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Thursday 21st of October Home Square Roots and Radical Expressions Solving Radical Equations Simplifying Radical Expressions Irrational Numbers in General and Square Roots in Particular Roots of Polynomials Simplifying Radical Expressions Exponents and Radicals Products and Quotients Involving Radicals Roots of Quadratic Equations Radical Expressions Radicals and Rational Exponents Find Square Roots and Compare Real Numbers Radicals Radicals and Rational Exponents Theorems on the Roots of Polynomial Equations SYNTHETIC DIVISION AND BOUNDS ON ROOTS Simplifying Radical Expressions Exponents and Radicals Properties of Exponents and Square Roots Solving Radical Equations Rational Exponents and Radicals,Rationalizing Denominators Rational Exponents and Radicals,Rationalizing Denominators Quadratic Roots Exponents and Roots Multiplying Radical Expressions Exponents and Radicals Solving Radical Equations Solving Quadratic Equations by Factoring and Extracting Roots Newton's Method for Finding Roots Roots of Quadratic Equations Studio Roots, Radicals, and Root Functions Review division factoring and Root Finding Radicals Simplifying Radical Expressions Multiplying and Simplifying Radical Expressions LIKE RADICALS Multiplication and Division of Radicals Radical Equations BOUNDING ROOTS OF POLYNOMIALS
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# Irrational Numbers in General and Square Roots in Particular

In Set 2 you learned that every rational number has a decimal expansion which either
terminates or repeats. Thus if there exist any decimal expansions which do not
terminate and do not repeat, the numbers represented by these decimal expansions
cannot be rational.

1. Consider for instance the following number:

r = :123456789101112131415161718192021222324252627 ...

What is the pattern for the digits of this number? What will the next six digits be?
Since this number does not have a repeating pattern, it is not rational. In other
words, if m and n are any integers, then is never exactly equal to r . (However might be extremely close to r . For instance is very close
to r. (why?))

Numbers which are not rational - such as r - are called irrational numbers.

You might wonder why anybody would need irrational numbers. Indeed, rational
numbers are good enough to do a whole lot with. Since the rational numbers are
closed under addition, subtraction, multiplication, and division (except for division
by 0), each of the following equations has a solution which is a rational number: Verify the above statement by solving all of the equations given and noticing that the
solutions are rational numbers.

Not all equations involving integers have solutions which are rational numbers,
however. In this set we will show that if N is an integer which is not a \perfect
square" (such as 9, 16, 25, 36, 49, or 64) then the solution to the equation x2 = N
is not a rational number. You are going to see that if a and b are any integers then is never exactly equal to N , so that is never exactly equal to .

(Actually, if N > 0 then the equation x2 = N has two solutions: and − .
What we are going to show is the is always either an integer { in which case N
is a perfect square { or an irrational number.)

As preparation for the proof, we will notice something about squaring fractions which
are in lowest terms.

2. Using the fact that find the prime factorizations for 1002, 912, 2472 , 6672, and 10,1272 .
Based on these examples, state a general rule for the prime factorization of the square
of a number.

Which of the following fractions are in lowest terms? If a and b are integers, which of the following possibilities can occur?
(1) and are both in lowest terms.
(2) is in lowest terms but is not.
(3) is not in lowest terms but is in lowest terms.
(4) and are both not in lowest terms.

Explain why!

3. We want to show that if a and b are integers such that is not an integer and
if furthermore N is an integer, then . To start with. let's think about
when a fraction can be an integer.

Which of the following rational numbers are integers? Which of these fractions is in lowest terms?

If a and b are integers, what has to be true in order for to be in lowest terms and
also to be an integer?
If a and b are integers, when is it possible for to be in lowest terms and for to
be an integer?

If is a fraction in lowest terms, which of the following are possible values
for ? How does your answer change if is not in lowest terms?

4. If N is an integer and a and b are integers and , explain why has to be an integer. Explain why N has to be a perfect square.

Explain why if N is not a perfect square then has to be irrational.

Which of the numbers listed above in have irrational square roots?