Number systems TWSSP Thursday

Thursday Agenda Another technique for proving a number is irrational Define algebraic and transcendental numbers Explore closure of algebraic and transcendental numbers Choose something about number to explore and prepare to share with the group

Thursday Agenda Questions for today: How can we prove a number is irrational? What are algebraic and transcendental numbers? Under what operations are algebraic numbers closed? Transcendentals? Learning targets: The rational root theorem can be applied to prove that a number is irrational Algebraic numbers are roots of polynomials with integer coefficients Transcendental numbers are numbers which are not roots of polynomials with integer coefficients The algebraic numbers are closed under ______ The transcendental numbers are closed under ______ Success criteria: I can prove that a number is irrational. I can determine if a number is algebraic or transcendental. I can determine closure of the algebraic and transcendental numbers.

Left over from yesterday Suppose we have two irrational numbers (call them α and β) whose sum (α + β) is rational. What can you say about α – β? What about α + 2β?

Proving irrational numbers There’s another way – and you probably already know the tool we will use! The Rational Root Theorem: For any polynomial with integer coefficients, 𝑐 𝑛 𝑥 𝑛 + 𝑐 𝑛−1 𝑥 𝑛−1 + …+ 𝑐 2 𝑥 2 + 𝑐 1 𝑥 1 + 𝑐 0 if the polynomial has a rational root, then the numerator of the root is a divisor of 𝑐 0 and the denominator is a divisor of 𝑐 𝑛 Just to remind yourself, what are all of the possible roots of 2 𝑥 3 −9 𝑥 2 +10𝑥 −3=0?

Choose 2… 4 13 −3 6 Prove that the following are irrational 3 5 13 5 91 4 13 −3 6 15 3 − 2

Algebraic Numbers A number is algebraic if it is a root of a polynomial with integer coefficients A number is transcendental if it is not algebraic (i.e., it is not a root of a polynomial with integer coefficients) Are rational numbers algebraic, transcendental, or neither? Is there a neither? Is 2 algebraic or transcendental? What about 𝑛 𝑎 for 𝑎 and 𝑛 integers, 𝑛≠0?

Transcendental numbers Some transcendental numbers include: 𝜋 2 2 log 2 0.101001000100001.. e The why is beyond the scope of what we can do in 4 days

Closure Under which of the four operations, if any, are the algebraic numbers closed? Under which of the four operations, if any, are the transcendental numbers closed? What happens if we use the four operations on one algebraic number and one transcendental (i.e., what if we add a transcendental to an algebraic)? What kind of number do we get?

Putting it all together Create a visual organizer to represent the relationships between all of the types of numbers we have discussed (naturals, integers, rationals, irrationals, reals, algebraics, and transcendentals), and which also indicates closure of the set under the four arithmetic operations Use a think-go around-discuss, and whiteboard your end result.

Conduct your own exploration Choose something about number to explore further and prepare to share your results with the whole group Anything you might have lingering questions or ideas about Possibilities Sizes of infinity Are there infinitely many primes? Why is the Fundamental Theorem of Arithmetic true? Approximating irrationals with rationals That whole closure of subsets and closure of the larger set thing Any finite subsets of ℕ, ℤ, or ℚ which are closed under the operations? Solve some fun problems about primes Solve some fun problems about divisibility