Algebraic and Transcendental Numbers

1
Algebraic and Transcendental Numbers

• Dr. Dan Biebighauser

2
Outline

• Countable and Uncountable Sets

3
Outline

• Countable and Uncountable Sets
• Algebraic Numbers

4
Outline

• Countable and Uncountable Sets
• Algebraic Numbers
• Existence of Transcendental Numbers

5
Outline

• Countable and Uncountable Sets
• Algebraic Numbers
• Existence of Transcendental Numbers
• Examples of Transcendental Numbers

6
Outline

• Countable and Uncountable Sets
• Algebraic Numbers
• Existence of Transcendental Numbers
• Examples of Transcendental Numbers
• Constructible Numbers

7
Number Systems

• N natural numbers 1, 2, 3,

8
Number Systems

• N natural numbers 1, 2, 3,
• Z integers , -2, -1, 0, 1, 2,

9
Number Systems

• N natural numbers 1, 2, 3,
• Z integers , -2, -1, 0, 1, 2,
• Q rational numbers

10
Number Systems

• N natural numbers 1, 2, 3,
• Z integers , -2, -1, 0, 1, 2,
• Q rational numbers
• R real numbers

11
Number Systems

• N natural numbers 1, 2, 3,
• Z integers , -2, -1, 0, 1, 2,
• Q rational numbers
• R real numbers
• C complex numbers

12
Countable Sets

• A set is countable if there is a one-to-one
  correspondence between the set and N, the natural
  numbers

13
Countable Sets

• A set is countable if there is a one-to-one
  correspondence between the set and N, the natural
  numbers

14
Countable Sets

• N, Z, and Q are all countable

15
Countable Sets

• N, Z, and Q are all countable

16
Uncountable Sets

• R is uncountable

17
Uncountable Sets

• R is uncountable
• Therefore C is also uncountable

18
Uncountable Sets

• R is uncountable
• Therefore C is also uncountable
• Uncountable sets are bigger

19
Algebraic Numbers

• A complex number is algebraic if it is the
  solution to a polynomial equation
• where the ais are integers.

20
Algebraic Number Examples

• 51 is algebraic x 51 0

21
Algebraic Number Examples

• 51 is algebraic x 51 0
• 3/5 is algebraic 5x 3 0

22
Algebraic Number Examples

• 51 is algebraic x 51 0
• 3/5 is algebraic 5x 3 0
• Every rational number is algebraic
• Let a/b be any element of Q. Then a/b is a
  solution to bx a 0.

23
Algebraic Number Examples

• is algebraic x2 2 0

24
Algebraic Number Examples

• is algebraic x2 2 0
• is algebraic x3 5 0

25
Algebraic Number Examples

• is algebraic x2 2 0
• is algebraic x3 5 0
• is algebraic x2 x 1 0

26
Algebraic Number Examples

• is algebraic x2 1 0

27
Algebraic Numbers

• Any number built up from the integers with a
  finite number of additions, subtractions,
  multiplications, divisions, and nth roots is an
  algebraic number

28
Algebraic Numbers

• Any number built up from the integers with a
  finite number of additions, subtractions,
  multiplications, divisions, and nth roots is an
  algebraic number
• But not all algebraic numbers can be built this
  way, because not every polynomial equation is
  solvable by radicals

29
Solvability by Radicals

• A polynomial equation is solvable by radicals if
  its roots can be obtained by applying a finite
  number of additions, subtractions,
  multiplications, divisions, and nth roots to the
  integers

30
Solvability by Radicals

• Every Degree 1 polynomial is solvable

31
Solvability by Radicals

• Every Degree 1 polynomial is solvable

32
Solvability by Radicals

• Every Degree 2 polynomial is solvable

33
Solvability by Radicals

• Every Degree 2 polynomial is solvable

34
Solvability by Radicals

• Every Degree 2 polynomial is solvable
• (Known by ancient Egyptians/Babylonians)

35
Solvability by Radicals

• Every Degree 3 and Degree 4 polynomial is solvable

36
Solvability by Radicals

• Every Degree 3 and Degree 4 polynomial is
  solvable
• del Ferro Tartaglia
  Cardano Ferrari
• (Italy, 1500s)

37
Solvability by Radicals

• Every Degree 3 and Degree 4 polynomial is
  solvable
• Cubic Formula
• Quartic Formula

38
Solvability by Radicals

• For every Degree 5 or higher, there are
  polynomials that are not solvable

39
Solvability by Radicals

• For every Degree 5 or higher, there are
  polynomials that are not solvable
• Ruffini (Italian)
  Abel (Norwegian)
• (1800s)

40
Solvability by Radicals

• For every Degree 5 or higher, there are
  polynomials that are not solvable
• is not solvable by radicals

41
Solvability by Radicals

• For every Degree 5 or higher, there are
  polynomials that are not solvable
• is not solvable by radicals
• The roots of this equation are algebraic

42
Solvability by Radicals

• For every Degree 5 or higher, there are
  polynomials that are not solvable
• is solvable by
  radicals

43
Algebraic Numbers

• The algebraic numbers form a field, denoted by A

44
Algebraic Numbers

• The algebraic numbers form a field, denoted by A
• In fact, A is the algebraic closure of Q

45
Question

• Are there any complex numbers that are not
  algebraic?

46
Question

• Are there any complex numbers that are not
  algebraic?
• A complex number is transcendental if it is not
  algebraic

47
Question

• Are there any complex numbers that are not
  algebraic?
• A complex number is transcendental if it is not
  algebraic
• Terminology from Leibniz

48
Question

• Are there any complex numbers that are not
  algebraic?
• A complex number is transcendental if it is not
  algebraic
• Terminology from Leibniz
• Euler was one of the first to
• conjecture the existence of
• transcendental numbers

49
Existence of Transcendental Numbers

• In 1844, the French mathematician Liouville
  proved that some complex numbers are
  transcendental

50
Existence of Transcendental Numbers

• In 1844, the French mathematician Liouville
  proved that some complex numbers are
  transcendental

51
Existence of Transcendental Numbers

• His proof was not constructive, but in 1851,
  Liouville became the first to find an example of
  a transcendental number

52
Existence of Transcendental Numbers

• His proof was not constructive, but in 1851,
  Liouville became the first to find an example of
  a transcendental number

53
Existence of Transcendental Numbers

• Although only a few special examples were known
  in 1874, Cantor proved that there are
  infinitely-many more transcendental numbers than
  algebraic numbers

54
Existence of Transcendental Numbers

• Although only a few special examples were known
  in 1874, Cantor proved that there are
  infinitely-many more transcendental numbers than
  algebraic numbers

55
Existence of Transcendental Numbers

• Theorem (Cantor, 1874) A, the set of algebraic
  numbers, is countable.

56
Existence of Transcendental Numbers

• Theorem (Cantor, 1874) A, the set of algebraic
  numbers, is countable.
• Corollary The set of transcendental numbers must
  be uncountable. Thus there are infinitely-many
  more transcendental numbers.

57
Existence of Transcendental Numbers

• Proof Let a be an algebraic number, a solution of

58
Existence of Transcendental Numbers

• Proof Let a be an algebraic number, a solution
  of
• We may choose n of the smallest possible degree
  and assume that the coefficients are relatively
  prime

59
Existence of Transcendental Numbers

• Proof Let a be an algebraic number, a solution
  of
• We may choose n of the smallest possible degree
  and assume that the coefficients are relatively
  prime
• Then the height of a is the sum

60
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.

61
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• Let a have height k. Let n be the degree of the
  polynomial for a in the definition of as height.

62
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• Let a have height k. Let n be the degree of the
  polynomial for a in the definition of as height.
• Then n cannot be bigger than k, by definition.

63
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• Also,
• implies that there are only finitely-many choices
  for the coefficients of the polynomial.

64
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• So there are only finitely-many choices for the
  coefficients of each polynomial of degree n
  leading to a height of k.

65
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• So there are only finitely-many choices for the
  coefficients of each polynomial of degree n
  leading to a height of k.
• Thus there are finitely-many polynomials of
  degree n that lead to a height of k.

66
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• This is true for every n less than or equal to k,
  so there are finitely-many polynomials that have
  roots with height k.

67
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• This means there are finitely-many such roots to
  these polynomials, i.e., there are finitely-many
  algebraic numbers of height k.

68
Existence of Transcendental Numbers

• Claim Let k be a positive integer. Then the
  number of algebraic numbers that have height k is
  finite.
• This means there are finitely-many such roots to
  these polynomials, i.e., there are finitely-many
  algebraic numbers of height k.
• This proves the claim.

69
Existence of Transcendental Numbers

• Back to the theorem We want to show that A is
  countable.

70
Existence of Transcendental Numbers

• Back to the theorem We want to show that A is
  countable.
• For each height, put the algebraic numbers of
  that height in some order

71
Existence of Transcendental Numbers

• Back to the theorem We want to show that A is
  countable.
• For each height, put the algebraic numbers of
  that height in some order
• Then put these lists together, starting with
  height 1, then height 2, etc., to put all of the
  algebraic numbers in order

72
Existence of Transcendental Numbers

• Back to the theorem We want to show that A is
  countable.
• For each height, put the algebraic numbers of
  that height in some order
• Then put these lists together, starting with
  height 1, then height 2, etc., to put all of the
  algebraic numbers in order
• The fact that this is possible proves that A is
  countable.

73
Existence of Transcendental Numbers

• Since A is countable but C is uncountable, there
  are infinitely-many more transcendental numbers
  than there are algebraic numbers

74
Existence of Transcendental Numbers

• Since A is countable but C is uncountable, there
  are infinitely-many more transcendental numbers
  than there are algebraic numbers
• The algebraic numbers are spotted over the plane
  like stars against a black sky the dense
  blackness is the firmament of the
  transcendentals.
• E.T. Bell, math historian

75
Examples of Transcendental Numbers

• In 1873, the French mathematician Charles Hermite
  proved that e is transcendental.

76
Examples of Transcendental Numbers

• In 1873, the French mathematician Charles Hermite
  proved that e is transcendental.

77
Examples of Transcendental Numbers

• In 1873, the French mathematician Charles Hermite
  proved that e is transcendental.
• This is the first number proved to be
  transcendental that was not constructed for such
  a purpose

78
Examples of Transcendental Numbers

• In 1882, the German mathematician Ferdinand von
  Lindemann proved that
• is transcendental

79
Examples of Transcendental Numbers

• In 1882, the German mathematician Ferdinand von
  Lindemann proved that
• is transcendental

80
Examples of Transcendental Numbers

• Still very few known examples of transcendental
  numbers

81
Examples of Transcendental Numbers

• Still very few known examples of transcendental
  numbers

82
Examples of Transcendental Numbers

• Still very few known examples of transcendental
  numbers

83
Examples of Transcendental Numbers

• Still very few known examples of transcendental
  numbers

84
Examples of Transcendental Numbers

• Open questions

85
Constructible Numbers

• Using an unmarked straightedge and a collapsible
  compass, given a segment of length 1, what other
  lengths can we construct?

86
Constructible Numbers

• For example, is constructible

87
Constructible Numbers

• For example, is constructible

88
Constructible Numbers

• The constructible numbers are the real numbers
  that can be built up from the integers with a
  finite number of additions, subtractions,
  multiplications, divisions, and the taking of
  square roots

89
Constructible Numbers

• Thus the set of constructible numbers, denoted by
  K, is a subset of A.

90
Constructible Numbers

• Thus the set of constructible numbers, denoted by
  K, is a subset of A.
• K is also a field

91
Constructible Numbers
92
Constructible Numbers

• Most real numbers are not constructible

93
Constructible Numbers

• In particular, the ancient question of squaring
  the circle is impossible

94
The End!

• References on Handout