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AM-GM Inequality Calculator

Online AM-GM verifier: compare the arithmetic and geometric means of two positive numbers and check the equality condition.

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(a+b)/2≥√(ab), equality a=b
Positive
Positive

📖 Tutorial | AM-GM

1. Definition
For positive a, b, the arithmetic mean is at least the geometric mean: (a+b)/2≥√(ab), with equality iff a=b.
2. Formula & Notation
AM=(a+b)/2Arithmetic mean
GM=√abGeometric mean
3. Properties & Laws
  • AM≥GM, equality a=b;
  • Extends to n positives;
  • Fixed sum → max product at equality;
  • ab≤((a+b)/2)².
4. Steps
  1. Enter two positives;
  2. Click "Verify";
  3. Or click "Load Example".
5. Worked Examples
Example: 3, 5: AM=4, GM=√15≈3.873.
6. Common Pitfalls
Must be positive;
Do not miss a=b;
Check equality is attainable.

❓ FAQ | AM-GM

Statement?
AM≥GM.
Condition?
Positive.
Equality?
a=b.
Uses?
Optimization, proofs.
Fixed sum?
Equal terms maximize product.
Three numbers?
Still holds.
GM?
√(ab).