a. Explanation In order to convert the numeral to base 3 follows the given steps below, The positional value in the base 3 system are as follows, ... , 3 3 , 3 2 , 3 , 1 or ... , 27 , 9 , 3 , 1 . The number is 1. 1 in base 3 is 1 3 . The number is 2. 2 in base 3 is 2 3 . The number is 3. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 3 is 3. Then, 3 ÷ 3 = 1 with remainder 0 0 ÷ 1 = 0 with remainder 0 Since 0 is less than the base 3, no further division required. The numeral 3 with base 3 is expressed as follows, 3 = ( 1 × 3 ) + ( 0 × 1 ) = ( 10 ) 3 3 in base 3 is ( 10 ) 3 . The number is 4 in base computed as follows, From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 4 is 3. Then, 4 ÷ 3 = 1 with remainder 1 1 ÷ 1 = 1 with remainder 0 Since 1 is less than the base 3, no further division required. The numeral 4 with base 3 is expressed as follows, 4 = ( 1 × 3 ) + ( 1 × 1 ) = ( 11 ) 3 4 in base 3 is ( 11 ) 3 . The number is 5. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 5 is 3. Then, 5 ÷ 3 = 1 with remainder 2 2 ÷ 1 = 2 with remainder 0 Since 2 is less than the base 3, no further division required. The numeral 5 with base 3 is expressed as follows, 5 = ( 1 × 3 ) + ( 2 × 1 ) = ( 12 ) 3 5 in base 3 is ( 12 ) 3 . The number is 6. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 6 is 3. Then, 6 ÷ 3 = 2 with remainder 0 0 ÷ 1 = 0 with remainder 0 Since 0 is less than the base 3, no further division required. The numeral 6 with base 3 is expressed as follows, 6 = ( 2 × 3 ) + ( 0 × 1 ) = ( 20 ) 3 6 in base 3 is ( 20 ) 3 . The number is 7. Then, From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 7 is 3. 7 ÷ 3 = 2 with remainder 1 1 ÷ 1 = 1 with remainder 0 Since 1 is less than the base 3, no further division required. The numeral 7 with base 3 is expressed as follows, 7 = ( 2 × 3 ) + ( 1 × 1 ) = ( 21 ) 3 7 in base 3 is ( 21 ) 3 . The number is 8. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 8 is 3. Then, 8 ÷ 3 = 2 with remainder 2 2 ÷ 1 = 2 with remainder 0 Since 2 is less than the base 3, no further division required. The numeral 8 with base 3 is expressed as follows, 8 = ( 2 × 3 ) + ( 2 × 1 ) = ( 22 ) 3 8 in base 3 is ( 22 ) 3 . The number is 9. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 9 is 3 2 or 9. Then, 9 ÷ 9 = 1 with remainder 0 0 ÷ 3 = 0 with remainder 0 0 ÷ 1 = 0 with remainder 0 Since 0 is less than the base 3, no further division required. The numeral 9 with base 3 is expressed as follows, 9 = ( 1 × 9 ) + ( 0 × 3 ) + ( 0 × 1 ) = ( 100 ) 3 9 in base 3 is ( 100 ) 3 . The number is 10. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 10 is 3 2 or 9. Then, 10 ÷ 9 = 1 with remainder 1 1 ÷ 3 = 0 with remainder 1 1 ÷ 1 = 1 with remainder 0 Since 1 is less than the base 3, no further division required. The numeral 10 with base 3 is expressed as follows, 10 = ( 1 × 9 ) + ( 0 × 3 ) + ( 1 × 1 ) = ( 101 ) 3 10 in base 3 is ( 101 ) 3 . The number is 11. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 11 is 3 2 or 9. Then, 11 ÷ 9 = 1 with remainder 2 2 ÷ 3 = 0 with remainder 2 2 ÷ 1 = 2 with remainder 0 Since 2 is less than the base 3, no further division required. The numeral 11 with base 3 is expressed as follows, 11 = ( 1 × 9 ) + ( 0 × 3 ) + ( 2 × 1 ) = ( 102 ) 3 11 in base 3 is ( 102 ) 3 . The number is 12. From above positional values in the base 3 it is seen that the highest power of base 3 is less than or equal to 12 is 3 2 or 9. Then, 12 ÷ 9 = 1 with remainder 3 3 ÷ 3 = 1 with remainder 0 0 ÷ 1 = 0 with remainder 0 Since 0 is less than the base 3, no further division required. The numeral 12 with base 3 is expressed as follows, 12 = ( 1 × 9 ) + ( 1 × 3 ) + ( 0 × 1 ) = ( 110 ) 3 12 in base 3 is ( 110 ) 3 b. To determine To find: The next numeral after 222 3 .

Survey of Mathematics with Applications, A, a la Carte edition plus NEW MyLab Math with Pearson eText (10th Edition)

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ISBN: 9780134112237

Author: Allen R. Angel, Christine D. Abbott, Dennis Runde

Publisher: PEARSON

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Chapter 4.3, Problem 67E

To determine

To write: The first 20 numbers in the base 3 numeration system by using the numerals 0, 1 and 2.

To determine

To find: The next numeral after 2223 .

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Chapter 4.3, Problem 66E

Chapter 4.3, Problem 68E

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Survey of Mathematics with Applications, A, a la Carte edition plus NEW MyLab Math with Pearson eText (10th Edition)

Ch. 4.1 - Warm Up Exercises In Exercises 1-8, fill in the...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Warm Up Exercises In Exercises 1-8, fill in the...Ch. 4.1 - Warm Up Exercises In Exercises 1-8, fill in the...Ch. 4.1 - Warm Up Exercises In Exercises 1-8, fill in the...Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Practice The Skills In Exercises 9-14: write the...

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The first 20 numbers in the base 3 numeration system by using the numerals 0, 1 and 2.

Solution Summary: The author explains how to convert the first 20 numbers in the base 3 numeration system by using the numerals 0, 1 and 2.