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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,
𝑐𝑛 𝑥 𝑛 + 𝑐𝑛−1 𝑥 𝑛−1 + … + 𝑐2 𝑥 2 + 𝑐1 𝑥 1 + 𝑐0
if the polynomial has a rational root, then the numerator
of the root is a divisor of 𝑐0 and the denominator is a divisor
of 𝑐𝑛
• Just to remind yourself, what are all of the possible roots
of 2𝑥 3 − 9𝑥 2 + 10𝑥 − 3 = 0?
Choose 2…
• Prove that the following are irrational
•
3
13
•
•
5
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?
2 algebraic or transcendental?
𝑛
• What about 𝑎 for 𝑎 and 𝑛 integers, 𝑛 ≠ 0?
• Is
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