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RATIONAL AND IRRATIONAL NUMBERS • A rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. • An irrational number is a number that cannot be made into a fraction. Decimals that do not repeat or end are irrational numbers. Pi is an irrational number. • Any fraction can be changed into a decimal and any decimal can be changed into a fraction. This is because a decimal is based on the place values of tenths, hundredths, and thousandths etc., and most fractions can be changed to have a denominator of ten, hundred or thousand etc. o For the fractions that cannot easily be changed into 10, 100, or 1000, simple division will change the fraction into a decimal. o By being able to change fractions into decimals and vice versa, fractions and decimals can be compared easily. This enables fractions and decimals to be ordered. o When ordering fractions and decimals, it is most common to change the fractions to decimals and then put them in the correct order. • Mixed numbers are numbers that are the sum of a whole number and a fraction such as 1 4/5. o A mixed number can be changed into a fraction by multiplying the denominator by the whole number and then adding the numerator, this number becomes the new numerator and the denominator stays the same. The fraction, 1 4/5 would become 9/5. o A fraction in which the numerator is greater than the denominator is called an improper fraction. To change an improper fraction into a mixed decimal, the numerator is divided by the denominator to get the whole number and the remainder. With the whole number, the remainder becomes the numerator and the denominator stays the same. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

• A square root of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. A square root does not have to be a whole number. The square root of 1.44 is 1.2. How to use rational and irrational numbers: • Most fractions can be written as a decimal as long as the denominator can be changed to 10, 100, 1000 etc. • To change a denominator to 10, 100, or 1000, simply multiply the denominator by a factor of the number; if the denominator is 2, multiply 2 times 5 to get a new denominator of 10. Once the factor is found, multiply the numerator by this factor to get the new numerator, and then change this new fraction into a decimal. Example: Fraction: 3/5 multiply by 2: 6/10 changes into decimal: 0.6 • If a fraction does not easily change into 10, 100, 1000 etc., simple division will be used to change it into a decimal. The fraction, 12/16, can be made into a decimal by division. In order to divide, zeros must be added after a decimal point. Example: ___.75 12 → 16 ) 12.00 ←add zeros 16 - 112 80 - 80 0 • To change a decimal into a fraction, simply change the number into the numerator and the place value into the denominator. So .78 would change into the fraction 78/100. The new fraction should also then be put into lowest terms, 78/100 would equal 39/50. • When comparing fractions, all fractions should be changed into decimals. To compare .45 to 3/4, change 3/4 into .75 and then compare. The decimal, .45 < 3/4. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

• Mixed numbers can be changed into improper fractions. Example: 4 3/5 = (5)(4) + 3 = 23 (new denominator) or 23/5 • Improper fractions can be changed into mixed numbers. Example: Improper fraction, 78/7 → 7 √78 = 11 R1 → mixed number, 11 1/7 • Square roots are used in many different ways. They can be used to find the length of the sides of a square if the area is given. They can also be used to determine if a triangle is a right triangle. To determine which two numbers a square root is between, look for the closest two perfect squares. The √75 is between 8 and 9 because 8² is 64 and 9² is 81. When evaluating square roots, the answers may be rational or irrational numbers. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

Try This! Change the following into decimals: 6/20 __________ 8/12 ____________ 9/13 _________________ Change the following into fractions: .33333 _________ .825 __________ .125 ______________ Order the fractions and decimals from least to greatest: .45, 2/6, 3/5, .22 _______________ Change the following into improper fractions: 4 5/6 _____________ 3 7/8 ___________ 2 16/21 _____________ Change the following into mixed numbers: 45/10 ___________ 38/4 _____________ 29/3 ______________ Which of the following numbers are rational and which are irrational? 3/8, .252525, .87524136…, .6543 Compute the square roots of these numbers: √169________ √625___________ √2.25____________ What two numbers is √24 between? _____________ © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.