Math Gems

Math Gems An assortment of mathematical marvels. 12345679 ´9 111111111 ´ 111111111 12345678987654321 1 9

142857 ´ 2 = 285714 142857 ´ 3 = 428571 142857 ´ 4 = 571428 142857 ´ 5 = 714285 142857 ´ 6 = 857142 1 7

= .111111K

Fun arithmetic with the number nine.

16

3

2

13

5

10 11

8

9

6

7

12

4

15 14

1

= .142857K

A magic square. All rows, columns, and diagonals have the same sum.

Fun arithmetic with the number seven.

n! » ( ne )nÖ 2pn n

n

åi = 1 + 2 + 3 + L + n i =1

Õi = 1´ 2 ´ 3 ´ L´ n

G(n + 1) = n!

i =1

= n!

G( 12 ) = Ö p

The sum of the numbers from 1 to n.

The product of the numbers from 1 to n is n factorial.

Stirling's approximation of n factorial. Euler's gamma function gives factorials for integers but has surprising values for fractions.

n n ( x + y) n = å ( k ) x n - k y k

1

n (n + 1) = 2

k =0

n!

( nk ) = k ! (n - k )! The binomial theorem expands powers of sums. The binomial coefficient is the number of ways to choose k objects from a set of n objects, regardless of order. j=

1

1

1

1

1 4

1 3

2 6

1 3

1 4

1

1

5 10 10 5 1 6 15 20 15 6 1 ...

Pascal's triangle shows the binomial coefficients.

® ( mn ) 2 = 2 ® n 2 = 2m 2 ® n 2 is even ® n is even ® n 2 is divisible by 4 ® m 2 is even ® m is even ® mn is not reduced ® Ö 2 is not rational

Proof that the square root of two is irrational.

x = log y

e = 2.71828K Napier's constant, e, is the base of natural logarithms and exponentials. e is transcendental.

pr

p = 1 - 13 + 15 - 17 + 19 - L 4

2

p 2 2 4 4 6 = ´ ´ ´ ´ ´L 2 1 3 3 5 5

The ratio of the circumference of a circle to its diameter is pi. Pi is transcendental, i.e., irrational and non-algebraic.

s3

pr 2h

p 2 r h 3

4 pr 3 3

Pi, expressed as an infinite series and an infinite product.

Area and volume formulas. Archimedes solved the sphere.

2, 3, 5, 7, 11, 13, 17, 19, ... 2

3

4

6 5

8

9

10 12

7

p( x ) »

A prime number is divisible only by one and itself. The sieve of Eratosthenes finds primes.

z ( s) =

¥

1

å ns = Õ n =1

The prime number theorem of Gauss and Legendre approximates the number of primes less than x. c

a

2

ax 2 + bx + c = 0 - b ±Ö b - 4ac 2a 2

The quadratic equation defines a parabola.

2

a +b =c

b

x=

x log x

2

a2

The golden rectangle, a classical aesthetic ideal. Cutting off a square leaves another golden rectangle. A logarithmic spiral is inscribed.

d x e = ex dx

ò e dx = e x

The pentagram contains many pairs of line segments that have the golden ratio.

e = lim (1 + n1 ) x

Calculus, developed by Newton and Leibniz, is based on derivatives (slopes) and integrals (areas) of curves. The derivative of ex is ex. The integral of ex is ex.

n

n ®¥

e = 1 + 11! + 21! + 31! + 41! + L

e, expressed as a limit and an infinite series.

1

sin 2 q + cos 2 q = 1 The Pythagorean theorem. A proof by rearrangement.

1

The trigonometric functions. Another form of the Pythagorean theorem.

Fn =

1 + 1+1L

The golden ratio, expressed as a continued fraction.

Each Fibonacci number is the sum of the previous two. The number of spirals in a sunflower or a pinecone is a Fibonacci number.

e x Ö -1 = cos(x ) + Ö -1 sin( x ) e pÖ -1 = -1 y= Euler's formula relating exponentials to sine waves. A special case relating the numbers pi, e, and the imaginary square root of -1.

a c b cos q = c a tan q = b

( ac )2 + ( bc )2 = 1

lim

1 + Ö5 j= 2

1

ps

sin q =

c q b y = sin x

c2

b2

a

n ®¥

1+

1 1-

The zeta function of Euler and Riemann, expressed as an infinite series and a curious product over all primes.

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

j =1+

p

11

a+b a = b a

The golden ratio, phi. The ratio of a whole to its larger part equals the ratio of the larger part to the smaller. phi is irrational and algebraic.

1 bh 2

p = 3.14159...

Suppose Ö 2 were rational ® Ö 2 = mn , reduced

1 j =1+ j 1 j = 1.618K = 0.618K j

y = ex

s

2

1 x -m s

- ( 1 e 2 sÖ 2p

2

)

Fn +1 =j Fn

j n - ( -j1)n

Ö5

The ratio of successive Fibonacci numbers approaches the golden ratio. An exact formula for the nth Fibonacci number. r i r r A ´B = Ax Bx r¶ Ñ = i ¶x

r r j k k Ay Az i j By Bz r¶ r¶ + j ¶y + k ¶z r r ÑU Ñ × V Ñ ´ V

Ñ2U

2

U

The Gaussian or Gibbs's vector cross product. normal probability distribution Del operates on scalar and vector fields in 3D, quad in 4D. is a bell-shaped curve. Imagine listing all real numbers between 0 and 1 in any order.

1® 2® 3® 4®

v -e+f =2

v -e+f -c=0

The five regular polyhedra. Euler's formula for the number of vertices, edges, and faces of any polyhedron.

The hypercube. Schläfli's formula for vertices, edges, faces, and cells of any 4-dimensional polytope.

.8 .1 .1 .3

4 7 0 5

9 9 3 6

7 3 4 1

3 8 2 2

8 ... 0 ... 1 ... 2 ...

You can always make an unlisted real number by changing every digit on the diagonal, e.g., change .8731... to .9842...

The Möbius strip has only one side. The Klein bottle's inside is its outside.

Fractals of Mandelbrot, Koch, and Sierpinski have infinite levels of detail.

Cantor's proof that the infinity of real numbers is greater than the infinity of integers.

($y)(x) ~ Dem (x, y) É (x) ~ Dem (x, sub (n, 13, n)) [from Nagel and Newman, Gödel's Proof]

Gödel proved that if arithmetic is consistent, it must be incomplete, i.e., it has true propositions that can never be proved.

To find out more, look it up on the web or in the library. © 2007 Keith Enevoldsen thinkzone.wlonk.com