Questions: Converting Between Binary, Decimal, and Hexadecimal

Question 1

What is the hexadecimal equivalent of the binary number 1011 1010?

• A. 0x188
• B. 0x1BA
• C. 0xBA
• D. 0x11BA

Answer: (C)

Group the 8 bits into two groups of four from the right: 1011 and 1010. Convert each group separately: 1011 = 8+2+1 = 11 = B in hex; 1010 = 8+2 = 10 = A in hex. Result: 0xBA. This is the key advantage of the binary↔hex relationship — no arithmetic across digit boundaries is needed. Each 4-bit group translates independently to one hex digit. Options like 0x188 or 0x1BA arise from incorrect groupings or positional calculations; 0x11BA has too many digits for 8 bits.

Question 2

Why is hexadecimal preferred over binary for displaying memory addresses in debuggers and hex dumps?

• A. Hexadecimal arithmetic is faster for computers to perform than binary arithmetic
• B. Hexadecimal is a human-readable form compatible with standard decimal notation
• C. One hex digit represents exactly four bits, so hex compresses binary by 4:1 without any arithmetic conversion
• D. Memory addresses are stored internally as hexadecimal values, not binary

Answer: (C)

Computers store everything in binary — hex is purely a display convention for humans. The reason hex is preferred is the lossless 4:1 compression: one hex character represents exactly four bits, so an 8-bit byte becomes two hex characters, a 32-bit address becomes 8 hex characters. This is compact without requiring any arithmetic: the translation is a direct symbol substitution. Decimal would require arithmetic (repeated division) to recover binary, and binary itself would be an unmanageably long string of 0s and 1s.

Question 3

To convert between binary and hexadecimal, you is expected to first convert both values to decimal as an intermediate step.

• T. True
• F. False

Answer: False

Binary and hex share a direct relationship because 16 = 2⁴. Every hex digit maps to exactly four binary digits, and vice versa, with no arithmetic required — you simply substitute symbols. For example, 0xA = 1010 and 0xF = 1111. Going through decimal would be slower and unnecessary. Decimal is base 10, which has no power-of-2 relationship with binary or hex, so decimal conversions genuinely do require arithmetic. But binary↔hex is a direct, table-based lookup.

Question 4

In the repeated-division method for converting a decimal number to another base, the remainders are read from the last computed to the first — that is, bottom to top — to form the result.

• T. True
• F. False

Answer: True

Each division step produces the next least-significant digit — the first remainder is the units digit (lowest place value), the second is the 'base¹' digit, and so on. When you read the remainders in the order they were computed (top to bottom), you get the number in reverse. Reading bottom to top gives the correct most-significant-digit-first representation. For example, converting 42 to binary: remainders are 0,1,0,1,0,1 (in order), read bottom to top: 101010.

Question 5

Explain why converting between binary and hexadecimal requires no arithmetic, while converting between either of these and decimal does require calculation.

Model answer: Binary and hex are both powers of 2 (base 2¹ and base 2⁴). Because 16 = 2⁴, exactly four binary digits correspond to one hex digit — the mapping is a fixed, arithmetic-free lookup table. You group bits in fours and substitute symbols. Decimal is base 10, which shares no power-of-2 relationship with binary, so there is no digit-group correspondence. To convert to or from decimal, you must either sum positional weights (digit × base^position) or repeatedly divide by the target base — both involve arithmetic across all digit positions.

This distinction is practically important. Fluency with the direct binary↔hex substitution lets you read machine code and memory dumps without calculation. When you see 0xFF, you should immediately see 11111111 without computing. That immediacy comes from memorizing the 16 4-bit patterns, not from understanding base conversion in the abstract.