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Algebraic and Transcendental Numbers

Algebraic and Transcendental Numbers. Dr. Dan Biebighauser. Outline. Countable and Uncountable Sets. Outline. Countable and Uncountable Sets Algebraic Numbers. Outline. Countable and Uncountable Sets Algebraic Numbers Existence of Transcendental Numbers. Outline.

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Algebraic and Transcendental Numbers

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  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

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