Question 1

Binary Coded Decimal (BCD) — in digital electronics, BCD represents each decimal digit 0–9 independently. Each single decimal digit is encoded as which fixed-length binary pattern?

Accepted Answer

Introduction / Context: Binary Coded Decimal (BCD) is a widely used number representation in digital systems where each individual decimal digit is encoded separately. Unlike pure binary, BCD preserves the decimal digit boundaries, which simplifies human-readable I/O in calculators, digital meters, and financial devices. Given Data / Assumptions: We are encoding a single decimal digit 0–9. Standard BCD uses “8421” weighting per digit position. We are not discussing packed formats beyond the per-digit code width. Concept / Approach: In BCD, each decimal digit is represented by a 4-bit binary code (weights 8, 4, 2, 1). Thus 0→0000, 1→0001, …, 9→1001. Multi-digit decimal numbers are formed by concatenating these 4-bit nibbles. This differs from straight binary, where the entire number is encoded as a single binary value with place weights 2^0, 2^1, 2^2, etc. BCD keeps conversion to and from decimal trivial at the digit level and avoids rounding issues in some financial computations. Step-by-Step Solution: Identify the question scope: one decimal digit (0–9).Recall the standard BCD mapping: 0–9 use 4-bit codes from 0000 to 1001.Conclude that the fixed length per digit in standard BCD is 4 bits.Reject longer widths (8-bit, 16-bit) since they are unnecessary for a single digit. Verification / Alternative check: Check the digit 9: 1001 is a 4-bit pattern. Digits 10–15 do not exist in decimal, so the six remaining 4-bit combinations (1010–1111) are “invalid” in basic BCD and may be unused or reserved in packed formats. Why Other Options Are Wrong: 8-bit/16-bit: excessive width for a single decimal digit; used for bytes/words, not BCD per digit.ASCII: encodes characters; the ASCII code for '3' is 0x33, not a BCD digit code.Binary floating-point: represents real numbers with mantissa/exponent, unrelated to BCD digit encoding. Common Pitfalls: Confusing “binary representation of the whole number” with “BCD per digit”; forgetting that some 4-bit patterns are invalid in basic BCD. Final Answer: 4-bit binary code

Question 2

Unit-distance coding — which binary code progresses so that only one bit changes between any two successive code words (minimizing transition errors)?

Accepted Answer

Introduction / Context: When digital systems encode ordered states (such as shaft angles or counter positions), changing multiple bits at once can create transient errors if not all lines switch simultaneously. A “unit-distance” code mitigates this by ensuring that adjacent values differ by only one bit. Given Data / Assumptions: We need a code where successive values differ by Hamming distance 1. Use case includes encoders, ADCs, and state machines where transition glitches must be minimized. We consider common codes: weighted BCD, complements, and Gray. Concept / Approach: Gray code (reflected binary code) is constructed so that each successive word differs from the previous one by exactly one bit. This property reduces ambiguity during transitions because even if lines switch at slightly different times, at most one bit is in flux, which minimizes decoding errors. Weighted codes like 8421 BCD can change multiple bits between successive numbers (e.g., 7→8 toggles four bits), increasing the risk of transient misreads. Step-by-Step Solution: Define unit distance: Hamming distance of 1 between neighbors.Recall Gray code sequence: 00, 01, 11, 10 for 2-bit; generalizes by reflection.Check weighted codes (8421): observe multi-bit toggles between sequential decimal values.Conclude that Gray code uniquely satisfies the one-bit-change criterion. Verification / Alternative check: Create a small 3-bit table and compare transitions: Gray code shows single-bit steps while straight binary shows multi-bit steps such as 3 (011) to 4 (100), which toggles three bits. Why Other Options Are Wrong: 9's complement/excess-3: intended for decimal arithmetic and self-complementing properties, not unit-distance transitions.8421 BCD: weighted; adjacent decimal digits often change several bits.ASCII: character code, not numeric progression code. Common Pitfalls: Confusing “nonweighted” with “unit distance”; some codes are nonweighted yet not strictly single-bit transition for all sequences. Final Answer: Gray code

Question 3

Bit significance — in a fixed-width binary number, changing which bit causes the largest change in numeric value for a single-bit toggle?

Accepted Answer

Introduction / Context: Binary numbers assign positional weights that are powers of 2. Understanding bit significance is essential for error analysis, resolution calculations, and designing counters or DACs/ADCs where bit toggles directly map to step sizes in value or voltage. Given Data / Assumptions: A fixed binary word length is used (e.g., 8-bit). Each position weight doubles as you move toward more significant bits. We toggle exactly one bit and compare its impact on numeric value. Concept / Approach: The weight of each bit position is 2^n, with n increasing toward the MSB. Therefore, toggling the MSB changes the number by the largest magnitude among single-bit changes since its increment or decrement equals 2^(N−1) for an N-bit unsigned word. Conversely, toggling the LSB changes the value by only 1. This logic holds for signed magnitude or two’s complement as well, where the MSB influences sign or large magnitude changes. Step-by-Step Solution: Identify bit weights: LSB = 2^0 = 1; next = 2; …; MSB = 2^(N−1).Compare magnitudes: 2^(N−1) is greater than any other single-bit weight.Therefore, toggling the MSB produces the largest value step.Confirm with an example: in 8-bit, MSB toggle changes value by 128. Verification / Alternative check: Consider DAC resolution: the MSB contributes half of the full-scale range, reinforcing its dominant effect on output changes. Why Other Options Are Wrong: LSB: smallest unit change (1) only.Duty cycle/frequency/parity bits: not positional weights in numeric representation; they are control/metadata concepts. Common Pitfalls: Confusing bit roles in data frames (parity, control) with positional weights in numeric binary representation; mixing two’s complement sign effects with magnitude analysis without considering absolute value change. Final Answer: Most Significant Bit (MSB)

Question 4

Foundational base — digital electronics, logic gates, and computer arithmetic are fundamentally built on which numbering system?

Accepted Answer

Introduction / Context: Modern digital hardware represents information using discrete voltage levels mapped to symbols. The most robust and economical choice is two stable states, enabling noise-immune switching and straightforward implementation with transistors. This naturally aligns with the binary numbering system. Given Data / Assumptions: Logic circuits use two distinct states (low/high). Voltage thresholds separate 0 from 1 to provide noise margins. Other radices may be used for human representation but not for core hardware operation. Concept / Approach: The binary system (base 2) uses digits 0 and 1. In hardware, these correspond to two voltage ranges (e.g., 0 V and +V). Boolean algebra maps logic operations (AND, OR, NOT) to simple transistor networks. While hexadecimal and octal are convenient shorthand for grouping bits (4 and 3 respectively), they are not the physical basis; they simply compress binary for readability. Decimal is used in human interfaces and BCD storage when necessary, but internal logic remains binary. Step-by-Step Solution: Identify physical implementation: two-state devices (transistors, flip-flops).Relate states to symbols: assign 0 and 1 to voltage ranges.Conclude the fundamental numbering system is binary.Note that other bases are derived not native for hardware switching. Verification / Alternative check: Observe that memory cells, registers, and buses store and transmit bits. Groupings (nibbles/bytes) are powers of 2, reinforcing the binary foundation. Why Other Options Are Wrong: Decimal: human-centric; inefficient for hardware implementation.Hex/Octal: compact encodings of binary, not the physical basis.Balanced ternary: interesting academically but not standard in modern hardware. Common Pitfalls: Assuming hex is “native” because of its prevalence in programming; it merely maps to 4-bit groups. Final Answer: Binary (base 2)

Question 5

Nonweighted, unit-distance identification — which of the following is a nonweighted binary code that also changes only one bit between successive values (a unit-distance code)?

Accepted Answer

Introduction / Context: Binary codes are often categorized by whether they are weighted (each position has a fixed weight) or nonweighted (no simple positional weights). In applications like position encoders and certain counters, a “unit-distance” property—successive code words differ by one bit—is highly desirable to minimize transition errors. Given Data / Assumptions: We need a code that is nonweighted. We also need only a single-bit change between successive values. Codes under consideration include Gray and Excess-3. Concept / Approach: Gray code (reflected binary) is both nonweighted and unit-distance. It is constructed recursively to ensure that adjacent code words differ by exactly one bit, which reduces decoding ambiguity during changes. Excess-3, while nonweighted and self-complementing, does not guarantee single-bit transitions between successive decimal digits and is designed for decimal arithmetic, not transition robustness. Weighted codes like 8421 BCD explicitly use positional weights and are therefore not nonweighted. Step-by-Step Solution: List properties: Gray → nonweighted + unit-distance.Check Excess-3: nonweighted, but not unit-distance for general sequences.Eliminate weighted code (8421) immediately.Select Gray code as the only option satisfying both criteria. Verification / Alternative check: Construct a short Gray sequence (3-bit): 000, 001, 011, 010, 110, 111, 101, 100 — observe one-bit transitions throughout. Compare to Excess-3 of decimal digits and note multiple-bit changes frequently occur. Why Other Options Are Wrong: Excess-3: nonweighted, but not unit-distance by design goal.Multi-bit parity/minival: not standard nonweighted unit-distance codes for ordered numeric progression.8421 BCD: weighted, so fails the nonweighted requirement. Common Pitfalls: Equating “nonweighted” with “unit-distance”; they are independent properties. Gray happens to possess both, which is why it is common in encoders. Final Answer: Gray code (reflected binary)

Question 6

Excess-3 (XS-3) BCD — which code represents each decimal digit by its 4-bit binary value after adding 3 to that digit (i.e., code = binary(digit + 3))?

Accepted Answer

Introduction / Context: Several BCD variants exist to simplify arithmetic and error detection. Excess-3 (also written XS-3) is a self-complementing code that shifts each decimal digit by 3 before encoding it in 4-bit binary. This property eases 9’s complement and subtraction operations in certain hardware designs. Given Data / Assumptions: Per-digit BCD with 4-bit fields. Mapping formula: codeword = binary(digit + 3). Valid for digits 0–9 (so 3–12 in binary before mapping; 10–12 codes are still 4-bit patterns). Concept / Approach: By adding 3 to the decimal digit, Excess-3 ensures that the 9’s complement of a number corresponds to bitwise complement of its code, simplifying subtraction via complements. For example, digit 0 maps to 0011, 1→0100, …, 9→1100. This is not a weighted code and differs from 8421 BCD, which maps 0→0000 through 9→1001 directly without the +3 offset. Step-by-Step Solution: Start with digit d in 0–9.Compute d + 3.Encode the result as a 4-bit binary value.Thus obtain the Excess-3 codeword for that digit. Verification / Alternative check: Take d = 5: 5 + 3 = 8 → 1000; d = 9: 9 + 3 = 12 → 1100. The mapping matches standard XS-3 tables. Why Other Options Are Wrong: 9's complement: arithmetic concept, not the code itself.8421: direct weighted BCD without the +3 shift.Gray: unit-distance code for sequencing, not decimal encoding.Two's complement: signed binary integer representation, not per-digit BCD. Common Pitfalls: Confusing Excess-3 with 8421 due to both using 4 bits; overlooking its self-complementing advantage for subtraction operations. Final Answer: Excess-3 code

Question 7

Sign representation basics: In common signed-number encodings (signed magnitude, one’s complement, two’s complement), what sign bit indicates a positive value?

Accepted Answer

Introduction / Context: Digital systems need consistent conventions for signed binary values. Several representations exist—signed magnitude, one’s complement, and two’s complement—but they agree on the meaning of the single sign bit: 0 for positive (and zero), 1 for negative. This question corrects a common misunderstanding about which sign-bit value indicates positivity. Given Data / Assumptions: Single sign-bit formats are considered (no biased or excess-N encodings). Magnitude bits represent the absolute value in some form. We exclude floating-point encodings here. Concept / Approach: Across the common signed integer encodings, the most significant bit (MSB) is designated the sign bit. Conventionally, sign = 0 indicates non-negative; sign = 1 indicates negative. The exact mapping of magnitude differs among formats, but the sign-bit meaning remains consistent, particularly for two’s complement, the dominant representation in modern processors. Step-by-Step Solution: Identify the sign bit: the MSB in an n-bit word.Recall the convention: sign=0 → positive/zero; sign=1 → negative.Reject statements that invert this convention. Verification / Alternative check: Inspect two’s complement ranges: for 8-bit, 0x00..0x7F are non-negative (MSB=0), 0x80..0xFF are negative (MSB=1). Signed magnitude and one’s complement follow the same sign-bit convention even though zero handling differs. Why Other Options Are Wrong: Positive is 1: Inverts the established convention.Depends on format: While magnitudes differ, sign-bit meaning does not.Two sign bits claim: Not true for standard binary integer encodings. Common Pitfalls: Confusing sign-bit meaning with bias encodings (e.g., excess-127 in floating point) or with sign extension behavior on shifts. Final Answer: Incorrect — the sign bit 0 denotes positive; sign bit 1 denotes negative

Question 8

Core idea of digital systems: Do digital systems operate exclusively on discrete symbols (digits), such as codes representing numbers, letters, or control symbols, rather than continuously variable quantities?

Accepted Answer

Introduction / Context: Digital electronics encodes information as discrete symbols—usually two levels labelled 0 and 1, but sometimes multi-level codes. These symbols represent numbers, letters, and control characters, enabling robust processing with noise margins. This question confirms the fundamental distinction between digital and analog domains. Given Data / Assumptions: Digital signals are quantized into discrete states. Analog signals vary continuously and require conversion to be processed digitally. Codes like ASCII or BCD map symbols to binary patterns. Concept / Approach: Digital systems internally manipulate discrete symbols using logic gates, registers, and finite-state machines. Even when interfacing with the analog world (sensors/actuators), conversion stages (ADC/DAC) bridge the domains so the core system still processes discrete values, not continuous waveforms directly. Step-by-Step Solution: Identify representations: numbers, letters, symbols → binary codes.Note processing: operations occur on codes via arithmetic/logic circuits.Acknowledge interfaces: ADC/DAC handle analog interactions at the boundaries. Verification / Alternative check: Inspect a microcontroller diagram: ALU, registers, data buses, and instruction decoders all operate on fixed-width binary words—discrete symbols. Only dedicated analog blocks (e.g., ADC) handle continuous signals. Why Other Options Are Wrong: Incorrect: Contradicts the definition of digital processing.Arithmetic-only: Control logic also operates on discrete states.ADCs are analog: They are boundary components; the system’s core remains digital. Common Pitfalls: Assuming PWM or DAC outputs imply internal analog processing; they are digital methods to approximate analog behavior at interfaces. Final Answer: Correct

Question 9

Opcode width reality check: Do computer manufacturers commonly use only 3-bit codes to represent operations/instructions across their processors?

Accepted Answer

Introduction / Context: An instruction opcode must uniquely identify a processor operation (and often a class of operations). While encodings vary across ISAs, practical opcodes are far wider than 3 bits to support meaningful instruction sets and addressing modes. This question addresses a misconception that “3-bit codes” suffice generally. Given Data / Assumptions: ISAs (e.g., 8051, x86, ARM, RISC-V) define many operations. Opcodes must be wide enough to encode instruction types and sometimes subfields. Some micro-ISAs may embed smaller subfields, but not entire instruction sets in 3 bits. Concept / Approach: With only 3 bits, at most 8 distinct opcodes are possible. Real-world ISAs require far more than eight instructions to be practical. Even minimalist microcontrollers exceed this count, and complex ISAs include dozens to hundreds of operations with variable-length encodings. Step-by-Step Solution: Compute capacity: 3 bits → 2^3 = 8 possible opcodes.Compare to typical ISA sizes: commonly tens or hundreds of instruction mnemonics.Conclude 3 bits are insufficient for “manufacturers of computers.” Verification / Alternative check: Examine any ISA reference: even highly reduced RISC cores define more than 8 operations. Many instructions also include function fields, register specifiers, and immediate encodings far beyond a trivial 3-bit opcode. Why Other Options Are Wrong: 3-bit standard/certain subsystems: Not representative of full ISAs.8051/RISC-only claims: Both use more than 3 unique operations. Common Pitfalls: Confusing opcode fields with function fields or microcoded control signals; cherry-picking tiny educational CPUs that are not representative of commercial processors. Final Answer: Incorrect — real processors use wider opcodes than 3 bits

Question 10

Error detection basics: Is a parity bit (for example, odd or even parity) used to detect errors in data transmission that may be caused by noise?

Accepted Answer

Introduction / Context: Communication systems often add redundancy to detect data corruption introduced by noise or interference. The simplest redundancy mechanism is a single parity bit appended to each data word. This question verifies that you understand parity as a detection mechanism, not a correction mechanism, and that it is sensitive primarily to odd numbers of bit errors. Given Data / Assumptions: Parity can be “even” (total ones count even) or “odd” (total ones count odd). One parity bit is appended to each word or character. Transmission errors may flip bits. Concept / Approach: At the receiver, the system recomputes parity on the received data and compares it with the received parity bit. A mismatch indicates that an odd number of bits flipped (most commonly a single-bit error). Parity cannot localize the error nor correct it; it merely signals that corruption likely occurred. Step-by-Step Solution: Choose parity type (odd/even) and compute the parity bit before transmission.Transmit data + parity.On reception, recompute parity and compare to the received parity bit.If mismatch, flag an error and request retransmission or discard, depending on protocol. Verification / Alternative check: Consider a single flipped bit: parity changes and a mismatch is detected. Consider two flipped bits: parity may match (missed detection), demonstrating the limitation of single-bit parity. Why Other Options Are Wrong: Incorrect: Parity is precisely an error-detection method.Two-bit only / correction claim: Parity detects odd-number-of-bit errors and does not correct. Common Pitfalls: Overestimating parity’s capability; for stronger protection use checksums or CRCs which detect more error patterns. Final Answer: Correct

Question 11

Scope of ASCII: Does the ASCII code represent all possible alphanumeric data (across languages and symbol sets), or is it limited to a specific basic set?

Accepted Answer

Introduction / Context: ASCII (American Standard Code for Information Interchange) is a 7-bit character code defining 128 values: control characters, digits, uppercase and lowercase English letters, and basic punctuation. It is a foundational code in computing, but it does not represent every possible alphanumeric symbol or the characters of all languages. This question challenges the overbroad claim that ASCII encodes “all alphanumeric data.” Given Data / Assumptions: ASCII defines 7-bit codes; many systems store them in 8-bit bytes with the MSB set to 0. Extended “ASCII” byte values (128–255) are not standardized globally and vary by code page. Modern global text interchange relies on Unicode/UTF encodings. Concept / Approach: ASCII reliably represents English A–Z/a–z, digits 0–9, and basic symbols. It cannot represent accented letters, non-Latin scripts, or the vast array of symbols required worldwide. Therefore, while ASCII is a “special code” historically, its scope is limited and not universal. Step-by-Step Solution: Identify ASCII’s size: 128 codes → limited character repertoire.Contrast with global needs: thousands of characters in world scripts and symbols.Conclude that ASCII does not cover “all” alphanumeric data. Verification / Alternative check: Check that ASCII lacks letters like é, ñ, and non-Latin scripts (Greek, Cyrillic, Devanagari, Han). Unicode was introduced to provide a universal mapping, proving ASCII’s limitations. Why Other Options Are Wrong: “ASCII equals Unicode”: Factually incorrect; Unicode subsumes ASCII as a tiny subset.“16-bit extension”: Even 16-bit sets do not automatically equal ASCII; UTF encodings map Unicode code points. Common Pitfalls: Confusing vendor-specific “extended ASCII” code pages with a single global standard; such pages are mutually incompatible. Final Answer: Incorrect — ASCII covers a basic English alphanumeric set, not “all” alphanumeric data

Question 12

Parity example with ASCII “2B” (hex for the “+” character): If we add an odd-parity bit as the most significant bit to the 7-bit ASCII code for 0x2B, does the resulting 8-bit pattern equal 10101011?

Accepted Answer

Introduction / Context: Parity examples help solidify bit-level reasoning. The ASCII code 0x2B corresponds to the “+” character. Classic ASCII is a 7-bit code; many systems transmit an additional parity bit to detect errors. With odd parity, the total number of 1s in the transmitted byte (data bits + parity bit) must be odd. This question verifies that adding an odd-parity bit to ASCII 0x2B yields the pattern 10101011. Given Data / Assumptions: 7-bit ASCII for “+” is 0x2B (binary 0101011 in 7 bits; 00101011 if shown as 8 bits with leading 0). Parity bit is placed as the most significant bit in this example. Odd parity means total count of 1s (including the parity bit) is odd. Concept / Approach: Count the number of 1s in the data bits, then choose the parity bit to make the total odd. The 7-bit 0x2B has four 1s when represented as 00101011 (the leading 0 does not change parity). Since four is even, the parity bit must be 1 to yield an odd total of five. Placing that 1 as the MSB produces 10101011. Step-by-Step Solution: Write ASCII “+” as binary: 0x2B → 00101011 (8-bit representation with leading zero).Count ones: there are 4 ones → even.Odd parity requires total ones odd → add parity bit = 1.Place parity as MSB → result = 1 00101011 → 10101011. Verification / Alternative check: Recount ones in 10101011: there are 5 ones, which is odd. Therefore, odd parity is satisfied. If even parity had been required, the parity bit would be 0 (yielding 00101011). Why Other Options Are Wrong: Incorrect: Contradicts the direct parity calculation.Even parity claim: Would require parity bit 0, not 1, and would not match 10101011.LSB placement claim: Parity placement can vary by protocol, but the example explicitly uses the MSB convention. Common Pitfalls: Mixing 7-bit and 8-bit representations when counting ones, or forgetting which end hosts the parity bit in a specific protocol. Final Answer: Correct

Question 13

Fundamentals of number systems in digital electronics: In computer architecture and digital circuit design, is the binary (base-2) number system considered the foundational representation used by all digital computers for data and instructions?

Accepted Answer

Introduction / Context: Digital electronics and computer systems represent information using discrete levels. The binary number system (base 2) maps naturally onto this two-level hardware, with logic 0 and logic 1 corresponding to defined voltage ranges. This question checks your understanding of why binary is considered the foundational number system for all digital computers. Given Data / Assumptions: A digital computer is built from logic gates that operate with two stable states. Memory cells, registers, and buses store and move information in bits. Higher-level number systems (decimal, hexadecimal) are human conveniences and are translated to/from binary in hardware or software. Concept / Approach: Because hardware devices like MOSFETs implement two-state logic reliably, data is encoded as sequences of bits. Arithmetic units perform operations on binary operands; instruction sets encode opcodes and addresses in binary; storage elements hold binary states. Even when designers use hexadecimal or octal for compact notation, these are just grouped representations of the underlying binary bits (4-bit nibbles for hex, 3-bit groups for octal). Step-by-Step Solution: Identify hardware basis: gates and flops are inherently two-state devices.Map states to numbers: 0 and 1 form base-2 digits (bits).Build larger structures: bytes (8 bits), words, addresses, instructions—all binary.Conclude: binary is the fundamental representation across digital computers. Verification / Alternative check: Review any ISA encoding (for example, an instruction format diagram): fields are defined as bit positions. Memory dumps and bus analyzers display binary or its grouped forms (hex), confirming the binary foundation. Why Other Options Are Wrong: “Incorrect” contradicts the core design of digital logic. “Applies only to analog computers” is irrelevant; analog computers are not digital. Restricting binary to microcontrollers or memory ignores that CPUs and logic also operate on bits. Common Pitfalls: Confusing human-readable bases (decimal/hex) with machine representation; assuming that because code is written in text, the machine stores characters directly without binary encoding. Final Answer: Correct

Question 14

Binary-to-hexadecimal conversion technique: When converting a binary value to hexadecimal, is it valid to pad the left side (most significant side) with leading zeros to form complete groups of 4 bits before translating each group to a hex digit?

Accepted Answer

Introduction / Context: Hexadecimal is a compact way to express binary numbers by grouping bits in sets of four, since 16 = 2^4. This question tests whether you may add leading zeros on the most significant side of a binary number to complete 4-bit groups prior to conversion to hexadecimal digits. Given Data / Assumptions: A binary number may not have a length that is a multiple of 4. Leading zeros do not change the value of an unsigned number. Hex conversion maps each 4-bit nibble to one hex digit (0–F). Concept / Approach: Adding zeros to the left (toward the MSB) simply increases the bit width without altering numeric value. Grouping bits into 4-bit nibbles aligns perfectly with hexadecimal, enabling a one-to-one mapping from each nibble to a hex character. This is standard practice in debugging, memory dumps, and digital design documentation. Step-by-Step Solution: Take the binary sequence and partition it into 4-bit chunks from the LSB side.If the leftmost chunk has fewer than 4 bits, prepend zeros until it has 4.Translate each 4-bit group to its hex equivalent (0000→0, 1111→F).Join the hex digits to form the final hexadecimal value. Verification / Alternative check: Convert back from the hex digits to binary and verify that dropping any leading zeros restores the original bit pattern; the numeric value remains unchanged. Why Other Options Are Wrong: Signedness does not affect leading zeros for magnitude; requiring MSB = 1 is unnecessary; octal uses 3-bit groups, not 4-bit groups. Common Pitfalls: Mistaking leading zeros for sign extension (that applies to two’s complement when extending sign with 1s for negative numbers); confusing octal grouping (3 bits) with hex grouping (4 bits). Final Answer: Correct

Question 15

Hexadecimal base identification: Does the hexadecimal numbering system use base 15, or is it defined as a base-16 system with digits 0–9 and A–F?

Accepted Answer

Introduction / Context: Number systems are characterized by their base (radix), which determines how many distinct digit symbols can appear in any position. This question checks whether hexadecimal is base 15 or base 16 and reinforces the digit set used for hex representation in digital systems. Given Data / Assumptions: Hexadecimal is commonly used to compress binary notation. The digits used in hex are 0–9 followed by letters A–F. Each hex digit corresponds to exactly 4 binary bits. Concept / Approach: Because hexadecimal has 16 distinct symbols, its base is 16. The mapping to binary is 1 hex digit ↔ 4 bits since 16 = 2^4. A base-15 system would only have 15 symbols and would not align with a 4-bit grouping. Using letters does not change the base; it simply provides symbols for values 10–15. Step-by-Step Solution: List hex symbols: 0..9 (10 symbols), A..F (6 symbols) → total 16 symbols.Compute power-of-two relationship: 16 = 2^4, so 4 bits per hex digit.Confirm that base-15 would not map neatly to a fixed bit group.Conclude that hexadecimal is base 16, not base 15. Verification / Alternative check: Convert a binary nibble like 1010 to hex A; this validates the 4-bit ↔ 1 hex digit mapping that depends on base 16. Why Other Options Are Wrong: Endianness affects byte order, not radix; “base 12 if letters are used” is arbitrary; base 10 with letter aliases would be decimal, which it is not. Common Pitfalls: Confusing symbol choices with base; thinking that the presence of letters implies a different numbering base. Final Answer: Incorrect

Question 16

Terminology and storage granularity: Do most computers “store binary data in groups of 32 bits called double words (DWORDs),” or is 32-bit grouping platform-dependent while the fundamental storage unit remains the 8-bit byte?

Accepted Answer

Introduction / Context: Computers organize data in multiple granularities: bits, bytes, words, and wider registers. This question clarifies a common misconception by examining whether 32-bit “double words” are universally the primary storage grouping, or whether such terminology is architecture-specific while the byte remains fundamental. Given Data / Assumptions: A byte universally equals 8 bits in modern systems. Word size varies by architecture (16-bit, 32-bit, 64-bit, etc.). “DWORD” is a conventional term in some ecosystems (for example, Windows/x86) meaning 32 bits. Concept / Approach: The fundamental addressable unit in memory is the byte (8 bits). While some platforms historically used 32-bit words and popularized the term “double word,” modern systems are often 64-bit, and other embedded systems may be 8-, 16-, or 32-bit. Therefore, claiming that most computers store data in 32-bit groups called DWORDs is overly general and inaccurate across the industry. The correct general statement is that computers store data as bytes, words, and larger units depending on architecture and data type. Step-by-Step Solution: Recognize the universal 8-bit byte as the base addressable unit.Note that “word” size depends on architecture; 32-bit was common, 64-bit is now typical on desktops/servers.Understand that “DWORD” (32-bit) is platform-specific jargon, not a universal rule.Conclude the blanket statement is inaccurate; storage granularity is not always 32 bits. Verification / Alternative check: Inspect ABI documentation for different CPUs: x86-64 defines 64-bit general-purpose registers and native word sizes; many MCUs define 8-, 16-, or 32-bit words. Why Other Options Are Wrong: Saying “Correct” generalizes improperly; restricting to legacy 16-bit CPUs or claiming it is true only for memory or networking ignores broader architecture diversity. Common Pitfalls: Confusing a compiler’s data type sizes with hardware word size; assuming Windows terminology applies to all systems. Final Answer: Incorrect

Question 17

Definition check for BCD (binary-coded decimal): Do BCD digit encodings use only the 4-bit patterns from 0000 through 1001 (representing decimal 0–9), with the remaining 4-bit combinations considered invalid or unused for a single BCD digit?

Accepted Answer

Introduction / Context: Binary-coded decimal (BCD) represents each decimal digit individually using a 4-bit group. Understanding which 4-bit patterns are valid BCD is essential when designing arithmetic units that operate on decimal digits or when validating user input in firmware and digital systems. Given Data / Assumptions: Each BCD digit encodes a single decimal digit 0–9. A single 4-bit group is used per digit. Unused patterns must be detected or avoided in strict BCD contexts. Concept / Approach: Since there are 10 decimal digits and 16 possible 4-bit combinations, only 0000 through 1001 (0–9 decimal) are defined as valid BCD. The remaining combinations 1010 through 1111 do not represent single decimal digits in strict BCD and may be treated as invalid or used for special flags in some variants. Many hardware blocks implement decimal correction logic to ensure digit results remain within 0–9 during arithmetic operations like addition. Step-by-Step Solution: List valid decimal digits: 0–9 → need 10 encodings.Map to 4-bit range: use 0000..1001 for 0..9.Identify invalids: 1010..1111 are not single-digit BCD.Apply checks or corrections after operations that may overflow a digit. Verification / Alternative check: Examine BCD adders: they perform binary add and then add 6 (0110) when the digit exceeds 1001 to restore a valid BCD range. Why Other Options Are Wrong: Including 1010 or 1011 as valid single digits contradicts strict BCD; limiting to 0000..0111 would exclude 8 and 9; two’s complement is unrelated to BCD digit validity. Common Pitfalls: Confusing BCD with packed formats that contain multiple digits per byte; forgetting to validate digits after arithmetic operations. Final Answer: Correct

Question 18

Bit–byte relationship: Given that 1 byte = 8 bits, does a string of 64 bits correspond to 9 bytes, or is the correct conversion 8 bytes?

Accepted Answer

Introduction / Context: Converting between bits and bytes is a foundational skill in digital electronics and computer engineering. This question verifies your ability to convert a 64-bit length to its byte count accurately and to avoid overcounting due to framing or encoding misconceptions. Given Data / Assumptions: 1 byte = 8 bits, by standard definition. No additional parity or framing bits are included unless explicitly stated. The term “string” here denotes a contiguous sequence of bits. Concept / Approach: The conversion formula is straightforward: bytes = bits / 8. Applying this to 64 bits yields exactly 8 bytes. Any claim of 9 bytes would imply additional overhead bits (for example, parity, start/stop in serial links), which are not part of the raw bit count unless separately specified. Step-by-Step Solution: Use the identity: 1 byte = 8 bits.Compute: 64 bits / 8 = 8 bytes.Confirm no overhead specified → result stands.Conclude: 64 bits correspond to 8 bytes, not 9. Verification / Alternative check: Consider an 8-byte register or 64-bit CPU general-purpose register; it contains 64 bits, matching 8 bytes exactly. Why Other Options Are Wrong: “Correct” would misstate the math; parity or framing are separate layers and do not change the base conversion; Unicode or character encodings do not alter the bits-per-byte identity. Common Pitfalls: Mixing physical-layer framing with payload size; assuming all 64-bit data types have hidden overhead bits. Final Answer: Incorrect

Question 19

Range of a fixed-width unsigned binary value: What is the maximum decimal value representable by a 4-bit binary number (using all bits for magnitude)?

Accepted Answer

Introduction / Context: Understanding representable ranges for a given bit width is essential for sizing registers, counters, and fields in digital designs. This question focuses on the maximum value a 4-bit unsigned binary number can represent when all bits are used for magnitude (no sign bit). Given Data / Assumptions: Bit width N = 4 bits. Unsigned interpretation (no sign or bias). All 4 bits can take values 0 or 1 independently. Concept / Approach: For an unsigned N-bit number, the value range is 0 through (2^N − 1). Substituting N = 4 gives 0 through 15. In binary, the maximum is 1111 which equals 8 + 4 + 2 + 1 = 15. This rule generalizes to any bit width and underpins field sizing in hardware description and microarchitecture design. Step-by-Step Solution: Apply formula: max = 2^N − 1.Compute: 2^4 − 1 = 16 − 1 = 15.Check binary: 1111b = 8 + 4 + 2 + 1 = 15.Conclude: maximum representable value is 15. Verification / Alternative check: Count all 4-bit patterns (16 of them) from 0000 to 1111 → values 0..15 inclusive; the highest is 15. Why Other Options Are Wrong: 16 requires 5 bits; 14 corresponds to 1110; 7 corresponds to 3 bits; 31 requires 5 bits (11111). Common Pitfalls: Confusing the number of distinct patterns (2^N) with the maximum value (2^N − 1); mixing unsigned with signed two’s-complement ranges. Final Answer: Correct (maximum value is 15)

Question 20

Bitwise complementation: Is the 1’s complement of a binary number obtained by inverting every bit (changing all 0s to 1s and all 1s to 0s) across the entire width?

Accepted Answer

Introduction / Context: Complement systems are used in arithmetic and error detection. The 1’s complement and 2’s complement are closely related but distinct operations. This question confirms the exact transformation for obtaining a 1’s complement from a given binary pattern. Given Data / Assumptions: A fixed bit width is implied (for example, 8, 16, 32 bits). All bit positions are subject to the operation. No arithmetic carry is involved in 1’s complement itself. Concept / Approach: The 1’s complement is defined as a bitwise NOT operation across the entire width: every 0 becomes 1 and every 1 becomes 0. In contrast, 2’s complement is obtained by taking the 1’s complement and then adding 1 to the result, which represents the negative of a number in signed arithmetic. Understanding this distinction avoids mistakes in sign inversion and bit manipulations. Step-by-Step Solution: Given value X, compute ~X (bitwise NOT) to get the 1’s complement.Invert each bit position independently.If a 2’s complement is required, then compute ~X + 1 afterward.Keep width fixed to avoid losing leading inversion results. Verification / Alternative check: Example: for 8-bit 0101 1000, 1’s complement = 1010 0111; performing +1 yields 2’s complement = 1010 1000, demonstrating the relation but confirming that inversion alone defines 1’s complement. Why Other Options Are Wrong: Inverting only the MSB or trailing zeros is not the definition; adding 1 is part of 2’s complement, not 1’s complement. Common Pitfalls: Forgetting to keep a fixed width (leading to incorrect inversion on the MSB); confusing 1’s complement with 2’s complement in signed math. Final Answer: Correct

Number Systems and Codes Questions