NUMBER SYSTEMS TWSSP Wednesday

Wednesday Agenda Finish our work with decimal expansions Define irrational numbers Prove the existence of irrationals Explore closure of irrationals Establish multiple techniques for proving irrationality of a number

Wednesday Agenda Questions for today: How can we prove a number is irrational? Under what operations are irrational numbers closed? Learning targets: Real numbers are either rational or irrational The irrational numbers are closed under__________ Success criteria: I can prove that a number is irrational. Given a subset of the real numbers, I can determine if that subset is closed under an operation.

How do you know if it’s rational? Any rational number can be written as a terminating or infinitely periodic decimal; conversely, any terminating or infinitely periodic decimal is a rational number Terminating: a finite number of digits in the decimal expansion Infinitely periodic: an infinite number of digits, but digits repeating in a fixed pattern First: Suppose a decimal is terminating. How do you know it is rational, by the definition? Next: How do you know if a rational number (written in fraction form) will have a terminating decimal expansion?

How do you know if it’s rational?

Real Numbers The real numbers are defined to be the collection of all numbers associated with a point on a continuous number line We denote the reals ℝ Under this definition, how do our previously defined sets of numbers relate to the real numbers?

Irrational Numbers Can you create a decimal which does not terminate or repeat? An irrational number is a real number which is not rational. That is, if a number cannot be written as a/d for a, d integers and d not 0, that number is irrational Every real number can either be represented by a terminating decimal, or by a unique infinite decimal. WHY?

Proving irrationality

Any other irrational numbers?

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

Choose 4…

Rational or irrational?

Approximating with rationals

Inequalities

Exit Ticket (sort of…)