Proof that the Area of a Triangle = bh/2
Summer Math Series: Week 1
Notes by David Kosbie
1. Area of a Triangle = bh/2
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2. Pythagorean Theorem: Euclid’s Windmill Proof
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3. Pythagorean Theorem: Chinese Proof (or perhaps the Indian mathematician Bhaskara’s)
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4. Pythagorean Theorem: President Garfield’s Trapezoid Proof
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5. The Distance Formula: derived from Pythagorean Theorem!
6. Fermat’s Last Theorem: xn + yn = zn has no positive integral solutions for n>2.
Proven recently by Andrew Wiles (omitted here for lack of room in the margin).
7. Hypotenuses in a “square root spiral” are of length sqrt 2, sqrt 3, sqrt 4, sqrt 5,…
(Inductive proof)
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8. The square root of 2 is irrational.
a. (p/q) 2 = 2 ( p2 = 2q2, then apply Fundamental Theorem of Arithmetic ( lhs has even # of prime factors, rhs has odd #, QED.
b. (p/q) 2 = 2 ( p2 = 2q2 ( p is even ( … ( q is even, QED.
9. “Nearly all” real numbers are irrational!
a. The integers are countable (as are evens, primes, powers of 10, …)
b. Integer pairs – Z2 – are countable (dovetailing!)
c. Integer triplets, etc – Z3 , Z4,… – are countable.
d. Rationals are countable.
e. Algebraics are countable.
f. Reals are not countable (diagonalization!)
g. Thus, “nearly all” reals are irrational (even non-algebraic, hence transcendental!)
Summer Math Series: Week 2
10. More on cardinality
a. From last week: the following sets are all countable (“denumerable”): Natural numbers (N), Integers (Z), Evens (E), Primes (P), Powers of 10, …, integer pairs (Z2), integer triplets, etc (Z3 , Z4,…), rationals (Q), and algebraics (A).
b. We say that |N| = |E| = |P| = |Z| = |Zk| = |Q| = |A| = א0
c. Reals (R) – actually, just the Reals in (0,1) – are not countable (diagonalization!).
d. |R| = |reals in (0,1)| = c (where c > א0)
Use y = (2x – 1) / (x – x2)
e. |R| = |R2| = |Rk| = c (shuffling!)
f. Cantor’s Theorem: |P[A]| > |A| (diagonalization! See p.277)
(the power set of any infinite set A – written as P[A] or 2A – has greater cardinality than the original set)
g. Thus, |N|= א0 < |P[N]| < |P[P[N]]| < |P[P[P[R]]]| < ….
h. Cantor’s Paradox: There is no Universal Set U = Set of All Sets (as |P[U]| > |U|)
i. |R| = c = |P[N]|
i. |P[N]| ................
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