Rational vs. Irrational Numbers


 

Unit Overview

In this unit, students will be able to:

·        Differentiate different number sets

·        Solve square root and cube root equations

 

Key Concepts

·        Real numbers

·        Rational numbers

·        Irrational numbers

·        Integers

·        Whole numbers

·        Radicals

 

Ohio’s Learning Standards

·        8.NS.1 Know that real numbers are either rational or irrational. Understand informally that every number has a decimal expansion which is repeating, terminating, or is non-repeating and non-terminating.

·        8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, e.g., π². For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

·        8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

 

Calculators

·        Here is a link to the Desmos scientific calculator that is provided for the Ohio State Test for 8th Grade Mathematics in the spring.

·        You are strongly encouraged to use the Texas Instruments TI-30XIIS handheld scientific calculator.  It is extremely user friendly.

 

Number sets

·        As stated in the previous unit, real numbers are broken into two subsets:

 


·        Rational numbers have two subsets that we will examine:

o   Whole numbers:  0, 1, 2, 3,…

o   Integers:  whole numbers and their opposites

§  …, -3, -2, 1, 0, 1, 2, 3,…

§  The set of whole numbers is a subset (belongs to) the set of integers

·        All whole #s are integers, but not all integers are whole numbers

 

 

 

 

 

·        Example A set: Which sets does the number belong to? 

 

 

 

 

Here are the answers with a few notes:

 

 

 

For further explanation and practice on number sets:

 

 

Let’s practice classifying numbers into what sets that they belong to. 

 

 

 

 

 

 

Solving Square Root Equations

In Unit 3, we learned that a key step to solve most any algebraic equation is the property of equality, which allows one to perform the same operation to both sides of equation.  This method is used to get the variable by itself on one side of the equation by eliminating constants and coefficients.  Use the inverse operation to eliminate those numbers. 

·        Addition is the inverse operation of subtraction

 

·        Multiplication is the inverse operation of division

 

Example B set:  Look at these basic examples that illustrate the four Properties of Equality: 

x + 4 = 6                                c – 5 = 8                                                                   6h = -18

     x + 4 – 4 = 6 – 4                   c – 5 + 5 = 6 + 5                                           6h ÷ 6 = -18 ÷ 6

                   x = 2                                      c = 11                              y = 48                                      h = 3

 

This same principle can be applied to solve root equations:

·        Finding the square root is the inverse operation of finding the square

 

·        Finding the cube root is the inverse operation of finding the cube

 

Example C: 

x2 = 9  → You probably know that 32 = 9, but remember that (-3)2 = 9 as well.

The solutions are x = 3 or x = -3  →  this can also be written as x = ±3

 

·        To solve a basic equation involving a square:

o   Take the square root of both sides to undo the square

 

o   Also include the ± symbol when taking the square root to include both solutions

     x2 = 9

                                                    

                  x = ±3  →  32 = 3 • 3 = 9  and  (-3)2 = (-3) • (-3) = 9

 

 

Example D set:  Solving basic x2 (quadratic) equations

x2 = 36                             v2 + 8 = 10.56                            5c2 = 100                        n2 = -81

                                    -8        -8                                ÷5       ÷5                  

            x = ±6                                      v2 = 2.56                               c2 = 20                       n = no solutions

                                                                                                     

                                                        v = ±1.4                                 c = ±4.47 (rounded)       

                                                                                                        c = ± (exact)

 

Example E set:  Solving basic x3 (cubic) equations

x3 = 8  → You may know that 23 = 2•2•2 = 8

(-2)• (-2)•(-2) = -8 →-2 is not a solution for x3 = 8

·        To solve a basic equation involving a cube:

o   Take the cube root of both sides to undo the square → there is only 1 solution

x3 = 64                                         c3 = -8                                    n3 = -100

                                                                                                               

 x = 4                                             c = -2                                     n = -4.64 (rounded)         

                                                                                                             n =  (exact)

 

For further explanation and practice on solving root equations:

 

 

 

 

Let’s practice.