Section 7.1: Classifying Real Numbers

Which of the following are real numbers? Select all that apply. Select all that apply: √-196 -√81 -√49 √-121

-√81 -√49 The square root of a negative number is not a real number. Therefore, −81−−√ and −49−−√ are the only real numbers in this list.

Which of the following is a whole number? Select all that apply: 13 −10 −6 7 0 29/4

13 7 0 Remember that whole numbers are non-negative integers (including 0), so the correct answers are 0, 7, and 13.

Which of the following is a whole number? −9 54 √6 16 0 −3

16 0 Remember that whole numbers are non-negative integers (including 0), so the correct answers are 0 and 16.

Which of the following is an integer?

9 Remember that integers are all of the whole numbers and their negatives (...,−3,−2,−1,0,1,2,3,...), so the correct answer is 9.

Which of the following is a whole number?

0,4,11 Remember that whole numbers are non-negative integers (including 0), so the correct answers are 0, 4, and 11.

Which of the following is an integer? Select all that apply: −10 62/7 32/7 −4 3.1232... 58/7

-10 -4 Remember that integers are all of the whole numbers and their negatives (...,−3,−2,−1,0,1,2,3,...), so the correct answers are −4 and −10.

Write −12 as a ratio of two integers.Enter your answer as a reduced improper fraction.

-12/1 We can write any integer as a fraction with a denominator of 1. It follows that −12=−12/1

Write 9.279 as a ratio of two integers.Enter your answer as a reduced improper fraction.

9279/1000

Write 4.399 as a ratio of two integers.Enter your answer as a reduced improper fraction.

4399/1000

Which of the following is an integer? Select the correct answer below: 9 32/5 48/5 −7.1 13/5 √-4

9 Remember that integers are all of the whole numbers and their negatives (...,−3,−2,−1,0,1,2,3,...), so the correct answer is 9.

Which of the following is a whole number? Select all that apply: 25/9 0 √679- 58/9 4 11

0,4,11 Remember that whole numbers are non-negative integers (including 0), so the correct answers are 0, 4, and 11.

Given the numbers 0.29 , 0.816 (repeat bar over 6) , 2.515115111... , 2.63(repeat bar over 3) , 0.125, and 0.418302... , select all of the rational numbers. Select all that apply:

0.29 0.816 (repeat bar over 6) , 2.63(repeat bar over 3) 0.125 The numbers 0.418302... and 2.515115111... are the only irrational numbers because they do not repeat and do not stop.

Given the numbers 0.29 , 0.816¯¯¯ , 2.515115111... , 2.63¯¯¯ , 0.125, and 0.418302... , select all of the rational numbers. Select all that apply: 0.29 0.816¯¯¯(repeat bar on 6) 2.515115111... 2.63¯¯¯(repeat bar on 3) 0.125 0.418302...

0.29 0.816¯¯¯(repeat bar on 6) 2.63¯¯¯(repeat bar on 3) 0.125 The numbers 0.418302... and 2.515115111... are the only irrational numbers because they do not repeat and do not stop.

Which of the numbers given are rational? Select all that apply: 0.6560 29.46660 4.381293... 31.7408¯¯¯(repeat bar on 8) 21.90062¯¯¯¯¯(repeat bar on 62)

0.6560 29.46660 31.7408¯¯¯(repeat bar on 8) 21.90062¯¯¯¯¯(repeat bar on 62) Let's take a closer look at each of the answer options. Since the decimal part of 0.6560 terminates, 0.6560 is rational. Since the decimal part of 29.46660 terminates, 29.46660 is rational. The decimal part of 4.381293... has no repeating block of digits and does not terminate. So, 4.381293... is irrational. The decimal part of 31.7408¯¯¯ that is underneath the line repeats, so 31.7408¯¯¯ is rational. The decimal part of 21.90062¯¯¯¯¯ that is underneath the line repeats, so 21.90062¯¯¯¯¯ is rational.

Which of the numbers given are rational? Select all that apply: 1.421490... 18.12270¯¯¯¯¯(repeat bar on 70) 32.74178¯¯¯¯¯(repeat bar on 78) 30.346203... 33.2406¯¯¯(repeat bar on 6)

18.12270¯¯¯¯¯(repeat bar on 70) 32.74178¯¯¯¯¯(repeat bar on 78) 33.2406¯¯¯(repeat bar on 6) Let's take a closer look at each of the answer options. The decimal part of 1.421490... has no repeating block of digits and does not terminate. So, 1.421490... is irrational. The decimal part of 18.12270¯¯¯¯¯ that is underneath the line repeats, so 18.12270¯¯¯¯¯ is rational. The decimal part of 32.74178¯¯¯¯¯ that is underneath the line repeats, so 32.74178¯¯¯¯¯ is rational. The decimal part of 30.346203... has no repeating block of digits and does not terminate. So, 30.346203... is irrational. The decimal part of 33.2406¯¯¯ that is underneath the line repeats, so 33.2406¯¯¯ is rational.

Which of the numbers given are rational? 18.12760¯¯¯¯¯(repeat bar on 60) 25.4777 √121 19.268198¯¯¯¯¯¯¯¯(repeat bar on 198) 12.15167¯¯¯¯¯¯¯¯(repeat bar on 167)

18.12760¯¯¯¯¯(repeat bar on 60) √121 25.4777 19.268198¯¯¯¯¯¯¯¯(repeat bar on 198) 12.15167¯¯¯¯¯¯¯¯(repeat bar on 167) Let's take a closer look at each of the answer options. The decimal part of 18.12760¯¯¯¯¯ that is underneath the line repeats, so 18.12760¯¯¯¯¯ is rational. Recognize that 121 is a perfect square since 112=121. So, 121−−−√=11, and it follows that 121−−−√ is rational. Since the decimal part of 25.4777 terminates, 25.4777 is rational. The decimal part of 19.268198¯¯¯¯¯¯¯¯ that is underneath the line repeats, so 19.268198¯¯¯¯¯¯¯¯ is rational. The decimal part of 12.15167¯¯¯¯¯¯¯¯ that is underneath the line repeats, so 12.15167¯¯¯¯¯¯¯¯ is rational.

Which of the numbers given are rational? 19.30764¯¯¯¯¯(repeat on 64) 35.45449 10.210033... 18.145393... √1

19.30764¯¯¯¯¯(repeat on 64) 35.45449 Let's take a closer look at each of the answer options. The decimal part of 19.30764¯¯¯¯¯ that is underneath the line repeats, so 19.30764¯¯¯¯¯ is rational. Since the decimal part of 35.45449 terminates, 35.45449 is rational. The decimal part of 10.210033... has no repeating block of digits and does not terminate. So, 10.210033... is irrational. The decimal part of 18.145393... has no repeating block of digits and does not terminate. So, 18.145393... is irrational. Recognize that 1 is a perfect square since 12=1. So, 1-√=1, and it follows that 1-√ is rational.

Which of the following numbers are irrational numbers? 21.10675..., 89, √121 ,0.011, √526, −9.72827...

21.10675... √526 −9.72827... The irrational include all real numbers that are not rational. Since the rational numbers include all integers, as well as fractions and decimals that stop or repeat, we conclude that the numbers 21.10675..., 526−−−√, and −9.72827... are irrational numbers.

Which of the following numbers are irrational numbers? 21.10675... 89 √121 0.011 √526 −9.72827...

21.10675... √526 −9.72827... The irrational include all real numbers that are not rational. Since the rational numbers include all integers, as well as fractions and decimals that stop or repeat, we conclude that the numbers 21.10675..., √526, and −9.72827... are irrational numbers.

Which of the numbers given are rational? 6.87441 √3,100 √49 28.788712... √144

6.87441 √49 √144 Let's take a closer look at each of the answer options. Since the decimal part of 6.87441 terminates, 6.87441 is rational. Remember that 552=3,025 and 562=3,136, so 3,100 is not a perfect square. Therefore, the decimal form of 3,100−−−−√ will never repeat and never stop, so 3,100−−−−√ is irrational. Recognize that 49 is a perfect square since 72=49. So, 49−−√=7, and it follows that 49−−√ is rational. The decimal part of 28.788712... has no repeating block of digits and does not terminate. So, 28.788712... is irrational. Recognize that 144 is a perfect square since 122=144. So, 144−−−√=12, and it follows that 144−−−√ is rational.

Write as the ratio of two integers: 8.41

841/100 We want to write 8.41 as the ratio of two integers. First, write it as a mixed number. Remember that 8 is the whole number and the decimal part, 0.41, indicates hundredths. 8 and 41/100 Then convert to an improper fraction. 841/100

Which of the following sets does the number 265 belong to? Select all that apply: Integers Irrational numbers Rational numbers Real numbers Whole numbers

Integers Rational numbers Real numbers Whole numbers The whole numbers are 0, 1, 2, 3, ..., so the number 265 is a whole number. Since any whole number is an integer, any integer is a rational number, and any rational number is a real number we conclude that 265 belongs to the set of Whole numbers, Integers, Rational numbers, and Real numbers.

Question: For each number given, identify whether it is a real number or not a real number: a. √-169. b.√-64.

Solution: a. There is no real number whose square is −169. Therefore, √-169 is not a real number. b. Since the negative is in front of the radical, −√64 is −8. Since −8 is a real number, −√64 is a real number.

Which of the following sets does the number −8 belong to?

integers rational numbers real numbers The whole numbers are 0,1,2,3,..., so −8 does not belong to the set of whole numbers.The integers are the whole numbers and their opposites, so −8 does belong to the set of integers.All integers are rational, so −8 does belong to the set of rational numbers.Because −8 is rational, it does not belong to the set of irrational numbers.All rational numbers are real numbers, so −8 does belong to the set of real numbers.In summary, −8 belongs to the following sets: integers, rational numbers, and real numbers.

Which of the following sets does the number -√3 belong to? Select all that apply: whole numbers integers rational numbers irrational numbers real numbers none of the above

irrational numbers real numbers The whole numbers are 0,1,2,3,..., so -√3 does not belong to the set of whole numbers.The integers are the whole numbers and their opposites, so -√3 does not belong to the set of integers.Rational numbers include decimals that repeat or stop. Because -√3=−0.732050808..., it does not repeat and does not belong to the set of rational numbers.Because -√3 written as a decimal number does not stop or repeat, it belongs to the set of irrational numbers.All irrational numbers are real numbers, so -√3 does belong to the set of real numbers.In summary,-√3 belongs to the following sets: irrational numbers and real numbers.

Which of the following sets does the number −√3 belong to? whole numbers integers rational numbers irrational numbers real numbers none of the above

irrational numbers real numbers The whole numbers are 0,1,2,3,..., so −3-√ does not belong to the set of whole numbers.The integers are the whole numbers and their opposites, so −3-√ does not belong to the set of integers.Rational numbers include decimals that repeat or stop. Because −3-√=−0.732050808..., it does not repeat and does not belong to the set of rational numbers.Because −3-√ written as a decimal number does not stop or repeat, it belongs to the set of irrational numbers.All irrational numbers are real numbers, so −3-√ does belong to the set of real numbers.In summary, −3-√ belongs to the following sets: irrational numbers and real numbers.

Which of the following sets does the number 2 and 1/3 belong to? Select all that apply: whole numbers integers rational numbers irrational numbers real numbers none of the above

rational numbers real numbers The whole numbers are 0,1,2,3,..., so 2 and 1/3 does not belong to the set of whole numbers.The integers are the whole numbers and their opposites, so 2 and 1/3 does not belong to the set of integers.Rational numbers include decimals that repeat or stop. Because 2and 1/3=2.3¯¯¯(repeat bar on 3), it does belong to the set of rational numbers.Because 2 and 1/3 is rational, it does not belong to the set of irrational numbers.All rational numbers are real numbers, so 213 does belong to the set of real numbers.In summary, 2 and 1/3 belongs to the following sets: rational numbers and real numbers.

Which of the following sets does the number 2 belong to?

whole numbers integers rational numbers real numbers The whole numbers are 0,1,2,3,..., so 2 belongs to the set of whole numbers.The integers are the whole numbers and their opposites, so 2 belongs to the set of integers.Rational numbers include decimal numbers that repeat or stop. Because 2 stops, it belongs to the set of rational numbers.Irrational numbers do not stop or repeat when written as decimal numbers, so 2 does not belong to the set of irrational numbers.All rational numbers are real numbers, so 2 belongs to the set of real numbers.In summary, 2 belongs to the following sets: whole numbers, integers, rational numbers, and real numbers.

Which of the following sets does the number 2 belong to? whole numbers integers rational numbers irrational numbers real numbers none of the above

whole numbers integers rational numbers real numbers The whole numbers are 0,1,2,3,..., so 2 belongs to the set of whole numbers.The integers are the whole numbers and their opposites, so 2 belongs to the set of integers.Rational numbers include decimal numbers that repeat or stop. Because 2 stops, it belongs to the set of rational numbers.Irrational numbers do not stop or repeat when written as decimal numbers, so 2 does not belong to the set of irrational numbers.All rational numbers are real numbers, so 2 belongs to the set of real numbers.In summary, 2 belongs to the following sets: whole numbers, integers, rational numbers, and real numbers.

Which of the following sets does the number √49 belong to?

whole numbers integers rational numbers real numbers The whole numbers are 0,1,2,3,..., so 49−−√=7 belongs to the set of whole numbers.The integers are the whole numbers and their opposites, so 49−−√ belongs to the set of integers.Rational numbers include decimal numbers that repeat or stop. Because 49−−√=7 stops, it belongs to the set of rational numbers.Irrational numbers do not stop or repeat when written as decimal numbers, so 49−−√ does not belong to the set of irrational numbers.All rational numbers are real numbers, so 49−−√ belongs to the set of real numbers.In summary, 49−−√ belongs to the following sets: whole numbers, integers, rational numbers, and real numbers.

Which of the following is an integer? Select all that apply: −4 1.765¯¯¯¯¯¯¯¯(repeat on 765) 6 56/9 −2 10/9

−4 6 −2 Remember that integers are all of the whole numbers and their negatives (...,−3,−2,−1,0,1,2,3,...), so the correct answers are −4, −2, and 6.

Which of the following are real numbers? Select all that apply. √-196 −√81 -√49 √-121

−√81 -√49 The square root of a negative number is not a real number. Therefore, −81−−√ and −49−−√ are the only real numbers in this list.

Which of the numbers given are rational? Select all that apply: 19.891317... 18.441649... √16 15.567 27.20058

√16 15.567 27.20058 Let's take a closer look at each of the answer options. The decimal part of 19.891317... has no repeating block of digits and does not terminate. So, 19.891317... is irrational. The decimal part of 18.441649... has no repeating block of digits and does not terminate. So, 18.441649... is irrational. Recognize that 16 is a perfect square since 42=16. So, 16−−√=4, and it follows that 16−−√ is rational. Since the decimal part of 15.567 terminates, 15.567 is rational. Since the decimal part of 27.20058 terminates, 27.20058 is rational.

Which of the numbers given are rational? Select all that apply: 19.891317... 18.441649... √16 15.567 27.20058

√16 15.567 27.20058 Let's take a closer look at each of the answer options. The decimal part of 19.891317... has no repeating block of digits and does not terminate. So, 19.891317... is irrational. The decimal part of 18.441649... has no repeating block of digits and does not terminate. So, 18.441649... is irrational. Recognize that 16 is a perfect square since 42=16. So, 16−−√=4, and it follows that √16 is rational. Since the decimal part of 15.567 terminates, 15.567 is rational. Since the decimal part of 27.20058 terminates, 27.20058 is rational.

Which of the numbers given are rational? √16 16.636291 17.85199¯¯¯¯¯(repeat bar on 99) 23.901212¯¯¯¯¯¯¯¯(repeat bar on 212) 18.788042

√16 17.85199¯¯¯¯¯(repeat bar on 99) 23.901212¯¯¯¯¯¯¯¯(repeat bar on 212) Let's take a closer look at each of the answer options. Recognize that 16 is a perfect square since 4^2=16. So, √16=4, and it follows that √16 is rational. The decimal part of 16.636291... has no repeating block of digits and does not terminate. So, 16.636291... is irrational. The decimal part of 17.85199¯¯¯¯¯ that is underneath the line repeats, so 17.85199¯¯¯¯¯ is rational. The decimal part of 23.901212¯¯¯¯¯¯¯¯ that is underneath the line repeats, so 23.901212¯¯¯¯¯¯¯¯ is rational. The decimal part of 18.788042... has no repeating block of digits and does not terminate. So, 18.788042... is irrational.

Which of the numbers given are rational? Select all that apply: √169 25.66830 33.989361 5.50233¯¯¯¯¯¯¯¯ (repeat bar on 233) 20.146209

√169 25.66830 5.50233¯¯¯¯¯¯¯¯ (repeat bar on 233) Let's take a closer look at each of the answer options. Recognize that 169 is a perfect square since 13^2=169. So,√169 =13, and it follows that √169 is rational. Since the decimal part of 25.66830 terminates, 25.66830 is rational. The decimal part of 33.989361... has no repeating block of digits and does not terminate. So, 33.989361... is irrational. The decimal part of 5.50233¯¯¯¯¯¯¯¯ that is underneath the line repeats, so 5.50233¯¯¯¯¯¯¯¯ is rational. The decimal part of 20.146209... has no repeating block of digits and does not terminate. So, 20.146209... is irrational.

Which of the numbers given are rational? Select all that apply: √25 1.54422 √200 8.21677 √49

√25 1.54422 8.21677 √49 Let's take a closer look at each of the answer options. Recognize that 25 is a perfect square since 52=25. So, 25−−√=5, and it follows that 25−−√ is rational. Since the decimal part of 1.54422 terminates, 1.54422 is rational. Remember that 142=196 and 152=225, so 200 is not a perfect square. Therefore, the decimal form of 200−−−√ will never repeat and never stop, so 200−−−√ is irrational. Since the decimal part of 8.21677 terminates, 8.21677 is rational. Recognize that 49 is a perfect square since 72=49. So, 49−−√=7, and it follows that 49−−√ is rational.

Which of the numbers given are rational? √25 1.54422 √200 8.21677 √49

√25 √49 1.54422 8.21677 Let's take a closer look at each of the answer options. Recognize that 25 is a perfect square since 5^2=25. So, √25=5, and it follows that √25 is rational. Since the decimal part of 1.54422 terminates, 1.54422 is rational. Remember that 14^2=196 and 15^2=225, so 200 is not a perfect square. Therefore, the decimal form of √200 will never repeat and never stop, so √200 is irrational. Since the decimal part of 8.21677 terminates, 8.21677 is rational. Recognize that 49 is a perfect square since 7^2=49. So, √49=7, and it follows that √49 is rational.

Given the numbers √36, √44, √81, and √17, **select all of the irrational numbers.**

√44, (ends up in a long decimal answer) √17, (ends up in a long decimal answer) The numbers 36−−√and 81−−√ are the only rational numbers because they simplify to 6 and 9, respectively.

Which of the numbers given are rational? Select all that apply: √49 31.460145¯¯¯¯¯¯¯¯(repeat on 145) 10.611184... √261 0.754448...

√49 31.460145¯¯¯¯¯¯¯¯(repeat on 145) Let's take a closer look at each of the answer options. Recognize that 49 is a perfect square since 72=49. So, 49−−√=7, and it follows that 49−−√ is rational. The decimal part of 31.460145¯¯¯¯¯¯¯¯ that is underneath the line repeats, so 31.460145¯¯¯¯¯¯¯¯ is rational. The decimal part of 10.611184... has no repeating block of digits and does not terminate. So, 10.611184... is irrational. Remember that 162=256 and 172=289, so 261 is not a perfect square. Therefore, the decimal form of 261−−−√ will never repeat and never stop, so 261−−−√ is irrational. The decimal part of 0.754448... has no repeating block of digits and does not terminate. So, 0.754448... is irrational.

Which of the numbers given are rational? 26.384954... 1.261091... √81 15.46274 √486

√81 15.46274 Let's take a closer look at each of the answer options. The decimal part of 26.384954... has no repeating block of digits and does not terminate. So, 26.384954... is irrational. The decimal part of 1.261091... has no repeating block of digits and does not terminate. So, 1.261091... is irrational. Recognize that 81 is a perfect square since 92=81. So, 81−−√=9, and it follows that 81−−√ is rational. Since the decimal part of 15.46274 terminates, 15.46274 is rational. Remember that 222=484 and 232=529, so 486 is not a perfect square. Therefore, the decimal form of 486−−−√ will never repeat and never stop, so 486−−−√ is irrational.

Which of the following is a whole number? 5 0 10 −9.0 19/3 −7

5,0,10 Remember that whole numbers are non-negative integers (including 0), so the correct answers are 0, 5, and 10.

Which of the numbers given are rational? Select all that apply: 1.421490... 18.12270¯¯¯¯¯(repeat bar over 70) 32.74178¯¯¯¯¯(repeat bar over 78) 30.346203... 33.2406¯¯¯(repeat bar over 6)

18.12270¯¯¯¯¯(repeat bar over 70) 32.74178¯¯¯¯¯(repeat bar over 78) 33.2406¯¯¯(repeat bar over 6) Let's take a closer look at each of the answer options. The decimal part of 1.421490... has no repeating block of digits and does not terminate. So, 1.421490... is irrational. The decimal part of 18.12270¯¯¯¯¯ that is underneath the line repeats, so 18.12270¯¯¯¯¯ is rational. The decimal part of 32.74178¯¯¯¯¯ that is underneath the line repeats, so 32.74178¯¯¯¯¯ is rational. The decimal part of 30.346203... has no repeating block of digits and does not terminate. So, 30.346203... is irrational. The decimal part of 33.2406¯¯¯ that is underneath the line repeats, so 33.2406¯¯¯ is rational.

Write 3.19 as a ratio of two integers.Enter your answer as a reduced improper fraction.

319/100

Which of the following sets does the number −12.12532... belong to? Select all that apply: whole numbers integers rational numbers irrational numbers real numbers none of the above

irrational numbers real numbers The whole numbers are 0,1,2,3,..., so −12.12532... does not belong to the set of whole numbers.The integers are the whole numbers and their opposites, so −12.12532... does not belong to the set of integers.Rational numbers include decimals that repeat or stop. Because −12.12532... does not stop or repeat, it does not belong to the set of rational numbers.Irrational numbers do not stop or repeat when written as decimal numbers, so −12.12532... belongs to the set of irrational numbers.All irrational numbers are real numbers, so −12.12532... belongs to the set of real numbers.In summary, −12.12532... belongs to the following sets: irrational numbers and real numbers.

Which of the following sets does the number 0.1453¯¯¯¯¯ (repeat bar on 53) belong to? Select all that apply: whole numbers integers rational numbers irrational numbers real numbers none of the above

rational numbers real numbers The whole numbers are 0,1,2,3,..., so 0.1453¯¯¯¯¯ (repeat bar on 53)does not belong to the set of whole numbers. The integers are the whole numbers and their opposites, so 0.1453¯¯¯¯¯(repeat bar on 53) does not belong to the set of integers. Rational numbers include decimals that repeat or stop. Because 0.1453¯¯¯¯¯(repeat bar on 53) is a repeating decimal number, it belongs to the set of rational numbers. Because 0.1453¯¯¯¯¯(repeat bar on 53) is a rational number, it does not belong to the set of irrational numbers. All rational numbers are real numbers, so 0.1453¯¯¯¯¯ (repeat bar on 53) belongs to the set of real numbers.In summary, 0.1453¯¯¯¯¯(repeat bar on 53) belongs to the following sets: rational numbers and real numbers.

Which of the following sets does the number 12/5 belong to? Select all that apply: whole numbers integers rational numbers irrational numbers real numbers none of the above

rational numbers real numbers The whole numbers are 0,1,2,3,..., so 12/5 does not belong to the set of whole numbers.The integers are the whole numbers and their opposites, so 12/5 does not belong to the set of integers.Rational numbers include decimals that repeat or stop. Because 12/5=2.4, it does belong to the set of rational numbers.Because 12/5 is rational, it does not belong to the set of irrational numbers.All rational numbers are real numbers, so 12/5 does belong to the set of real numbers.In summary, 12/5 belongs to the following sets: rational numbers and real numbers.

Given the numbers √36, √44, √,81 and √17, select all of the irrational numbers.

√44, √17, The numbers 36−−√and 81−−√ are the only rational numbers because they simplify to 6 and 9, respectively.