NUMBER SYSTEMSTWSSP 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?• 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

• Now: Consider the decimal . How can you write this number as a fraction? In general, what does this suggest about periodic decimals?

• Next: In general, if a fraction does not have a terminating decimal expansion, why must it have a periodic expansion?

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• How do we know is irrational?• Proof by contradiction – suppose it is NOT irrational (i.e.,

it is rational), and show something goes horribly wrong

• Suppose is rational. Then there are some integers a and b, b ≠ 0, so that . Let’s choose a and b so that the fraction is in simplest form

• Then , so • This means must be even, and since the odd integers are

closed under multiplication, must be even, too

Proving irrationality

• Then there is some integer so that • But then • AND , so • But that means • So, is even• Which means must be, too.• But then and are both even, which means can’t be in

simplest form, a contradiction• This means our original assumption must be wrong, so

must be irrational.

Any other irrational numbers?• Choose at least 2 of the following:

• Prove that is irrational• Prove that is irrational• Prove that is irrational• Prove that is irrational• Prove that is irrational

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• There’s another way – and you probably already know the

tool we will use!

• The Rational Root Theorem:• For any polynomial with integer coefficients,

if the polynomial has a rational root, then the numerator of the root is a divisor of and the denominator is a divisor of

• Just to remind yourself, what are all of the possible roots of ?

Choose 4…• Prove that the following are irrational

Rational or irrational?

Approximating with rationals• Given an irrational number, like , we can find a rational

quantity that is as close as we want it to be to • In order to start doing this, we need to establish some

properties of inequalities.• When we write , what do we really mean? What other

inequalities are equivalent to this statement?

Inequalities• Prove each of the following:• If , and is any number, then • If , and is any number, then • If , and is any positive number, then • If , and is any positive number, then • If , and and are positive, then , but

Exit Ticket (sort of…)

• Without dividing: terminating or repeating, and how do you know?

• Convert to a fraction: 1.112112112…• Rational or irrational, and how do you know: • The irrational numbers are closed under: _________