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

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

Number Systems N = natural numbers = {1, 2, 3, …}

Number Systems N = natural numbers = {1, 2, 3, …} Z = integers = {…, -2, -1, 0, 1, 2, …}

Number Systems N = natural numbers = {1, 2, 3, …} Z = integers = {…, -2, -1, 0, 1, 2, …} Q = rational numbers

Number Systems N = natural numbers = {1, 2, 3, …} Z = integers = {…, -2, -1, 0, 1, 2, …} Q = rational numbers R = real numbers

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

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

Countable Sets N, Z, and Q are all countable

Countable Sets N, Z, and Q are all countable

Uncountable Sets R is uncountable

Uncountable Sets R is uncountable Therefore C is also uncountable

Uncountable Sets R is uncountable Therefore C is also uncountable Uncountable sets are “bigger”

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 51 is algebraic: x – 51 = 0

Algebraic Number Examples 51 is algebraic: x – 51 = 0 3/5 is algebraic: 5x – 3 = 0

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 Examples is algebraic: x2 – 2 = 0

Algebraic Number Examples is algebraic: x2 – 2 = 0 is algebraic: x3 – 5 = 0

Algebraic Number Examples is algebraic: x2 – 2 = 0 is algebraic: x3 – 5 = 0 is algebraic: x2 – x – 1 = 0

Algebraic Number Examples is algebraic: x2 + 1 = 0

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 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 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 Radicals Every Degree 1 polynomial is solvable:

Solvability by Radicals Every Degree 1 polynomial is solvable:

Solvability by Radicals Every Degree 2 polynomial is solvable:

Solvability by Radicals Every Degree 2 polynomial is solvable:

Solvability by Radicals Every Degree 2 polynomial is solvable: (Known by ancient Egyptians/Babylonians)

Solvability by Radicals Every Degree 3 and Degree 4 polynomial is solvable

Solvability by Radicals Every Degree 3 and Degree 4 polynomial is solvable del Ferro Tartaglia Cardano Ferrari (Italy, 1500’s)

Solvability by Radicals Every Degree 3 and Degree 4 polynomial is solvable Cubic Formula Quartic Formula

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

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

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

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

Algebraic Numbers The algebraic numbers form a field, denoted by A

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

Question Are there any complex numbers that are not algebraic?

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

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

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

Existence of Transcendental Numbers In 1844, the French mathematician Liouville proved that some complex numbers are transcendental

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

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

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 Numbers Proof: Let a be an algebraic number, a solution of

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

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

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

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

Examples of Transcendental Numbers In 1873, the French mathematician Charles Hermite proved that e is transcendental.

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

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

Examples of Transcendental Numbers Still very few known examples of transcendental numbers:

Examples of Transcendental Numbers Still very few known examples of transcendental numbers:

Examples of Transcendental Numbers Still very few known examples of transcendental numbers:

Examples of Transcendental Numbers Still very few known examples of transcendental numbers:

Examples of Transcendental Numbers Open questions:

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

Constructible Numbers For example, is constructible:

Constructible Numbers For example, is constructible:

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

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

Constructible Numbers

Constructible Numbers Most real numbers are not constructible

Constructible Numbers In particular, the ancient question of squaring the circle is impossible

The End! References on Handout