AM–GM inequality
The arithmetic mean of positive numbers is at least their geometric mean, with equality only when they are equal. History: 19th century.
How to work it out
- 1
The statement:
- 2
The arithmetic mean of positive numbers is at least their geometric mean, with equality only when they are equal.
- 3
Below, the numbers in the table are computed by this site from the definition, not copied.
LaTeX source of the formulas on this page
\[ \frac{a+b}{2} \ge \sqrt{ab} \]Pairs (computed)
| a, b | Arithmetic mean | Geometric mean |
|---|---|---|
| 1, 9 | 5 | 3 |
| 2, 8 | 5 | 4 |
| 4, 4 | 4 | 4 |
| 3, 12 | 7.5 | 6 |
| 1, 100 | 50.5 | 10 |
Questions
What is the am–gm inequality?
The arithmetic mean of positive numbers is at least their geometric mean, with equality only when they are equal.
Who is the AM–GM inequality associated with?
Augustin-Louis Cauchy.