Understanding Phi Function and Multiplicativity

Thread starter hanelliot
Start date Apr 29, 2009
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Phi

In summary, Phi, also known as the golden ratio, is a mathematical constant that appears in many natural and man-made structures. It is often associated with beauty, balance, and perfection. To calculate phi of a large number, you can use the formula phi = (1 + √5) / 2, which becomes more accurate as the number gets larger. Phi can be found in many natural phenomena and man-made designs, and its value remains constant at approximately 1.6180339887... It has unique mathematical properties and is often used in the Fibonacci sequence, golden triangle, and golden rectangle, making it an important tool in mathematics and design.

Apr 29, 2009

hanelliot

phi(20^100)
= phi(4^100 * 5^100)
= phi(2^200 * 5^100)
= (2^200 - 2^199)(5^100 - 5^99)
= 2^199(2-1) * 5^99(5-1)
= 2^199 * 5^99 * 4
= 2^201 * 5^99.

I don't understand line 4-7. Can anyone explain?

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Apr 29, 2009

CRGreathouse

Science Advisor

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phi(p^a) = (p-1)(p^(a-1)) for p prime

Apr 30, 2009

matticus

also since phi is multiplicative phi(a*b) = phi(a)*phi(b) when gcd(a,b)=1

What is the significance of "phi" in a large number?

Phi, also known as the golden ratio, is a mathematical constant that appears in many natural and man-made structures. It is often associated with beauty, balance, and perfection.

How do you calculate phi of a large number?

To calculate phi of a large number, you can use the formula phi = (1 + √5) / 2. This formula can be used for any positive integer, but it becomes more accurate as the number gets larger.

What are some real-life examples of phi in large numbers?

Phi can be found in many natural phenomena such as the spiral patterns of shells, the branching of trees, and the proportions of the human body. It is also seen in man-made designs, including architecture, art, and music.

Is phi of a large number always the same?

Yes, phi is a constant and its value does not change. It is approximately equal to 1.6180339887...

Why is phi of a large number important in mathematics?

Phi has many interesting and unique mathematical properties. It can be found in the Fibonacci sequence, in which each number is the sum of the two preceding numbers. It also has connections to the golden triangle and the golden rectangle. Its relationship to these structures makes it a valuable tool in mathematics and design.