What does this have to do with the Golden Ratio?" The answer is forthcoming. This sequence of numbers was first discovered by a man named Leonardo Fibonacci, and hence is known as Fibonacci's sequence. 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

Aha! Notice that as we continue down the sequence, the ratios seem to be converging upon one number (from both sides of the number)! 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! Generally, we round the Golden Ratio to 1.618. We work with another important irrational number in Geometry: pi, which is approximately 3.14. Since we don't want to make the Golden Ratio feel left out, we will give it its own Greek letter: phi. One more interesting thing about Phi is its reciprocal. If you take the ratio of any number in the Fibonacci sequence to the next number (this is the reverse of what we did before), the ratio will approach the approximation 0.618. This is the reciprocal of Phi: 1 / 1.618 = 0.618. It is highly unusual for the decimal integers of a number and its reciprocal to be exactly the same. In fact, I cannot name another number that has this property! This only adds to the mystique of the Golden Ratio and leads us to ask: What makes it so special?

You are now ready to go to and complete the .