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 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