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Each number in the Fibonacci sequence is the sum of the two previous numbers. The first two numbers are 1. If you add them together, you get 2. Then add 2 to one to get 3. 3+2=5, and so on, until you get the 8th value. It is important to remember that the first two ones are part of the sequence.1, 1, 2, 3, 5, 8, 13, 21
The "golden ratio" is the limit of the ratio between consecutive terms of the Fibonacci series. That means that when you take two consecutive terms out of your Fibonacci series and divide them, the quotient is near the golden ratio, and the longer the piece of the Fibonacci series is that you use, the nearer the quotient is. The Fibonacci series has the property that it converges quickly, so even if you only look at the quotient of, say, the 9th and 10th terms, you're already going to be darn close. The exact value of the golden ratio is [1 + sqrt(5)]/2
As you expand the Fibonacci series, each new value in proportion to the previous approaches the Golden Ratio.
In addition to popularizing the Fibonacci sequence, Leonardo Fibonacci is credited with introducing the concept of Hindu-Arabic numerals to Europe through his work "Liber Abaci." This numeral system, which includes the digits 0-9 and the concept of place value, revolutionized mathematics by replacing the less efficient Roman numeral system. His contributions laid the groundwork for modern arithmetic and algebra.
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