## Proof Exercise #2:

proving the Geomtric Mean - Arithmetic Mean Inequality for the most particular case

search term: geometric mean

search term: arithmetic mean

search term: inequality

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Definition 1. "fif" means "if, and only if,"

Definition 2. If each of x and y is a number, then the arithmetic mean of x and y is that number a such

that x + y = a + a. (In other words, the number that each original number can be swapped out with and

still have the number-list have the same sum.)

Theorem 1. If each of x and y is a number, then the arithmetic mean of x and y is (x + y)/2.

Proof: Left to the reader.

Definition 3. If each of x and y is a positive real number, then the geometric mean of x and y is that number g such

that xy = gg. (In other words, the number that each original number can be swapped out with and still have the

number-list have the same product.)

Theorem 2. If each of x and y is a positive real number, then the geometric mean of x and y is (xy)^(1/2).

Proof: Left to the reader.

Remark: The reason that we use the term "geometric mean" instead of "multiplicative mean" is because "multiplicative" is reserved to describe a certain type of function in Number Theory, and because the "geometric" quantities of greatest interest are of a multiplicative nature. For example, volume is equal to length times width times height. So, it's not so farfetched to use "geometric" to mean "multiplicative".

Remark: Much of the significance of the geometric mean is due to the fact that quantities that are of inherently positive measure, such as weight or age, are best compared multiplicatively. In particular, fund performance is measured multiplicatively. For example, if a fund start out with $100 and increases over the course of one year to

$150, we say that it has increased by 50%. Let us say that over the course of the next year the fund, which is now $150, increases by 10%. Since 10% of $150 is $15, the fund a the end of the second year is $165. Now here comes the interesting question: What constant growth rate would have resulted in this same amount of $165? The answer is the geometric mean of (1.5) and (1.1). This is not the same thing as their arithmetic mean. Is there any size relationship between the geometric mean and the arithmetic mean? Indeed there is: The geometric mean never exceeds the arithmetic mean. The following theorem is all about that.

Theorem 3. If each of x and y is a positive real number, then the geometric mean of x and y is less than or equal to the arithmetic mean of x and y.

There are various proofs of this. We will present one of them.

What, from the list below, do you think, will be our strategy?

choice #1: Take the cube root of a certain quadratic expression.

choice #2: Fully exploit the fact that every integer > 1 has a unique prime factorization.

choice #3: Work with the multiplicative reciprocals of the numbers instead of the numbers themselves.

choice #4: Fully exploit the fact that the square of a real number is never negative.

choice #5: Use the right triangle having legs equal in lengths to the two numbers.

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search term: arithmetic mean

search term: inequality

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Definition 1. "fif" means "if, and only if,"

Definition 2. If each of x and y is a number, then the arithmetic mean of x and y is that number a such

that x + y = a + a. (In other words, the number that each original number can be swapped out with and

still have the number-list have the same sum.)

Theorem 1. If each of x and y is a number, then the arithmetic mean of x and y is (x + y)/2.

Proof: Left to the reader.

Definition 3. If each of x and y is a positive real number, then the geometric mean of x and y is that number g such

that xy = gg. (In other words, the number that each original number can be swapped out with and still have the

number-list have the same product.)

Theorem 2. If each of x and y is a positive real number, then the geometric mean of x and y is (xy)^(1/2).

Proof: Left to the reader.

Remark: The reason that we use the term "geometric mean" instead of "multiplicative mean" is because "multiplicative" is reserved to describe a certain type of function in Number Theory, and because the "geometric" quantities of greatest interest are of a multiplicative nature. For example, volume is equal to length times width times height. So, it's not so farfetched to use "geometric" to mean "multiplicative".

Remark: Much of the significance of the geometric mean is due to the fact that quantities that are of inherently positive measure, such as weight or age, are best compared multiplicatively. In particular, fund performance is measured multiplicatively. For example, if a fund start out with $100 and increases over the course of one year to

$150, we say that it has increased by 50%. Let us say that over the course of the next year the fund, which is now $150, increases by 10%. Since 10% of $150 is $15, the fund a the end of the second year is $165. Now here comes the interesting question: What constant growth rate would have resulted in this same amount of $165? The answer is the geometric mean of (1.5) and (1.1). This is not the same thing as their arithmetic mean. Is there any size relationship between the geometric mean and the arithmetic mean? Indeed there is: The geometric mean never exceeds the arithmetic mean. The following theorem is all about that.

Theorem 3. If each of x and y is a positive real number, then the geometric mean of x and y is less than or equal to the arithmetic mean of x and y.

There are various proofs of this. We will present one of them.

What, from the list below, do you think, will be our strategy?

choice #1: Take the cube root of a certain quadratic expression.

choice #2: Fully exploit the fact that every integer > 1 has a unique prime factorization.

choice #3: Work with the multiplicative reciprocals of the numbers instead of the numbers themselves.

choice #4: Fully exploit the fact that the square of a real number is never negative.

choice #5: Use the right triangle having legs equal in lengths to the two numbers.

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