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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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outline
Outline
  • Countable and Uncountable Sets
outline3
Outline
  • Countable and Uncountable Sets
  • Algebraic Numbers
outline4
Outline
  • Countable and Uncountable Sets
  • Algebraic Numbers
  • Existence of Transcendental Numbers
outline5
Outline
  • Countable and Uncountable Sets
  • Algebraic Numbers
  • Existence of Transcendental Numbers
  • Examples of Transcendental Numbers
outline6
Outline
  • Countable and Uncountable Sets
  • Algebraic Numbers
  • Existence of Transcendental Numbers
  • Examples of Transcendental Numbers
  • Constructible Numbers
number systems
Number Systems
  • N = natural numbers = {1, 2, 3, …}
number systems8
Number Systems
  • N = natural numbers = {1, 2, 3, …}
  • Z = integers = {…, -2, -1, 0, 1, 2, …}
number systems9
Number Systems
  • N = natural numbers = {1, 2, 3, …}
  • Z = integers = {…, -2, -1, 0, 1, 2, …}
  • Q = rational numbers
number systems10
Number Systems
  • N = natural numbers = {1, 2, 3, …}
  • Z = integers = {…, -2, -1, 0, 1, 2, …}
  • Q = rational numbers
  • R = real numbers
number systems11
Number Systems
  • N = natural numbers = {1, 2, 3, …}
  • Z = integers = {…, -2, -1, 0, 1, 2, …}
  • Q = rational numbers
  • R = real numbers
  • C = complex numbers
countable sets
Countable Sets
  • A set is countable if there is a one-to-one correspondence between the set and N, the natural numbers
countable sets13
Countable Sets
  • A set is countable if there is a one-to-one correspondence between the set and N, the natural numbers
countable sets14
Countable Sets
  • N, Z, and Q are all countable
countable sets15
Countable Sets
  • N, Z, and Q are all countable
uncountable sets
Uncountable Sets
  • R is uncountable
uncountable sets17
Uncountable Sets
  • R is uncountable
  • Therefore C is also uncountable
uncountable sets18
Uncountable Sets
  • R is uncountable
  • Therefore C is also uncountable
  • Uncountable sets are “bigger”
algebraic numbers
Algebraic Numbers
  • A complex number is algebraic if it is the solution to a polynomial equation

where the ai’s are integers.

algebraic number examples
Algebraic Number Examples
  • 51 is algebraic: x – 51 = 0
algebraic number examples21
Algebraic Number Examples
  • 51 is algebraic: x – 51 = 0
  • 3/5 is algebraic: 5x – 3 = 0
algebraic number examples22
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.

algebraic number examples23
Algebraic Number Examples
  • is algebraic: x2 – 2 = 0
algebraic number examples24
Algebraic Number Examples
  • is algebraic: x2 – 2 = 0
  • is algebraic: x3 – 5 = 0
algebraic number examples25
Algebraic Number Examples
  • is algebraic: x2 – 2 = 0
  • is algebraic: x3 – 5 = 0
  • is algebraic: x2 – x – 1 = 0
algebraic number examples26
Algebraic Number Examples
  • is algebraic: x2 + 1 = 0
algebraic numbers27
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
algebraic numbers28
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
solvability by radicals
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
solvability by radicals30
Solvability by Radicals
  • Every Degree 1 polynomial is solvable:
solvability by radicals31
Solvability by Radicals
  • Every Degree 1 polynomial is solvable:
solvability by radicals32
Solvability by Radicals
  • Every Degree 2 polynomial is solvable:
solvability by radicals33
Solvability by Radicals
  • Every Degree 2 polynomial is solvable:
solvability by radicals34
Solvability by Radicals
  • Every Degree 2 polynomial is solvable:

(Known by ancient Egyptians/Babylonians)

solvability by radicals35
Solvability by Radicals
  • Every Degree 3 and Degree 4 polynomial is solvable
solvability by radicals36
Solvability by Radicals
  • Every Degree 3 and Degree 4 polynomial is solvable

del Ferro Tartaglia Cardano Ferrari

(Italy, 1500’s)

solvability by radicals37
Solvability by Radicals
  • Every Degree 3 and Degree 4 polynomial is solvable

Cubic Formula

Quartic Formula

solvability by radicals38
Solvability by Radicals
  • For every Degree 5 or higher, there are polynomials that are not solvable
solvability by radicals39
Solvability by Radicals
  • For every Degree 5 or higher, there are polynomials that are not solvable

Ruffini (Italian) Abel (Norwegian)

(1800’s)

solvability by radicals40
Solvability by Radicals
  • For every Degree 5 or higher, there are polynomials that are not solvable

is not solvable by radicals

solvability by radicals41
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

solvability by radicals42
Solvability by Radicals
  • For every Degree 5 or higher, there are polynomials that are not solvable

is solvable by radicals

algebraic numbers43
Algebraic Numbers
  • The algebraic numbers form a field, denoted by A
algebraic numbers44
Algebraic Numbers
  • The algebraic numbers form a field, denoted by A
  • In fact, A is the algebraic closure of Q
question
Question
  • Are there any complex numbers that are not algebraic?
question46
Question
  • Are there any complex numbers that are not algebraic?
  • A complex number is transcendental if it is not algebraic
question47
Question
  • Are there any complex numbers that are not algebraic?
  • A complex number is transcendental if it is not algebraic
  • Terminology from Leibniz
question48
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

existence of transcendental numbers
Existence of Transcendental Numbers
  • In 1844, the French mathematician Liouville proved that some complex numbers are transcendental
existence of transcendental numbers50
Existence of Transcendental Numbers
  • In 1844, the French mathematician Liouville proved that some complex numbers are transcendental
existence of transcendental numbers51
Existence of Transcendental Numbers
  • His proof was not constructive, but in 1851, Liouville became the first to find an example of a transcendental number
existence of transcendental numbers52
Existence of Transcendental Numbers
  • His proof was not constructive, but in 1851, Liouville became the first to find an example of a transcendental number
existence of transcendental numbers53
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
existence of transcendental numbers54
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
existence of transcendental numbers55
Existence of Transcendental Numbers
  • Theorem (Cantor, 1874):A, the set of algebraic numbers, is countable.
existence of transcendental numbers56
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.
existence of transcendental numbers57
Existence of Transcendental Numbers
  • Proof: Let a be an algebraic number, a solution of
existence of transcendental numbers58
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

existence of transcendental numbers59
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

existence of transcendental numbers60
Existence of Transcendental Numbers
  • Claim: Let k be a positive integer. Then the number of algebraic numbers that have height k is finite.
existence of transcendental numbers61
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.
existence of transcendental numbers62
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.
existence of transcendental numbers63
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.

existence of transcendental numbers64
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.
existence of transcendental numbers65
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.
existence of transcendental numbers66
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.
existence of transcendental numbers67
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.
existence of transcendental numbers68
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.
existence of transcendental numbers69
Existence of Transcendental Numbers
  • Back to the theorem: We want to show that A is countable.
existence of transcendental numbers70
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
existence of transcendental numbers71
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
existence of transcendental numbers72
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.
existence of transcendental numbers73
Existence of Transcendental Numbers
  • Since A is countable but C is uncountable, there are infinitely-many more transcendental numbers than there are algebraic numbers
existence of transcendental numbers74
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

examples of transcendental numbers
Examples of Transcendental Numbers
  • In 1873, the French mathematician Charles Hermite proved that e is transcendental.
examples of transcendental numbers76
Examples of Transcendental Numbers
  • In 1873, the French mathematician Charles Hermite proved that e is transcendental.
examples of transcendental numbers77
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
examples of transcendental numbers78
Examples of Transcendental Numbers
  • In 1882, the German mathematician Ferdinand von Lindemann proved that

is transcendental

examples of transcendental numbers79
Examples of Transcendental Numbers
  • In 1882, the German mathematician Ferdinand von Lindemann proved that

is transcendental

examples of transcendental numbers80
Examples of Transcendental Numbers
  • Still very few known examples of transcendental numbers:
examples of transcendental numbers81
Examples of Transcendental Numbers
  • Still very few known examples of transcendental numbers:
examples of transcendental numbers82
Examples of Transcendental Numbers
  • Still very few known examples of transcendental numbers:
examples of transcendental numbers83
Examples of Transcendental Numbers
  • Still very few known examples of transcendental numbers:
constructible numbers
Constructible Numbers
  • Using an unmarked straightedge and a collapsible compass, given a segment of length 1, what other lengths can we construct?
constructible numbers86
Constructible Numbers
  • For example, is constructible:
constructible numbers87
Constructible Numbers
  • For example, is constructible:
constructible numbers88
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
constructible numbers89
Constructible Numbers
  • Thus the set of constructible numbers, denoted by K, is a subset of A.
constructible numbers90
Constructible Numbers
  • Thus the set of constructible numbers, denoted by K, is a subset of A.
  • K is also a field
constructible numbers92
Constructible Numbers

Most real numbers are not constructible

constructible numbers93
Constructible Numbers
  • In particular, the ancient question of squaring the circle is impossible
the end
The End!
  • References on Handout