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Understanding Rational and Irrational Numbers

Learn to differentiate between rational and irrational numbers, identify irrational numbers, convert decimals to fractions, and practice with various examples.

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Understanding Rational and Irrational Numbers

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  1. Lesson 7 Rational and Irrational Numbers

  2. Numbers • Numbers can be classified as rational or irrational. What is the difference? • Rational • Integers- all positive, whole numbers, their opposites and zero. • Ratio- Any number that can be written in a ratio of 2 integers. • Terminating decimals- decimals that end • Repeating decimals- they have a digit that goes on forever.

  3. Then what’s irrational? • If a number can’t be written as the ratio of two integers, it is irrational. • All non-terminating, non-repeating decimals are irrational. • 3.14159265… we know this as pi. It is a non-repeating decimal. Its digits go on forever, but never repeat. • 1.7320508…is irrational because when written as a decimal its digits never end and never repeat. • There is no way to write non-terminating decimals as a ratio.

  4. Let’s Practice… • Identify all of the irrational numbers in the list below: • 3, ¼, 0, √8, √9 • First we need to figure out what √8 and √9 are equal to. √8 is about 2.82842712, and √9 is 3. • Which is irrational? • Is the decimal for √2 a repeating or non-repeating decimal? • Find its value.

  5. How do we make a repeating decimal into a fraction? • Usually when we change a decimal to a fraction, we put it over 10, 100 or 1,000. • .2= 2/10 or 1/5 • .57 = 57/100 • .649 = 649/1000 • How would we put .8(repeating) into a fraction? • Since it is repeating, there is not definite place value, what do we put it over? • The fraction. 1/9 has a repeating decimal of .1(repeating), so, the fraction we would write would be 8/9.

  6. Practice time! • What decimal represents √5? • What is the decimal equivalent to 4/9? • Which number is a rational number? 0.76, 0.83961257…, √10, √14 • What decimal is equivalent to 2/3? • An irrational number is….

  7. Last but not least • Look at the list of numbers below. Π, -0.005, -9/7, √12, √81 • Name two different irrational numbers in the list above. • Explain how you know that each number you chose is irrational.

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