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Numeration Systems
19 questions By Tony R. Kuphaldt
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Question 13 of 19
A numeration system often used as a “shorthand” way of writing large binary numbers is the octal, or base-eight, system.
Based on what you know of place-weighted numeration systems, describe how many valid ciphers exist in the octal system, and the respective “weights” of each place in an octal number.
Also, perform the following conversions:
- 358 into decimal:
- 1610 into octal:
- 1100102 into octal:
- 518 into binary:
Reveal answerThere are only eight valid ciphers in the octal system (0, 1, 2, 3, 4, 5, 6, and 7), with each successive place carrying eight times the “weight” of the place before it.
- 358 into decimal: 2910
- 1610 into octal: 208
- 1100102 into octal: 628
- 518 into binary: 1010012
Follow-up question: why is octal considered a “shorthand” notation for binary numbers?
Notes:There are many references from which students may learn to perform these conversions. You assistance should be minimal, as these procedures are simple to comprehend and easy to find.
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Question 14 of 19
A numeration system often used as a “shorthand” way of writing large binary numbers is the hexadecimal, or base-sixteen, system.
Based on what you know of place-weighted numeration systems, describe how many valid ciphers exist in the hexadecimal system, and the respective “weights” of each place in a hexadecimal number.
Also, perform the following conversions:
- 3516 into decimal:
- 3410 into hexadecimal:
- 111000102 into hexadecimal:
- 9316 into binary:
Reveal answerThere are sixteen valid ciphers in the hexadecimal system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F), with each successive place carrying sixteen times the “weight” of the place before it.
- 3516 into decimal: 5310
- 3410 into hexadecimal: 2216
- 111000102 into hexadecimal: E216
- 9316 into binary: 100100112
Follow-up question: why is hexadecimal considered a “shorthand” notation for binary numbers?
Notes:There are many references from which students may learn to perform these conversions. You assistance should be minimal, as these procedures are simple to comprehend and easy to find.
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Question 15 of 19
Complete this table, performing all necessary conversions between numeration systems:
Reveal answer
Notes:Lots of conversions to do here! I particularly like the “table” format shown here for practicing numeration system conversions, because it compresses a lot of practice into a small space on paper, and also because it allows students to use different methods of conversion. For example, in converting a decimal number to the other forms, a student might choose to convert first to binary, then from binary to octal and hex. Or, alternatively, a student may choose to convert from decimal into hex, then from hex into binary, then from binary to octal, the last two conversions being especially easy.
