IRRATIONAL NUMBERSClassifying and Ordering Numbers

TODAY’S OBJECTIVES

Students will be able to demonstrate an understanding of irrational numbers by:1. Representing, identifying, and simplifying irrational

numbers by sorting a set of numbers into rational and irrational sets and determining an approximate value of a given irrational number

2. Ordering irrational numbers by approximating the locations of irrational numbers on a number line, and ordering a set of irrational numbers on a number line

IRRATIONAL NUMBERS Rational numbers are numbers that can be written in the

form of a fraction or ratio, or more specifically as a quotient of integers

Any number that cannot be written as a quotient of integers is called an irrational number

∏ is one example of an irrational number…. Can you think of any more? √0.24, 3√9, √2, √1/3, 4√12, e

Some examples of rational numbers? √100, √0.25, 3√8, 0.5, 5/6, 7, 5√-32

Using your calculators, find the approximate decimal value of each of these numbers to 5 or 6 decimal places. What do you notice?

RATIONAL VS. IRRATIONAL NUMBERS

You should have noticed that the decimal representation of a rational number either terminates, or repeats 0.5, 1.25, 3.675 1.3333…., 2.14141414…..

The decimal representation of an irrational number neither terminates nor repeats 3.14159265358………..

Which of these numbers are rational numbers and which are irrational numbers? √1.44, √64/81, 3√-27, √4/5, √5 √1.44, √64/81, 3√-27, √4/5, √5

EXACT VALUES VS. APPROXIMATE VALUES When an irrational number is written as a radical, for example;

√2 or 3√-50, we say the radical is the exact value of the irrational number. When we use a calculator to find the decimal value, we say this is an approximate value

We can approximate the location of an irrational number on a number line

EXAMPLE

If we do not have a calculator, we can use perfect powers to estimate the value of an irrational number:

Locate 3√-50 on a number line. We know that 3√-27 = -3, and 3√-64 = -4 Guess: 3√-50 ≈ -3.6 Test: (-3.6)3 = -46.656 Guess 3√-50 ≈ -3.7 Test: (-3.7)3 = -50.653

This is close enough to represent on a number line.

SUMMARY OF NUMBER SETS

Rational Numbers

Whole Numbers

Natural Numbers

Irrational Numbers

Real Numbers

EXAMPLE 2

Order these numbers on a number line from least to greatest

3√13,√18,√9,4√27,3√-5 Solution: 3√13 ≈ 2.3513… √18 ≈ 4.2426… √9 = 3 4√27 ≈

2.2795… 3√-5 ≈ -1.7099… From least to greatest: 3√-5, 4√27, 3√13, √9, √18

REVIEW

Written as a decimal number, rational numbers either: Repeat Terminate

Rational numbers can be written as a quotient of integers

Written as a decimal number, irrational numbers neither repeat or terminate

Irrational numbers cannot be written as a quotient of integers

All rational and irrational numbers are included in the set of real numbers

HOMEWORK

Pg. 211-2133,4, 9, 10b, 15, 17-20, 22