📂Lemmas

Geometric Mean

Definition

The value for two positive numbers $a, b$ is called the geometric mean of $a$ and $b$.

$$\sqrt{ab}$$

Generalization

For $n$ positive numbers $a_{1}, \dots, a_{n}$, the following value is called the geometric mean of $a_{1}, \dots, a_{n}$.

$$\sqrt[n]{a_{1}a_{2}\cdots a_{n}}$$

Explanation

If one considers an extension to complex numbers, then $a_{i}$ do not necessarily have to be positive.

One who encounters the geometric mean for the first time might inevitably ponder the following question:

How is taking the square root of the product of two numbers related to geometry and an average?

Why is it an average?

When we use the term average, it usually implies the arithmetic mean. The arithmetic mean of two numbers $a$ and $b$ is as follows.

$$\dfrac{a+b}{2} = c$$

In school, the term average is probably most commonly used during exam periods. For instance, saying the average score is 87 points implies that one scored 87 in every subject. This is the crux of the concept of average—the total of all subject scores is equivalent to adding 87 the same number of times. The equation above can be expressed as follows.

$$a + b = 2c = c + c$$

What this expresses is, “The average of the sum of two numbers $(a, b)$ is the number $(c)$ that must be added twice to equal the sum of the numbers $(a+b)$.” Applying this directly to multiplication gives the following.

If a number, when multiplied by itself, equals the product of two other numbers, then that number is called the average of the two numbers.

$$ab = cc \implies c = \sqrt{ab}\ \text{ is (geometric) mean.}$$

Why is it geometric?

Simply put, multiplication is related to area. Consider the two numbers $a$ and $b$ as not merely numbers but as the lengths of the sides of a rectangle. If the geometric mean is $c = \sqrt{ab}$, then the following holds.

$$ab = cc = c^{2}$$

Therefore, the geometric mean of $a$ and $b$ is “the length of one side of a square having the same area as a rectangle with side lengths $a$ and $b$.” Historically, measuring or estimating land area was a significant issue (especially related to taxes). It would have been more intuitive to think of squares rather than rectangles when estimating size, leading to the natural use of the concept of geometric mean, hence the name. Just as one can gauge a house’s size by asking, “How many pyeong is it?” regardless of the structure, asking, “What is the geometric mean?” makes it easy to understand the area, regardless of how the field or farm is shaped.