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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

- 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

- Also,

implies that there are only finitely-many choices for the coefficients of the polynomial.

Existence of Transcendental Numbers

- 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

- 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

- 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

- 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

- 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

Most real numbers are not constructible

Constructible Numbers

- In particular, the ancient question of squaring the circle is impossible

The End!

- References on Handout

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