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Irrational Numbers

Irrational Numbers. Classifying and Ordering Numbers. Today’s Objectives. Students will be able to demonstrate an understanding of irrational numbers by:

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Irrational Numbers

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  1. Irrational Numbers Classifying and Ordering Numbers

  2. Today’s Objectives • Students will be able to demonstrate an understanding of irrational numbers by: • 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 • 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

  3. 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?

  4. 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

  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

  6. 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.

  7. Summary of Number Sets Real Numbers Irrational Numbers

  8. 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

  9. 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

  10. Homework • Pg. 211-213 • 3,4, 9, 10b, 15, 17-20, 22

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