Lesson OVERVIEW of lessons on Irrational numbers

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OVERVIEW of lessons on Irrational numbers


This file is an OVERVIEW of my lessons on Irrational numbers.
For your convenience,  it contains the list of the lessons and then the same list with short annotations to each lesson.
    - Proving irrationality of some real numbers
    - What number is greater? Comparing magnitude of irrational numbers
    - Calculations of expressions containing square roots
    - Prove that two irrational numbers are equal
    - Is this number rational or irrational?
    - HOW TO rationalize a fraction by making its denominator free of square roots
    - HOW TO rationalize a fraction by making its denominator free of cubic roots
    - Amazing calculations with fractions that contain quadratic irrationalities in denominators

List of lessons with short annotations

Proving irrationality of some real numbers

     Problems 1 - 3.  Prove that  sqrt%282%29  is an irrational number.              Prove that  sqrt%283%29  is an irrational number.              Prove that  sqrt%286%29  is an irrational number.

     Problems 4 - 6.  Prove that  root%283%2C2%29%29  is an irrational number.            Prove that  root%283%2C6%29%29  is an irrational number.            Prove that  root%283%2C9%29%29  is an irrational number.

     Problems 7 - 9.  Prove that  sqrt%282%29 + sqrt%283%29%29  is an irrational number.      Prove that  sqrt%281%2Bsqrt%282%29%29  is an irrational number.       Prove that  sqrt%282+%2B+root%283%2C2%29%29  is an irrational number.


What number is greater? Comparing magnitude of irrational numbers

     Problem 1.  What number is greater,  root%283%2C3%29  or  sqrt%282%29 ?

     Problem 2.  What number is greater,  root%285%2C5%29  or  sqrt%282%29 ?

     Problem 3.  Solve yourself without using a calculator:  what number is greater,  root%285%2C5%29  or  root%283%2C3%29 ?


Calculations of expressions containing square roots

     Problem 1.  Calculate   a)  sqrt%2813%5E2+-+12%5E2%29;   b)  sqrt%2825%5E2+-+24%5E2%29;   c)  sqrt%285%5E2+%2B+12%5E2%29;   d)  sqrt%288%5E2+%2B+15%5E2%29.

     Problem 2.  Calculate   a)  sqrt%284%2Bsqrt%2825%29%29;   b)  sqrt%289%2Bsqrt%2849%29%29;   c)  sqrt%283%2Bsqrt%2836%29%29;   d)  sqrt%287%2Bsqrt%2881%29%29;   e)  sqrt%287-sqrt%289%29%29;   f)  sqrt%28sqrt%2825%29%2Bsqrt%2816%29%29.

     Problem 3.  Simplify   a)  sqrt%28sqrt%2825%29-4%29;   b)  sqrt%28sqrt%2849%29-3%29;   c)  %28sqrt%285%29-1%29%2A%28sqrt%285%29%2B1%29.

     Problem 4.  Simplify   sqrt%282+%2B+sqrt%283%29%29 + sqrt%282+-+sqrt%283%29%29.      Problem 5.  Simplify   sqrt%282+%2B+sqrt%283%29%29 - sqrt%282+-+sqrt%283%29%29.

     Problem 6.  Solve yourself:   Simplify   a)   sqrt%283+%2B+2sqrt%282%29%29 + sqrt%283+-+2sqrt%282%29%29;   b)   sqrt%283+%2B+2sqrt%282%29%29 - sqrt%283+-+2sqrt%282%29%29;

     Problem 7.  Prove that   a)  sqrt%285+%2B+2sqrt%286%29%29 + sqrt%285+-+2sqrt%286%29%29= 2sqrt%283%29;   b)  sqrt%285+%2B+2sqrt%286%29%29 - sqrt%285+-+2sqrt%286%29%29= 2sqrt%282%29.

     Problem 8.  If  x = 7-4sqrt%283%29,  find the value   sqrt%28x%29 + 1%2Fsqrt%28x%29.


Prove that two irrational numbers are equal

     Problem 1.  Prove that   sqrt%283+%2B+2sqrt%282%29%29 = 1+%2B+sqrt%282%29.

     Problem 2.  Prove that   sqrt%287-4sqrt%283%29%29 = 2-sqrt%283%29.

     Problem 3.  Prove that   sqrt%282+%2B+sqrt%283%29%29 - sqrt%282+-+sqrt%283%29%29 = sqrt%282%29.

     Problem 4.  Prove that   sqrt%283+%2B+2sqrt%282%29%29 - sqrt%283+-+2sqrt%282%29%29 = 2.


Is this number rational or irrational?

     Problem 1.  Simplify   sqrt%2857+-+12sqrt%2821%29%29.

     Problem 2.  Is this number rational or irrational:   sqrt%2852+-+14sqrt%283%29%29 + sqrt%283%29 ?

     Problem 3.  Prove that this number is irrational:   sqrt%283%29 + sqrt%285%29 + sqrt%287%29.


HOW TO rationalize a fraction by making its denominator free of square roots

     Example 1.  Transform the fraction 1%2F%28sqrt%282%29-1%29 to make it free of the square root in the denominator.

     Example 2.  Transform the fraction %28sqrt%283%29%2B1%29%2F%28sqrt%283%29-1%29 to make it free of the square root in the denominator.

     Example 3.  Rationalize the fraction sqrt%285%29%2F%28sqrt%285%29%2B1%29.


HOW TO rationalize a fraction by making its denominator free of cubic roots

     Example 1.  Transform the fraction   1%2F%28root%283%2C2%29-1%29   to make it free of the cubic root in the denominator.

     Example 2.  Transform the fraction   1%2F%28root%283%2C2%29%2B1%29   to make it free of the cubic root in the denominator.

     Example 3.  Simplify the expression   1%2F%28root%283%2Ca%5E2%29+-+root%283%2Ca%29+%2B+1%29   +   1%2F%28root%283%2Ca%29+%2B+1%29.


Amazing calculations with fractions that contain quadratic irrationalities in denominators

     Problem 1.  Simplify the fraction  1%2F%28sqrt%282%29+%2B+1%29.

     Problem 2.  Simplify and find the value of the sum of fractions  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29.

     Problem 3.  Simplify and find the value of the sum of fractions  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29.

     Problem 4.  Simplify the sum of fractions  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29 + . . . + 1%2F%28sqrt%28n%29+%2B+sqrt%28n-1%29%29.

     Problem 5.  Simplify the fraction  1%2F%28sqrt%283%29+%2B+1%29.

     Problem 6.  Simplify and find the value of the sum of fractions  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29.

     Problem 7.  Simplify and find the value of the sum of fractions  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29.

     Problem 8.  Simplify the sum of fractions  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29 + . . . + 1%2F%28sqrt%282n%2B1%29+%2B+sqrt%282n-1%29%29.

     Problem 9.  Simplify the sum of fractions  1%2F%28sqrt%284%29+%2B+1%29 + 1%2F%28sqrt%287%29+%2B+sqrt%284%29%29 + 1%2F%28sqrt%2810%29+%2B+sqrt%287%29%29 + . . . + 1%2F%28sqrt%283n%2B1%29+%2B+sqrt%283%28n-1%29%2B1%29%29.


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