Quest of 

1. Quest of  Lecture Five

2. Outline • Review of number representation • Mathematical classification of numbers • Irrationality of • History of  • Methods of computing 

3. Number 13 in Various Representations ΔIII Egypt China Attic IΓ 13 Hindu-Arabic modern Babylon Greece XIII 11012 binary Maya Rome

4. Property of Numbers • The mathematical properties of numbers are independent of their representations. E.g., 13 is an odd number, no matter how it is represented. 2 + 2 = 4, which transcends cultures.

5. Natural Numbers N • The numbers 1, 2, 3, …,

6. Integers Z • The numbers 1, 2, 3, 4, … and the number 0, and negative numbers -1, -2, -3, -4, … • It took a long time for people to understand the necessity of 0 and negative numbers • Negative number -n has the property that (-n) + n = 0

7. Rational Numbers Q • The numbers of the form p/q where p and q are integers • The representation of rational numbers in this form is not unique, p/q represents the same number as (np)/(nq) for any n ≠ 0 • We can choose p and q such that GCD(p,q) = 1

8. Adding and Multiplying Rational Numbers • p/q  r/s = (pr) / (qs) • p/q + r/s = (ps + rq)/(qs) (We use the abbreviated notation ps = ps)

9. Irrationals • There are numbers that cannot be written as p/q. These numbers will be called irrational numbers • There are countably infinite rational numbers, but irrational numbers are not countable

10. Count the Rationals • p/q 1 2 3 4 5 6 7 8 … • 1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 … • 2 ½ 2/2 3/2 4/2 5/2 6/2 7/2 8/2 … • 3 1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3 • 4 ¼ 2/4 ¾ 4/4 5/4 6/4 7/4 8/4 • 5 1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 • 6 1/6 2/6 3/6 4/6 5/6 6/6 7/6 8/6 • 1/7 2/7 3/7 4/7 5/7 6/7 7/7 8/7 • 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 … • … 1 2 4 11 7 3 5 8 6 9 10

11. Proof that is irrational • Prove by contradiction • Assuming p & q are integers such that and GCD(p,q)=1, then • We use the fact that square of odd number is odd, square of even is even. So p2 is even, then p is even. • Let p = 2s, where s is any integer, so p2=4s2, and q2=p2/2=2s2. This says q is also even, which contradicts the assumption that GCD(p,q) =1 or

12. Square of an even number is even, and odd is odd • Represent integer n by its binary expansion • Clearly n is odd if a0=1, and even if a0=0. Then the last binary digit in n2 is a02=a0 (think of multiplication in binary) • Thus the parity of n2 is same as n.

13. Real Numbers R • The collection of rational numbers and irrational numbers forms the real numbers. The set of reals are uncountable. -2 -1 0 1 2 3 x The real line.

14. Algebraic and Transcendental Numbers • The algebraic number x is a real number that is a root of a polynomial equation with integer coefficients • A real number that is not an algebraic number is called a transcendental number. E.g., -1, ½, are algebraic, e and  are transcendental.

15. Complex Number C • A complex unit i has the property i2 = -1, or i = • A complex number is of the form a+ib where a and b are real numbers. • Solution of a polynomial equation of real coefficients can be a complex number.

16. Addition and Multiplication of Complex Numbers • (a + ib) + (c + id) = (a+c) + i(b+d) • (a + ib)  (c + id) = (ac +iad + ibc + i2bd) = (ac-bd) + i(ad + bc)

17. Complex Plane Im z z=x+iy =reiθ r θ Re z x is called real part, y imaginary part

18. Relation of Types of Numbers Complex Real Rational Integer

19. Definition of  circumference S Radius R S =  D = 2 R Diameter D

20. Area of a Circle A=R (S/2) =  R2 R Approximately S/2 R r dr

21. Volume of a Sphere Hand with Reflecting Globe, self-portrait by M. C. Escher, 1935.

22.  in the Bible “Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.” — Old Testament (I Kings vii.23) The Bible suggested that  = 3.

23. Egyptian  (Rhind Papyrus) • Each square is 9 units of area, each triangle is 9/2 units. Assuming that the shaded area is approximately equal to the area of circle, and 79=63 ≈ 64, then • (9/2)2 ≈ 64 = 82 or  ≈ 4 (8/9)2 = 3.16049… 3

24. Archimedes Of Syracuse (ca 287-212 BC) Archimedes is regarded as one of the greatest mathematician of all time. In physics, we have the Archimedes’ principle for hydrostatics.

25. Archimedes’ Method The perimeter of a regular polygon of n sides inscribed in a circle is smaller than the circumference of the circle, whereas the similar circumscribed polygon is greater than the circumference. With 96 sides, Archimedes found 3 10/71 <  < 3 1/7 or 3.14084 <  < 3.142858 n=6

26. Zu Chongzhi (429 – 500 AD) Zu Chongzhi (祖冲之) obtained accurate estimates of 3.1415926 <  < 3.1415927. This level of accuracy was not surpassed until early Renaissance in Europe. Methodology-wise, it is similar to Archimedes’.

27. Modern Methods based on Series Expansions • Let f(x) be some smooth function, then we can write f(x) as a sum of infinite terms:

28. Example of Series

29. The Arctangent Function y /2 Given a point (1,y) on the vertical line at x=1, the arctangent of y is the angle θ, i.e., arctan(y) = θ, –/2 < θ < /2. The angle θ is measured in radian, that is, a full circle is 2. (1,y) x  θ 0 (0,0) (1,0) 3/2

30. Arctangent Series at y=1 Although this formula for  is simple and easy to calculate, it requires large numbers of terms for a good estimate of it. The rest of the story in the 1700 to 1800 AD was to find faster convergent series.

31. John Machin (1680-1752) • Combining two arctangent series with small arguments gives faster convergence. Machin obtained 100 decimal places by:

32. Other Interesting Formula for  • William Brouncker (1620-1684 AD) continued fraction

33. S. Ramanujan (1877-1920 AD) Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on, elliptic functions, continued fraction, and infinite series.

34. Leonhard Euler (1707-1783) Swiss mathematician who in his numerous works made major contributions to virtually every branch of the mathematics of his day. Euler formula relates the most common mathematical constants in a mysterious way: • e = 2.71828182846… • = 3.14159265358979… i =

35. Computer Age • Borwein quartic convergence algorithm (1987) ak approaches 1/ as k goes to infinity.

36. Working of Borwein Iteration • Start with y0=0.41421356…, a0=0.34314575…, • k=0, compute y1=(1-(1-y04)1/4)/(1+ (1-y04)1/4) = 0.00373489, and a1= a0 (1+y1)4-23y1(1+y1+y12)=0.31831, 1/a1=3.1415926… • Set k=1, compute y2 and a2 and so on •  ≈ 1/ak

37. A Brief Summary of History of 

38. The 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548586327886593615338182796823030195203530185296 …

39. Summary • Various ways of representing numbers are equivalent. Mathematical properties are independent of number representation. • Numbers are classified according to their properties. • The history of computation of the value of  reflects the development of mathematics and computing power of the day.