Basic Math BS | Ordering And Identifying Numbers | Exercise Questions With Solutions | Exercise Real Numbers

Basic Math BS | Ordering and Identifying Numbers | Exercise questions with solutions | Exercise Real Numbers

QUESTIONS

1. Select the numbers which are rational numbers.

1a. 2.2525252525

1b. 0.5555555

1c. √2

1d. 4.97654321

1e. √-100

2. If the decimal expansion of a rational number terminates after 5 decimal places, what can you say about its prime factorization? 

3. Put the numbers in order from least to greatest 64/8, √47,7.3, 48/4

4. Which of the numbers are irrational 64/8, √47,7.3, 48/4

5. Order the number from greatest to least 3/2, 27/5, √25,√ 33, 5.89

6. Name three different types of numbers that are considered rational

7. True or false? √43 is a rational number because it is a perfect square

8. Classify the following numbers:

8a. π/2

8b. √36

8c. 2.25111...

8d. √-5

8e. 75/-5

SOLUTIONS WITH EXPLANATION

1. Rational numbers are the ones that can be expressed as a ratio of two integers (where the denominator is not zero).

1a. 2.2525252525 is a rational number, since it can be expressed as the ratio of the integers 22525252525 and 10000000000.

1b. 0.5555555 is a rational number, since it can be expressed as the ratio of the integers 5 and 9.

1c. √2 is an irrational number, since it cannot be expressed as the ratio of two integers.

1d. 4.97654321 is a rational number, since it can be expressed as the ratio of the integers 997308641 and 200000000.

1e. √100 is a rational number, since it is equal to 10, which can be expressed as the ratio of the integers 10 and 1.

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2. If the decimal expansion of a rational number terminates after a finite number of decimal places, then it can be expressed as the ratio of two integers where the denominator has only prime factors of 2 and 5. This is because any rational number with a finite decimal expansion can be written as a fraction with a denominator that is a power of 10 (i.e. 10, 100, 1000, etc.), which has only prime factors of 2 and 5. For example, the rational number 1/8 can be expressed as 0.125, which terminates after three decimal places and can be written as the fraction 1/8 = 125/1000.

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3. (64/8, √47,7.3, 48/4) To put these numbers in order from least to greatest, we need to compare them.

64/8 = 8

√47 is between 6 and 7 (since 6^2 = 36 and 7^2 = 49)

7.3 is between 7 and 8

48/4 = 12

So the order from least to greatest is: 8, √47, 7.3, 12.

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4. (3/2, 27/5, √25,√ 33, 5.89) Only √47 is an irrational number. The other numbers (64/8, 7.3, and 48/4) are all rational numbers.

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5. To order these numbers from greatest to least, we can first simplify the expressions involving square roots:

√25 = 5

√33 is between 5 and 6 (since 5^2 = 25 and 6^2 = 36)

Then we have:

5.89v

3/2 = 1.5

27/5 = 5.

So the order from greatest to least is: √33, 5.4, 5, 1.5, 5.89.

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6. Three different types of numbers that are considered rational are:

1. Integers (e.g., -5, 0, 1, 2, 3)

2. Fractions (e.g., 3/4, -2/5, 1/2)

3. Decimal numbers that terminate or repeat (e.g., 0.5, 1.234, -0.7777...)

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7. False. √43 is an irrational number because it cannot be expressed as a fraction of two integers and it also cannot be simplified to a terminating or repeating decimal. It is not a perfect square because there is no integer n for which n^2 = 43.

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8a. π/2 is an irrational number. It cannot be expressed as a ratio of two integers and it also cannot be simplified to a terminating or repeating decimal. π/2 is approximately equal to 1.570796327... which is a non-repeating, non-terminating decimal.

8b. √36: Rational number (as it is a perfect square and can be expressed as 6)

8c. 2.25111...: Irrational number (as it cannot be expressed as a fraction of two integers and it also cannot be simplified to a terminating or repeating decimal)

8d. √-5: Imaginary number (as it involves the square root of a negative number)

8e. 75/-5: Rational number (as it can be simplified to -15)

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