Summer Math Series: Week 1
Notes by David Kosbie
1. Area of a Triangle = bh/2
2. Pythagorean Theorem: Euclid’s Windmill Proof
3. Pythagorean Theorem: Chinese Proof (or perhaps the Indian mathematician Bhaskara’s)
4. Pythagorean Theorem: President Garfield’s Trapezoid Proof
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)
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]| <= |R|
Given a subset of integers, construct a real number by placing a 3 in the
ith decimal digit if the number i appears in the subset, otherwise insert a
7. The result is a unique real in (0,1).
ii. |R| <= |P[N]|
First, map the reals into (0,1), as above. Then, given a real in (0,1), write
the number in binary and include the integer i in the corresponding
subset iff the ith digit is 1.
j.
Cantor’s Continuum Hypothesis: No set S exists where ‫א‬0 < |S| < c
i. Godel proved the continuum hypothesis cannot be disproved!
ii. Cohen proved the continuum hypothesis cannot be proved!
iii. In this sense, it is like Euclid’s Parallel Postulate
k. We define ‫א‬1 as the smallest cardinality greater than ‫א‬0.
l.
So we know |R| = c > ‫א‬0, but: Does |R| = c = ‫א‬1? This is undecidable!
11. Derive the Quadratic Formula by Completing the Square (coming soon: cubics, quartics!)
12. The Locker Problem (coming soon: Fundamental Thm of Arithmetic, number theory)
13. Prove the Binomial Theorem (by induction!) (coming soon: more number theory!)
a. Hint: Using construction method of Pascal’s Triangle, find recursive defn of nCk
b. Application: prove: nC0 + nC1 + … + nCn= 2n
Summer Math Series: Week 3
14. Pascal’s Triangle and Pascal’s Binomial Theorem
a. nCk = kth value in nth row of Pascal’s Triangle! (Proof by induction)
b. Rows of Pascal’s Triangle == Coefficients in (x + a)n. That is:
15. The Circle Problem and Pascal’s Triangle
a. How many intersections of chords connecting N vertices?
b. How does this relate to Pascal’s Triangle?
16. Patterns in Pascal’s Triangle (see http://www.kosbie.net/lessonPlans/pascalsTriangle/)
a. Simple Patterns
i. Natural Numbers (1,2,3,4…)
ii. Triangular Numbers (1,3,6,10,…)
iii. Binomial Coefficients (nCk)
 Pascal’s Binomial Theorem
iv. Tetrahedral Numbers (1,4,10,20,…)
v. Pentatope Numbers (1,5,15,35,70…)
b. More Challenging Patterns
i. Powers of 2 (2,4,8,16,…)
ii. Hexagonal Numbers (1,6,15,28,…)
iii. Fibonacci Numbers (1,1,2,3,5,8,…)
 Prove This!
iv. Sierpinski’s Triangle
v. Catalan Numbers (1,2,5,14,42,…)
 Prove This!
vi. Powers of 11 (11, 121, 1331, 14641,…)
17. Applications of the Binomial Theorem
a. Find the coefficient of x3 in (x + 5) 3
b. Prove: nC0 + nC1 + … + nCn= 2n (Hint: 2 = 1+1, so what does 2n = ?)
Summer Math Series: Week 4
18. π = C/D (By observation! Since Babylonian times, where π =~ 3.125)
19. Area of a Polygon = ½ hQ (1/2 * apothem * perimeter)
20. Archimedes’ Proof that A = πr2
a. Approximate circle with inscribed (2n)-gons
b. Rephrasing of argument on page 93:
i. Apolygon = ½ hQ, but:
1. As sides  infinity, h  r (apothem  radius)
2. As sides  infinity, Q  C (perimeter  circumference)
ii. So:
As sides  infinity, Apolygon  ½ r C  Acircle
(area of polygon  area of circle)
c. Last step (p. 96): combine:
i. A = ½ r C
ii. C = πD = 2πr
21. Archimedes’ Approximation of π
a. Inscribe (2n)-gons
22. Newton’s Binomial Theorem
a. Generalization to negative integer powers:
b. (P + PQ)m/n = P m/n + (m/n)AQ + (m-n)/(2n) BQ + (m-2n)/(3n) CQ + …
where A,B,C,… represent the immediately preceding terms
so B = (m/n)AQ, C = (m-n)/(2n) BQ, …
c. After some algebra:
(1 + Q) m/n = 1 + (m/n)Q + (m/n)(m/n - 1)/2 Q2 + (m/n)(m/n - 1)(m/n - 2)/(3*2) Q3 + ...
d. That is:
23. Applications of Newton’s Binomial Theorem
a. 1 / (1 + x)3 = 1 – 3x + 6x2 – 10x3 + 15x4 – …
b. Sqrt(1 – x) = 1 – (1/2)x – (1/8)x2 – (1/16)x3 – (5/128)x4 – …
c. So, sqrt(7) = 3 sqrt(1 – 2/9)  fast approximation for square roots!
d. Also cube roots, etc, since (1 – x)1/3 can be expanded this way, too…
Summer Math Series: Week 5
24. Newton’s Calculus (“Fluxions” from De Analysi; see Dunham pp. 171-3)
a. f(x) = x2

f’(x) = 2x
i. What this means graphically (max/min of f(x) = x2)
b. f(x) = a g(x)

f’(x) = a g’(x)
c. f(x) = g(x) + h(x) 
f’(x) = g’(x) + h’(x)
d. f(x) = xa

f’(x) = a x(a-1)
e. General derivative of a polynomial:
f(x) = a0x0 + … + anxn  f’(x) = a1x0 + 2a2x1 + 3a3x2… + nanx(n-1)
f. Homework:
i. Prove the Product Rule:
f(x) = g(x) h(x)

f’(x) = g’(x)h(x) + h’(x)g(x)
ii. Prove the Chain Rule:
f(x) = g(h(x))

f’(x) = g’(h(x)) h’(x)
iii. Prove the Quotient Rule:
f(x) = g(x) / h(x)

f’(x) = (g’(x) h(x) – h’(x) g(x)) / h(x)2
-1
Hint: rewrite as f(x) = g(x) h(x) and use the Product and Chain Rules.
g. Some other useful derivatives:
i. f(x) = cos(x) 
f’(x) = -sin(x)
ii. f(x) = sin(x)

f’(x) = cos(x)
iii. f(x) = ex

f’(x) = ex
iv. Homework: Find the derivatives of tan(x), cot(x), sec(x), csc(x)
h. Integral calculus and Newton’s Physics
i. Constant acceleration: a(t) = a0 (9.8 m/s2)
ii. v’(t) = a(t)  v(t) = a0t + v0
1. Why did Newton assume that v’(t) = a(t)
2. Why does this imply that v(t) = a0t + v0?
3. How fast is a free-falling object moving after 5 seconds?
iii. s’(t) = v(t)  s(t) = ½ a0t2 + v0t + s0
1. Why did Newton assume that s’(t) = v(t)
2. Why does this imply that s(t) = ½ a0t2 + v0t + s0?
3. How far did that free-falling object travel in 5 seconds?
i.
Find local extrema by setting f’(x) = 0
i. Fence problem
ii. Vertex of a parabola
iii. Extrema of cubics
25. Newton’s Approximation of π (see Dunham, pp. 174-6)
26. Maclaurin Series (Taylor Series about f(0)):
a. f(x) = f(0) + f’(0) x + f’’(0)/2! x2 + f(3)(0)/3! x3 + … + f(n)(0)/n! xn + …
b. Homework: Derive the following, using Maclaurin Series expansions:
i. cos(x) = 1 – x2/2! + x4/4! – x6/6! + x8/8! - …
ii. sin(x) = x - x3/3! + x5/5! – x7/7! + x9/9! - …
iii. ex = 1 + x + x2/2! + x3/3! + x4/4! + …
iv. Using these, show: eix = cos(x) + isin(x), hence eiπ = -1
27. Euler’s Approximation of π (using inverse squares!!!) (see Dunham, pp. 215-22)
Glossary
Binomial coefficients (N choose K): The number of ways in which you can choose K elements
from a set of N elements. This equals n! / ( k! (n-k)! ).
Catalan numbers (1, 2, 5, 14, 42, ...): The number of ways you can divide a polygon with N
sides into triangles, using non-intersecting diagonals (a triangle has 1 way, a rectangle has 2
ways, a pentagon has 5 ways, a hexagon has 14 ways, and so on). The Catalan numbers can be
computed using the formula:
Fibonacci numbers (1, 1, 2, 3, 5, 8, ...): A series in which the first two numbers are 1 and each
subsequent number is the sum of the preceding two numbers.
Hexagonal numbers (1, 6, 15, 28, ...): Numbers that can be represented as the number of points
on the perimeter of a hexagon with a constant number of points on each edge. These are given by
the formula N * (2N-1), and can be seen in the following figure:
Pentatope numbers (1, 5, 15, 35, 70, ...) A figurate number (a number that can be represented by
a regular geometric arrangement of equally spaced points) given by:
Ptopn = (1/4)Tn(n+3) = (1/24) n (n+1) (n+2) (n+3)
for tetrahedral number Tn. Note: pentatopes are 4-dimensional analogs of tetrahedra.
Sierpinski's triangle: a famous fractal formed by connecting triangle midpoints as such:
Tetrahedral numbers (1, 4, 10, 20, ...): a figurate number formed by placing discrete points in a
tetrahedron (triangular base pyramid). The formula is given by: n(n+1)(n+2)/6.
Triangular numbers (1, 3, 6, 10, ...): The number of dots you need to form a triangle: