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Math · Intermediate

Inequalities

Some of the most useful facts in mathematics are inequalities. Starting from the humble fact that squares are never negative, discover the AM–GM inequality and its geometric proof, solve fencing and averaging puzzles, find shortest paths with reflections, and meet the Cauchy–Schwarz inequality.

7 lessons~35 min0/7 complete
1

Means

From squares to AM–GM.

  1. Squares Are Never NegativeThe root of all inequalities
  2. The AM–GM InequalityAverages and products
  3. A Family of MeansHarmonic, geometric, arithmetic
2

Geometry and Vectors

Triangles, reflections and dot products.

  1. The Triangle InequalityShortcuts and reflections
  2. Cauchy–SchwarzDot products are bounded
  3. The Rearrangement InequalityBig with big, small with small
  4. Bernoulli’s InequalityGrowth beats a straight line