Which of the following statements about number systems is true?
2024
Which of the following statements about number systems is true?
Answer: C. (1011)2 + (1001)2 = (20)10 — ConceptIn a positional numeral with base b, a digit d at position k contributes d × bk to the represented value. Integer positions use non-negative powers;…
- A.
The binary equivalent of (57.625)10 is (111000.101)2.
- B.
(145)6 = (75)10
- C.
(1011)2 + (1001)2 = (20)10
- D.
(65)10 = (220)5
Attempted by 13 students.
Show answer & explanation
Correct answer: C
Concept
In a positional numeral with base b, a digit d at position k contributes d × bk to the represented value. Integer positions use non-negative powers; fractional positions use negative powers.
To test an equality between numeral systems, expand each numeral in one common base, usually decimal. Binary addition can also be checked directly and then converted to decimal.
Application
For (57.625)10, the integer part is 57 = 32 + 16 + 8 + 1, so it is (111001)2. The fractional part is 0.625 = 1/2 + 1/8, so it is (.101)2. Therefore the full binary numeral is (111001.101)2.
For (145)6, positional expansion gives 1 × 62 + 4 × 61 + 5 × 60 = 36 + 24 + 5 = (65)10.
For the sum, (1011)2 = 8 + 2 + 1 = (11)10 and (1001)2 = 8 + 1 = (9)10. Thus their sum is (20)10.
For (220)5, positional expansion gives 2 × 52 + 2 × 51 + 0 × 50 = 50 + 10 = (60)10.
Cross-check
Direct binary addition gives (1011)2 + (1001)2 = (10100)2, and (10100)2 = 16 + 4 = (20)10.
Hence, the true statement is (1011)2 + (1001)2 = (20)10.