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 ai’s 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, 1500’s)

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)
(1800’s)

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 a’s 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 a’s 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