Named Inequalities

Stage 22 of 23 Strand 3 of 4 4 lessons

4 illustrated lessons, each teaching the why before the how.

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The Triangle Inequality #

A detour is never shorter than going straight.

Going from one point to another by way of a third is never shorter than going straight there

Walk straight from one corner to another along c, or go around by way of a and b.

The detour is never shorter, so a + b is at least c for any triangle at all.

Sides 5, 9 and 15 do not form a triangle: 5 + 9 = 14, which is less than 15.

Written with modulus bars, the same inequality covers numbers, vectors and complex numbers.

Now you

Do these three lengths close into a triangle?

Two sides of a triangle are 8 and 11. What is the largest whole number the third side could be?

The average never falls below the root product.

For two nonnegative numbers the arithmetic mean is never below the geometric mean, with equality only when they agree

Take 2 and 8. Their arithmetic mean is 5, and their geometric mean, the square root of their product, is 4.

The whole proof is one square: a square is never negative, so expand it and rearrange.

Halving both sides gives the usual form: the arithmetic mean is at least the geometric mean.

Equality needs that square to be 0, which happens only when a and b are equal.

Now you

Take a = 49 and b = 1. Which is the arithmetic mean?

Take a = 4 and b = 1. Which is the arithmetic mean?

A dot product cannot beat the two lengths.

A dot product can never exceed the product of the two lengths, because the cosine of the angle between them is at most 1

Two vectors meet at an angle, and the dot product is |a| |b| cos θ.

The dot product is the projection of b onto a, multiplied by the length of a.

A cosine never exceeds 1, so the dot product never exceeds the product of the lengths.

Equality needs |cos θ| = 1, so the two vectors have to be parallel.

Now you

Two vectors have lengths 2 and 3. What is the largest possible size of their dot product?

Two vectors have lengths 5 and 4. What is the largest possible size of their dot product?

Which of three families wins in the end.

In the long run an exponential outgrows every polynomial and every polynomial outgrows every logarithm

At n = 4 both n² and 2ⁿ equal 16. By n = 16, n² is 256 and 2ⁿ is 65536.

The three families rank in one order, and no choice of constants changes that order.

Even n¹⁰⁰ is overtaken by 2ⁿ in the end. A large power delays the crossing point but never removes it.

Written as limits, each ratio tends to 0, which is what the ranking means.

Now you

Which of these grows fastest as n gets large?

Why does raising the power in nᵏ not save it against 2ⁿ?