Lesson OVERVIEW of lessons on Irrational numbers

Source code of 'OVERVIEW of lessons on Irrational numbers'

This Lesson (OVERVIEW of lessons on Irrational numbers) was created by by ikleyn(52778)

: View Source, Show
About ikleyn:

<H2>OVERVIEW of lessons on Irrational numbers</H2>

This file is an OVERVIEW of my lessons on Irrational numbers.

For your convenience, &nbsp;it contains the list of the lessons and then the same list with short annotations to each lesson.

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Number-Line/Proving-irrationality-of-some-real-numbers.lesson>Proving irrationality of some real numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/real-numbers/What-number-is-greater-Comparing-magnitude-of-irrational-numbers.lesson>What number is greater? Comparing magnitude of irrational numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/real-numbers/Calculations-of-expressions-containing-square-roots.lesson>Calculations of expressions containing square roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/real-numbers/Prove-that-two-irrational-numbers-are-equal.lesson>Prove that two irrational numbers are equal</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/real-numbers/Is-this-number-rational-or-irrational.lesson>Is this number rational or irrational?</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/HOW-TO-make-the-denominator-of-a-fraction-free-of-square-roots.lesson>HOW TO rationalize a fraction by making its denominator free of square roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/How-to-rationalize-a-fraction-by-making-its-denominator-free-of-cubic-roots.lesson>HOW TO rationalize a fraction by making its denominator free of cubic roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Amazing-calculations-with-fractions-that-contain-quadratic-irrationalities-in-denominators.lesson>Amazing calculations with fractions that contain quadratic irrationalities in denominators</A>

<H3>List of lessons with short annotations</H3><A HREF=http://www.algebra.com/algebra/homework/Number-Line/Proving-irrationality-of-some-real-numbers.lesson>Proving irrationality of some real numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp; Problems 1 - 3. &nbsp;Prove that &nbsp;{{{sqrt(2)}}}&nbsp; is an irrational number. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Prove that &nbsp;{{{sqrt(3)}}}&nbsp; is an irrational number. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Prove that &nbsp;{{{sqrt(6)}}}&nbsp; is an irrational number.

&nbsp;&nbsp;&nbsp;&nbsp; Problems 4 - 6. &nbsp;Prove that &nbsp;{{{root(3,2))}}}&nbsp; is an irrational number. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Prove that &nbsp;{{{root(3,6))}}}&nbsp; is an irrational number. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Prove that &nbsp;{{{root(3,9))}}}&nbsp; is an irrational number.

&nbsp;&nbsp;&nbsp;&nbsp; Problems 7 - 9. &nbsp;Prove that &nbsp;{{{sqrt(2)}}} + {{{sqrt(3))}}}&nbsp; is an irrational number. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Prove that &nbsp;{{{sqrt(1+sqrt(2))}}}&nbsp; is an irrational number. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Prove that &nbsp;{{{sqrt(2 + root(3,2))}}}&nbsp; is an irrational number.

<A HREF=http://www.algebra.com/algebra/homework/Inequalities/Which-number-is-greater-Comparing-magnitude-of-irrational-numbers.lesson>What number is greater? Comparing magnitude of irrational numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp; Problem 1. &nbsp;What number is greater, &nbsp;{{{root(3,3)}}}&nbsp; or &nbsp;{{{sqrt(2)}}} ?

&nbsp;&nbsp;&nbsp;&nbsp; Problem 2. &nbsp;What number is greater, &nbsp;{{{root(5,5)}}}&nbsp; or &nbsp;{{{sqrt(2)}}} ?

&nbsp;&nbsp;&nbsp;&nbsp; Problem 3. &nbsp;Solve yourself without using a calculator: &nbsp;what number is greater, &nbsp;{{{root(5,5)}}}&nbsp; or &nbsp;{{{root(3,3)}}} ?

<A HREF=http://www.algebra.com/algebra/homework/real-numbers/Calculations-of-expressions-containing-square-roots.lesson>Calculations of expressions containing square roots</A>

&nbsp;&nbsp;&nbsp;&nbsp; Problem 1. &nbsp;Calculate &nbsp;&nbsp;a) &nbsp;{{{sqrt(13^2 - 12^2)}}}; &nbsp;&nbsp;b) &nbsp;{{{sqrt(25^2 - 24^2)}}}; &nbsp;&nbsp;c) &nbsp;{{{sqrt(5^2 + 12^2)}}}; &nbsp;&nbsp;d) &nbsp;{{{sqrt(8^2 + 15^2)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 2. &nbsp;Calculate &nbsp;&nbsp;a) &nbsp;{{{sqrt(4+sqrt(25))}}}; &nbsp;&nbsp;b) &nbsp;{{{sqrt(9+sqrt(49))}}}; &nbsp;&nbsp;c) &nbsp;{{{sqrt(3+sqrt(36))}}}; &nbsp;&nbsp;d) &nbsp;{{{sqrt(7+sqrt(81))}}}; &nbsp;&nbsp;e) &nbsp;{{{sqrt(7-sqrt(9))}}}; &nbsp;&nbsp;f) &nbsp;{{{sqrt(sqrt(25)+sqrt(16))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 3. &nbsp;Simplify &nbsp;&nbsp;a) &nbsp;{{{sqrt(sqrt(25)-4)}}}; &nbsp;&nbsp;b) &nbsp;{{{sqrt(sqrt(49)-3)}}}; &nbsp;&nbsp;c) &nbsp;{{{(sqrt(5)-1)*(sqrt(5)+1)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 4. &nbsp;Simplify &nbsp;&nbsp;{{{sqrt(2 + sqrt(3))}}} + {{{sqrt(2 - sqrt(3))}}}. &nbsp;&nbsp;&nbsp;&nbsp; Problem 5. &nbsp;Simplify &nbsp;&nbsp;{{{sqrt(2 + sqrt(3))}}} - {{{sqrt(2 - sqrt(3))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 6. &nbsp;Solve yourself: &nbsp;&nbsp;Simplify &nbsp;&nbsp;a) &nbsp;&nbsp;{{{sqrt(3 + 2sqrt(2))}}} + {{{sqrt(3 - 2sqrt(2))}}}; &nbsp;&nbsp;b) &nbsp;&nbsp;{{{sqrt(3 + 2sqrt(2))}}} - {{{sqrt(3 - 2sqrt(2))}}};

&nbsp;&nbsp;&nbsp;&nbsp; Problem 7. &nbsp;Prove that &nbsp;&nbsp;a) &nbsp;{{{sqrt(5 + 2sqrt(6))}}} + {{{sqrt(5 - 2sqrt(6))}}}= {{{2sqrt(3)}}}; &nbsp;&nbsp;b) &nbsp;{{{sqrt(5 + 2sqrt(6))}}} - {{{sqrt(5 - 2sqrt(6))}}}= {{{2sqrt(2)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 8. &nbsp;If &nbsp;{{{x}}} = {{{7-4sqrt(3)}}}, &nbsp;find the value &nbsp;&nbsp;{{{sqrt(x)}}} + {{{1/sqrt(x)}}}.

<A HREF=http://www.algebra.com/algebra/homework/real-numbers/Prove-that-two-irrational-numbers-are-equal.lesson>Prove that two irrational numbers are equal</A>

&nbsp;&nbsp;&nbsp;&nbsp; Problem 1. &nbsp;Prove that &nbsp;&nbsp;{{{sqrt(3 + 2sqrt(2))}}} = {{{1 + sqrt(2)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 2. &nbsp;Prove that &nbsp;&nbsp;{{{sqrt(7-4sqrt(3))}}} = {{{2-sqrt(3)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 3. &nbsp;Prove that &nbsp;&nbsp;{{{sqrt(2 + sqrt(3))}}} - {{{sqrt(2 - sqrt(3))}}} = {{{sqrt(2)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 4. &nbsp;Prove that &nbsp;&nbsp;{{{sqrt(3 + 2sqrt(2))}}} - {{{sqrt(3 - 2sqrt(2))}}} = {{{2}}}.

<A HREF=http://www.algebra.com/algebra/homework/real-numbers/Is-this-number-rational-or-irrational.lesson>Is this number rational or irrational?</A>

&nbsp;&nbsp;&nbsp;&nbsp; Problem 1. &nbsp;Simplify &nbsp;&nbsp;{{{sqrt(57 - 12sqrt(21))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 2. &nbsp;Is this number rational or irrational: &nbsp;&nbsp;{{{sqrt(52 - 14sqrt(3))}}} + {{{sqrt(3)}}} ?

&nbsp;&nbsp;&nbsp;&nbsp; Problem 3. &nbsp;Prove that this number is irrational: &nbsp;&nbsp;{{{sqrt(3)}}} + {{{sqrt(5)}}} + {{{sqrt(7)}}}.

<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/HOW-TO-make-the-denominator-of-a-fraction-free-of-square-roots.lesson>HOW TO rationalize a fraction by making its denominator free of square roots</A>

&nbsp;&nbsp;&nbsp;&nbsp; Example 1. &nbsp;Transform the fraction {{{1/(sqrt(2)-1)}}} to make it free of the square root in the denominator.

&nbsp;&nbsp;&nbsp;&nbsp; Example 2. &nbsp;Transform the fraction {{{(sqrt(3)+1)/(sqrt(3)-1)}}} to make it free of the square root in the denominator.

&nbsp;&nbsp;&nbsp;&nbsp; Example 3. &nbsp;Rationalize the fraction {{{sqrt(5)/(sqrt(5)+1)}}}.

<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/How-to-rationalize-a-fraction-by-making-its-denominator-free-of-cubic-roots.lesson>HOW TO rationalize a fraction by making its denominator free of cubic roots</A>

&nbsp;&nbsp;&nbsp;&nbsp; Example 1. &nbsp;Transform the fraction &nbsp;&nbsp;{{{1/(root(3,2)-1)}}} &nbsp;&nbsp;to make it free of the cubic root in the denominator.

&nbsp;&nbsp;&nbsp;&nbsp; Example 2. &nbsp;Transform the fraction &nbsp;&nbsp;{{{1/(root(3,2)+1)}}} &nbsp;&nbsp;to make it free of the cubic root in the denominator.

&nbsp;&nbsp;&nbsp;&nbsp; Example 3. &nbsp;Simplify the expression &nbsp;&nbsp;{{{1/(root(3,a^2) - root(3,a) + 1)}}}&nbsp;&nbsp; + &nbsp;&nbsp;{{{1/(root(3,a) + 1)}}}.

<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Amazing-calculations-with-fractions-that-contain-quadratic-irrationalities-in-denominators.lesson>Amazing calculations with fractions that contain quadratic irrationalities in denominators</A>

&nbsp;&nbsp;&nbsp;&nbsp; Problem 1. &nbsp;Simplify the fraction &nbsp;{{{1/(sqrt(2) + 1)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 2. &nbsp;Simplify and find the value of the sum of fractions &nbsp;{{{1/(sqrt(2) + 1)}}} + {{{1/(sqrt(3) + sqrt(2))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 3. &nbsp;Simplify and find the value of the sum of fractions &nbsp;{{{1/(sqrt(2) + 1)}}} + {{{1/(sqrt(3) + sqrt(2))}}} + {{{1/(sqrt(4) + sqrt(3))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 4. &nbsp;Simplify the sum of fractions &nbsp;{{{1/(sqrt(2) + 1)}}} + {{{1/(sqrt(3) + sqrt(2))}}} + {{{1/(sqrt(4) + sqrt(3))}}} + . . . + {{{1/(sqrt(n) + sqrt(n-1))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 5. &nbsp;Simplify the fraction &nbsp;{{{1/(sqrt(3) + 1)}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 6. &nbsp;Simplify and find the value of the sum of fractions &nbsp;{{{1/(sqrt(3) + 1)}}} + {{{1/(sqrt(5) + sqrt(3))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 7. &nbsp;Simplify and find the value of the sum of fractions &nbsp;{{{1/(sqrt(3) + 1)}}} + {{{1/(sqrt(5) + sqrt(3))}}} + {{{1/(sqrt(7) + sqrt(5))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 8. &nbsp;Simplify the sum of fractions &nbsp;{{{1/(sqrt(3) + 1)}}} + {{{1/(sqrt(5) + sqrt(3))}}} + {{{1/(sqrt(7) + sqrt(5))}}} + . . . + {{{1/(sqrt(2n+1) + sqrt(2n-1))}}}.

&nbsp;&nbsp;&nbsp;&nbsp; Problem 9. &nbsp;Simplify the sum of fractions &nbsp;{{{1/(sqrt(4) + 1)}}} + {{{1/(sqrt(7) + sqrt(4))}}} + {{{1/(sqrt(10) + sqrt(7))}}} + . . . + {{{1/(sqrt(3n+1) + sqrt(3(n-1)+1))}}}.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.