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
is an irrational number. Prove that
is an irrational number. Prove that
is an irrational number.
Problems 4 - 6
. Prove that
is an irrational number. Prove that
is an irrational number. Prove that
is an irrational number.
Problems 7 - 9
. Prove that
+
is an irrational number. Prove that
is an irrational number. Prove that
is an irrational number.
What number is greater? Comparing magnitude of irrational numbers
Problem 1
. What number is greater,
or
?
Problem 2
. What number is greater,
or
?
Problem 3
. Solve yourself without using a calculator: what number is greater,
or
?
Calculations of expressions containing square roots
Problem 1
. Calculate a)
; b)
; c)
; d)
.
Problem 2
. Calculate a)
; b)
; c)
; d)
; e)
; f)
.
Problem 3
. Simplify a)
; b)
; c)
.
Problem 4
. Simplify
+
.
Problem 5
. Simplify
-
.
Problem 6
. Solve yourself: Simplify a)
+
; b)
-
;
Problem 7
. Prove that a)
+
=
; b)
-
=
.
Problem 8
. If
=
, find the value
+
.
Prove that two irrational numbers are equal
Problem 1
. Prove that
=
.
Problem 2
. Prove that
=
.
Problem 3
. Prove that
-
=
.
Problem 4
. Prove that
-
=
.
Is this number rational or irrational?
Problem 1
. Simplify
.
Problem 2
. Is this number rational or irrational:
+
?
Problem 3
. Prove that this number is irrational:
+
+
.
HOW TO rationalize a fraction by making its denominator free of square roots
Example 1
. Transform the fraction
to make it free of the square root in the denominator.
Example 2
. Transform the fraction
to make it free of the square root in the denominator.
Example 3
. Rationalize the fraction
.
HOW TO rationalize a fraction by making its denominator free of cubic roots
Example 1
. Transform the fraction
to make it free of the cubic root in the denominator.
Example 2
. Transform the fraction
to make it free of the cubic root in the denominator.
Example 3
. Simplify the expression
+
.
Amazing calculations with fractions that contain quadratic irrationalities in denominators
Problem 1
. Simplify the fraction
.
Problem 2
. Simplify and find the value of the sum of fractions
+
.
Problem 3
. Simplify and find the value of the sum of fractions
+
+
.
Problem 4
. Simplify the sum of fractions
+
+
+ . . . +
.
Problem 5
. Simplify the fraction
.
Problem 6
. Simplify and find the value of the sum of fractions
+
.
Problem 7
. Simplify and find the value of the sum of fractions
+
+
.
Problem 8
. Simplify the sum of fractions
+
+
+ . . . +
.
Problem 9
. Simplify the sum of fractions
+
+
+ . . . +
.
Use this file/link
ALGEBRA-I - YOUR ONLINE TEXTBOOK
to navigate over all topics and lessons of the online textbook ALGEBRA-I.
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