Question 1

Binary to Octal — Group Bits in Threes Find the octal equivalent of the binary number 1101110101110110 by grouping into 3-bit sets from the right.

Accepted Answer

Introduction / Context: Converting binary to octal is efficient because 8 equals 2^3. Therefore, every three binary bits map to exactly one octal digit. This technique avoids full decimal conversion and reduces working time and errors. Given Data / Assumptions: Binary input: 1101110101110110. No sign or fractional portion. Left padding with zeros is permissible to complete a leading triplet if required. Concept / Approach: Partition the binary string into 3-bit groups starting at the least significant end. Translate each triplet to an octal digit using 000→0 through 111→7 and assemble the digits from most significant group to least significant group. Step-by-Step Solution: 1) From the right, group bits: 1 101 110 101 110 110 → add leading zeros to the lone 1 to form 001.2) Final grouping: 001 101 110 101 110 110.3) Translate: 001→1, 101→5, 110→6, 101→5, 110→6, 110→6.4) Combine: 156566 base 8, written as 1565668. Verification / Alternative check: Convert 1565668 back to binary: 1→001, 5→101, 6→110, 5→101, 6→110, 6→110. Concatenate to get 001101110101110110 and remove the leading zeros to return to 1101110101110110. Why Other Options Are Wrong: 6545218, 5565618, 1566568: These imply different triplet translations or wrong group alignment and do not reproduce the given binary when mapped back. Common Pitfalls: Grouping from the left side, skipping the leading zero padding for an incomplete initial group, or reversing the order of triplets when writing the octal digits. Final Answer: 1565668

Question 2

Alphanumeric Code Requirements — Minimum Character Set As a minimum, what should a general purpose alphanumeric code include for practical computer use and human readable I/O?

Accepted Answer

Introduction / Context: An alphanumeric code for computer input and output must cover not only letters and digits but also the punctuation and control characters required for text, data formatting, and communication protocols. A minimal but practical set ensures readable documents, program source files, and terminal operations. Given Data / Assumptions: The code is intended for general purpose computing. Both uppercase and lowercase alphabets are required. Decimal digits and common punctuation marks must be present, along with additional symbols for control and formatting. Concept / Approach: ASCII established a widely adopted baseline including letters A–Z and a–z, digits 0–9, a set of punctuation marks, and control characters. A useful minimum mirrors that breadth. Limiting a code to only letters and digits is insufficient for programming, text markup, and data interchange. Step-by-Step Reasoning: 1) Identify everyday needs: sentences, numbers, punctuation, and symbols.2) Ensure both cases for readability and programming identifiers.3) Include punctuation such as period, comma, colon, semicolon, question mark, exclamation mark, and hyphen or minus.4) Provide other symbols for operations and control, as typical in ASCII like space and basic control codes. Verification / Alternative check: Programming languages and configuration files rely on punctuation and symbols. A reduced set would make source code and data files impossible to express cleanly. Why Other Options Are Wrong: A: Mentions only letters and numbers, omitting punctuation and symbols. B: BCD and octal phrasing is not relevant to a general alphanumeric character set. D: Refers to numeric range capability and ignores letters and punctuation entirely. Common Pitfalls: Confusing numeric encoding range with character repertoire; overlooking that human readable text needs punctuation and additional symbols for clarity. Final Answer: the alphabet upper- and lowercase letters, the decimal digits, the seven punctuation marks, and other characters or symbols.

Question 3

Hex to Decimal — Positional Weights Convert the hexadecimal number 1CF16 to its decimal value by expanding each digit with powers of sixteen.

Accepted Answer

Introduction / Context: Hexadecimal to decimal conversion relies on positional weights where each position represents a power of sixteen. Understanding this expansion is necessary for interpreting addresses, constants, and register values in documentation and code. Given Data / Assumptions: Hex input: 1CF16. Digit values: 1=1, C=12, F=15. Rightmost position is 16^0, moving left increases the exponent by one. Concept / Approach: Multiply each hex digit value by its positional weight and sum. The formula is value = d216^2 + d116^1 + d016^0 for a three digit hex number d2 d1 d0. Step-by-Step Solution: 1) Write weights: 16^2 = 256, 16^1 = 16, 16^0 = 1.2) Map digits: 1→1, C→12, F→15.3) Compute partials: 1256 = 256; 1216 = 192; 151 = 15.4) Sum: 256 + 192 + 15 = 463. Verification / Alternative check: Convert 463 back to hex by repeated division by 16: 463 / 16 = 28 remainder 15 (F); 28 / 16 = 1 remainder 12 (C); 1 / 16 = 0 remainder 1. Reading remainders in reverse yields 1CF16. Why Other Options Are Wrong: 4033, 4049, 479: Each results from incorrect weights or arithmetic mistakes, not from the exact expansion of 1CF16. Common Pitfalls: Confusing C with 13 rather than 12, or using powers of 10 instead of 16 for the place values. Final Answer: 463

Question 4

Decimal to BCD — Digitwise Encoding Convert the decimal number 469 into straight BCD by encoding each decimal digit as a 4-bit binary code and concatenating the results.

Accepted Answer

Introduction / Context: Binary Coded Decimal (BCD) represents each decimal digit independently using a 4-bit code. This preserves decimal structure for display and financial calculations at the cost of storage efficiency. The task is to produce straight BCD, not packed with delimiters, by simple digitwise mapping. Given Data / Assumptions: Decimal input: 469. Use straight BCD mapping: 0→0000 through 9→1001. Concatenate digit codes in the same order as the original number. Concept / Approach: Break the decimal number into digits, encode each digit into its 4-bit BCD representation, and then join the codes. No arithmetic conversion to pure binary is required for BCD; this is a symbol by symbol mapping process. Step-by-Step Solution: 1) Split 469 into digits: 4, 6, 9.2) Encode: 4→0100, 6→0110, 9→1001.3) Concatenate: 0100 0110 1001 → 010001101001.4) Result is the straight BCD form. Verification / Alternative check: Confirm by reading back the 4-bit groups: 0100=4, 0110=6, 1001=9. The round trip returns the original decimal digits without ambiguity. Why Other Options Are Wrong: 100101101000, 100001101001, 100101100100: These are incorrect encodings or reordered groupings and do not map to 4, 6, 9 as required. Common Pitfalls: Using pure binary for the entire number instead of per digit mapping, or forgetting that codes 1010–1111 are invalid in straight BCD. Final Answer: 010001101001

Question 5

Foundations of Digital Electronics — Underlying Number System Digital electronics fundamentally operates using which numbering system for representing logic levels and data?

Accepted Answer

Introduction / Context: Logic circuits use two distinct voltage ranges to represent information. These two states correspond naturally to a base-2 numbering system, making binary the fundamental language of digital hardware, storage, and computation. Given Data / Assumptions: Logic low and logic high are the only stable states in standard digital families. Transistors switch between two regions to encode information reliably. Systems can present higher level abstractions, but the hardware building blocks are two state devices. Concept / Approach: Map the two physical states to symbols 0 and 1. Larger numerical bases like octal and hex are useful shorthand for compact human representation, but at the hardware level all computation decomposes into binary operations and bit patterns. Step-by-Step Reasoning: 1) Recognize that digital storage elements hold two states.2) Relate two states to base-2 digits.3) Understand that octal and hex are only groupings of binary bits for convenience.4) Conclude that the foundational numbering system is binary. Verification / Alternative check: Examine truth tables, Karnaugh maps, and Boolean algebra. All of these formalisms operate over two valued logic, directly reflecting binary representation. Why Other Options Are Wrong: Decimal: Used for human readability, not hardware level operation. Octal and hexadecimal: Helpful groupings of binary but not fundamental. Common Pitfalls: Assuming shorthand bases change the underlying hardware representation; in practice, everything ultimately maps to binary signals. Final Answer: binary

Question 6

Decimal to Hexadecimal — Division and Remainder Convert the decimal number 74 to its hexadecimal form, showing the division by 16 process or an equivalent mental shortcut.

Accepted Answer

Introduction / Context: Decimal to hexadecimal conversion is frequent when interpreting addresses, constants, or color codes. The process uses repeated division by 16 and collecting remainders, or a quick decomposition into 16s and units if the number is small. Given Data / Assumptions: Decimal input: 74. Hex digit values: 10→A, 11→B, ... 15→F. No sign and no fractional part. Concept / Approach: Compute the quotient and remainder when dividing by 16. The quotient gives the high hex digit and the remainder gives the low hex digit. Alternatively, write 74 = 416 + 10 and map 10 to A directly. Step-by-Step Solution: 1) Perform division: 74 / 16 = 4 remainder 10.2) Map remainder 10 to hex A.3) High digit is 4 and low digit is A.4) Combine to get 4A base 16, written as 4A16. Verification / Alternative check: Convert back: 4A16 = 416 + A = 64 + 10 = 74 decimal. The round trip confirms the value. Why Other Options Are Wrong: A416 and B416: These represent 164 and 180 decimal respectively, not 74. 4B16: Equals 4*16 + 11 = 75, not 74. Common Pitfalls: Forgetting that A denotes 10, misreading the remainder, or reversing the order of quotient and remainder when forming the hex digits. Final Answer: 4A16

Question 7

Hex to Binary — Four Bits per Hex Digit Write the binary equivalent of the hexadecimal value F3A16 by expanding each hex digit into its 4-bit binary group.

Accepted Answer

Introduction / Context: Mapping hexadecimal to binary is direct because a hex digit represents exactly four binary bits. This ability is essential for interpreting bit fields, control registers, and instruction encodings without intermediate decimal steps. Given Data / Assumptions: Hex input: F3A16. Digit mapping: 0→0000, 1→0001, ..., 9→1001, A→1010, ..., F→1111. No sign, integer only. Concept / Approach: Translate each hex digit independently to a 4-bit binary group and concatenate in order. This preserves magnitude and avoids arithmetic errors that sometimes occur when converting via decimal. Step-by-Step Solution: 1) Expand digits: F→1111, 3→0011, A→1010.2) Concatenate groups in the same order: 1111 0011 1010.3) Remove spaces for the final binary: 111100111010.4) Therefore, F3A16 corresponds to 111100111010 base 2. Verification / Alternative check: Convert back by grouping the binary in fours: 1111→F, 0011→3, 1010→A. This reconstructs F3A16 exactly. Why Other Options Are Wrong: 111100111110: The last nibble 1110 represents E, not A. 000000111010 and 000011000100: These introduce leading zeros or different nibble values that do not match F3A16. Common Pitfalls: Reversing nibble order, dropping a nibble, or confusing A with 1011, which is actually B. Final Answer: 111100111010

Question 8

Address bus width to addressable memory mapping If a typical PC CPU exposes a 20-bit address bus, how many bytes of memory can it directly address?

Accepted Answer

Introduction / Context: Address-bus width determines how many unique memory locations a CPU can directly select. Each distinct address corresponds to one addressable unit (typically a byte in byte-addressable systems). This question reinforces the exponential relationship between address lines and addressable memory size, a cornerstone idea in computer architecture. Given Data / Assumptions: Address bus width: 20 bits. Byte-addressable memory (1 address → 1 byte). No bank switching or paging tricks; we compute the flat physical address space. Concept / Approach: With N address lines, the CPU can form 2^N unique addresses. In a byte-addressable system, that equals 2^N bytes. Convert bytes to megabytes (MB) as needed (1 MB = 2^20 bytes in binary-mebibyte terms commonly used for classic PCs). Step-by-Step Solution: Compute total addresses: total = 2^20.Evaluate 2^20 = 1,048,576 addresses.In a byte-addressable design: 1,048,576 bytes total.Convert to MB: 1,048,576 bytes = 1 MB (binary MB convention). Verification / Alternative check: Historical 8086/8088 PCs used a 20-bit physical address, yielding a 1 MB address space (00000H–FFFFFH), which exactly matches the calculation above. Why Other Options Are Wrong: 20 MB and 10 MB: would require 24+ bits; they vastly overstate the range for 20 bits. 580 MB: grossly incorrect; would imply far more than 29 bits. Common Pitfalls: Confusing data-bus width (bits per transfer) with address-bus width (address count), or mixing decimal MB with binary powers. Always anchor the calculation on 2^N addresses for N address lines. Final Answer: 1 MB

Question 9

Binary conversion practice — base-10 to base-2 Convert the decimal number 213 (base 10) into its binary representation (base 2).

Accepted Answer

Introduction / Context: Converting from decimal to binary is fundamental in digital systems. The idea is to express the same quantity using powers of 2. There are two common manual methods: repeated subtraction of powers of 2 and repeated division by 2. We will apply the powers-of-2 method to convert 213 to binary cleanly and verify it afterward. Given Data / Assumptions: Input: 213 in base 10. Unsigned representation; width is minimal (no fixed 8/16-bit constraint unless formatting options require). We will present an 8-bit form to match the options. Concept / Approach: Find the largest power of 2 ≤ number, set that bit, subtract, and continue. Powers of 2 near 213 are 128 (2^7), 64 (2^6), 32 (2^5), 16 (2^4), 8 (2^3), 4 (2^2), 2 (2^1), 1 (2^0). Set a bit to 1 when using that power, else 0. Step-by-Step Solution: Start: 213 ≥ 128 → set bit 7 = 1; remainder = 213 - 128 = 85.85 ≥ 64 → set bit 6 = 1; remainder = 21.21 ≥ 32 → no; bit 5 = 0.21 ≥ 16 → set bit 4 = 1; remainder = 5.5 ≥ 8 → no; bit 3 = 0.5 ≥ 4 → set bit 2 = 1; remainder = 1.1 ≥ 2 → no; bit 1 = 0.1 ≥ 1 → set bit 0 = 1; remainder = 0.Concatenate bits 7..0 → 11010101. Verification / Alternative check: Evaluate 11010101 as decimal: 128 + 64 + 16 + 4 + 1 = 213. Match confirmed. Why Other Options Are Wrong: 11001101 = 128 + 64 + 8 + 4 + 1 = 205. 01111001 = 64 + 32 + 16 + 8 + 1 = 121. 11100011 = 128 + 64 + 32 + 2 + 1 = 227. Common Pitfalls: Forgetting a power, reversing bit order, or assuming fixed-width without matching the option formatting. Always re-sum to double-check the binary pattern. Final Answer: 11010101

Question 10

BCD encoding — two-digit example Represent the decimal value 11 in BCD (Binary-Coded Decimal) using two 4-bit nibbles.

Accepted Answer

Introduction / Context: BCD encodes each decimal digit independently into 4 bits. Thus, a multi-digit number is stored as a sequence of 4-bit nibbles, one per decimal digit. This preserves decimal digit boundaries, which simplifies display and certain arithmetic routines at the cost of storage efficiency versus pure binary. Given Data / Assumptions: Target number: 11 in decimal (two digits: 1 and 1). Standard BCD mapping: 0→0000, 1→0001, …, 9→1001. No sign nibble or decimal point involved. Concept / Approach: Convert each decimal digit independently to a 4-bit code and then concatenate the codes in the same order (most significant digit first). For 11, both digits are 1, so both nibbles are identical. Step-by-Step Solution: Digit 1 → 0001.Second digit 1 → 0001.Concatenate: 0001 0001 → 00010001. Verification / Alternative check: Reverse mapping: split 00010001 into 0001 and 0001 → digits 1 and 1 → 11 decimal. This confirms correct BCD representation. Why Other Options Are Wrong: 00001011 is the pure binary of decimal 11 (1011) padded to 8 bits, not BCD. 00001100 equals binary 12 padded, not BCD for 11. 00010010 corresponds to digits 1 and 2 (i.e., 12 in BCD), not 11. Common Pitfalls: Confusing BCD with binary of the entire number, or reversing digit order. Always encode each decimal digit separately in BCD problems. Final Answer: 00010001

Question 11

Binary to hexadecimal by nibble grouping Convert the binary number 111111110010 (base 2) to hexadecimal (base 16).

Accepted Answer

Introduction / Context: Hexadecimal maps cleanly onto binary because every 4 binary bits (a nibble) correspond to one hex digit. Grouping the binary string into nibbles from right to left enables fast, error-resistant conversion. We will apply that rule here and verify the result by mapping each nibble to its hex equivalent. Given Data / Assumptions: Binary input: 111111110010. Left padding with zeros is allowed to complete the leftmost nibble. Use standard hex digits 0–9 and A–F. Concept / Approach: Partition the bits into 4-bit groups from right to left, then translate each group using the table 0000→0 … 1111→F. Keep the order of nibbles the same (most significant nibble on the left). Step-by-Step Solution: Group: 1111 1111 0010 (already a multiple of 4 bits, so no padding needed).Translate: 1111→F, 1111→F, 0010→2.Concatenate hex digits: FF2.Write with base indicator as per options: FF216. Verification / Alternative check: Convert the last nibble 0010 back to confirm → 2. The leading two nibbles 1111→F and 1111→F are standard and unambiguous. Reassembling returns to the original bit pattern. Why Other Options Are Wrong: EE216 would require nibbles 1110 1110, not present here. 2FE16 implies nibble order 0010 1111 1110, which is incorrect ordering. FD216 requires 1111 1101 0010; the middle nibble would be 1101 (D), not 1111 (F), so it does not match. Common Pitfalls: Grouping from the left instead of the right, forgetting to pad the most significant nibble, or accidentally reordering nibbles. Always move right-to-left in binary when forming hex nibbles. Final Answer: FF216

Question 12

Signed formats — sign-magnitude on 8 bits Express the decimal number −37 using 8-bit sign-magnitude representation (MSB is the sign).

Accepted Answer

Introduction / Context: Several binary signed-number schemes exist: sign-magnitude, one's complement, and two's complement. In sign-magnitude, the most significant bit (MSB) encodes the sign (0 = positive, 1 = negative), and the remaining bits encode the magnitude as an ordinary unsigned number. This question asks for −37 in 8-bit sign-magnitude form. Given Data / Assumptions: Total width: 8 bits. Sign-magnitude convention: MSB is sign; 0 = positive, 1 = negative. Magnitude to encode: 37 in decimal. Concept / Approach: First convert the magnitude to binary using as many lower bits as needed (here 7 magnitude bits). Then set the MSB to 1 to indicate a negative number. No inversion or addition is performed in sign-magnitude (contrast with two's complement). Step-by-Step Solution: Convert 37 to binary: 37 = 32 + 4 + 1 → bits → 00100101 (8-bit view).Extract the 7-bit magnitude: 0100101 (from the lower 7 bits of 00100101).Set MSB as sign = 1 for negative.Form final 8-bit word: 1 (sign) + 0100101 (magnitude) = 10100101. Verification / Alternative check: Check that the magnitude portion equals 00100101 when viewed as 8 bits with MSB cleared, and that the sign bit is 1. No complements are involved in sign-magnitude. Why Other Options Are Wrong: 00100101: sign bit 0 → +37, not −37. 11011000 or 11010001: these patterns do not represent −37 in sign-magnitude; they resemble unrelated bit patterns and, if interpreted as two's complement, would encode different values. Common Pitfalls: Confusing sign-magnitude with two's complement (which requires inverting and adding 1), or forgetting that only the MSB encodes the sign while the magnitude stays uninverted. Final Answer: 10100101

Question 13

Octal conversion — base-10 to base-8 Convert the decimal number 39 (base 10) into octal (base 8).

Accepted Answer

Introduction / Context: Octal (base 8) is often used to compactly represent binary on systems aligned to 3-bit groups. Converting a decimal number to octal can be done via repeated division by 8, collecting remainders. We will convert 39 to octal and verify via back-conversion to decimal for confidence. Given Data / Assumptions: Input: 39 in base 10. Octal digits range: 0–7. We expect a two-digit octal result for this small number. Concept / Approach: Use repeated division by 8. The octal digits are the remainders read in reverse order (last remainder becomes the most significant digit). Alternatively, recognize that 39 = 48 + 7 at a glance. Step-by-Step Solution: 39 ÷ 8 = 4 remainder 7 → least significant octal digit is 7.4 ÷ 8 = 0 remainder 4 → next digit is 4; division stops at 0.Write digits high→low: 4 then 7 → 47 in base 8.Match option format with a trailing 8: 478. Verification / Alternative check: Convert 47_8 back: 48 + 7 = 39. The round trip confirms correctness. Why Other Options Are Wrong: 638 and 748: include digits 8 or 9? (not here, but their values don't map to 39 when interpreted as octal strings presented). 368: digits are valid but equals 364 + 68 + 8? (format confusion). The correct two-digit octal for 39 is simply 47. Common Pitfalls: Writing remainders in the wrong order or confusing base markers. Always read remainders from last to first when finishing the division chain. Final Answer: 478

Question 14

BCD encoding — three-digit number Convert the decimal number 127 into packed BCD (concatenate one 4-bit nibble per decimal digit).

Accepted Answer

Introduction / Context: Binary-Coded Decimal (BCD) stores each decimal digit in its own 4-bit field. For a 3-digit number like 127, we form three nibbles in order: one for 1, one for 2, and one for 7. This preserves decimal readability and simplifies display drivers and certain decimal arithmetic operations. Given Data / Assumptions: Target: 127 (decimal). BCD map: 1→0001, 2→0010, 7→0111. Packed representation: concatenate nibbles MS digit to LS digit without separators. Concept / Approach: Translate each decimal digit independently to a 4-bit nibble and concatenate in order (hundreds, tens, ones). No binary arithmetic is needed; it is a direct symbol-to-nibble mapping exercise. Step-by-Step Solution: Hundreds digit 1 → 0001.Tens digit 2 → 0010.Ones digit 7 → 0111.Concatenate: 0001 0010 0111 → 000100100111. Verification / Alternative check: Split 000100100111 into nibbles: 0001 (1), 0010 (2), 0111 (7) → 127, confirming the mapping. Why Other Options Are Wrong: 011100100001 and 001010111 scramble nibble order or digits. 111010001 is neither proper BCD for 127 nor a correct concatenation of the required digit codes. Common Pitfalls: Confusing BCD with binary of 127 (which would be 1111111), or reversing digit order (721). Always encode each decimal digit as a valid nibble 0000–1001. Final Answer: 000100100111

Question 15

Octal to binary via 3-bit groups Convert the octal number 5278 to binary (base 2) by mapping each octal digit to a 3-bit group.

Accepted Answer

Introduction / Context: Octal and binary interconvert neatly because one octal digit corresponds to exactly three binary bits. Thus, conversion requires only a look-up per digit followed by concatenation—no arithmetic needed. We will convert 527_8 into binary using this 3-bit grouping rule. Given Data / Assumptions: Octal input: 527 (base 8). Mapping: 0→000, 1→001, 2→010, 3→011, 4→100, 5→101, 6→110, 7→111. Leading zeros are kept per digit if needed. Concept / Approach: Translate each octal digit independently to its 3-bit binary equivalent, preserving order. Concatenate the results to obtain the final binary representation. Step-by-Step Solution: Digit 5 → 101.Digit 2 → 010.Digit 7 → 111.Concatenate: 101 010 111 → 101010111. Verification / Alternative check: Convert back by regrouping the binary into 3-bit chunks from the right: 101 010 111 → 5 2 7 in octal. Round-trip confirms correctness. Why Other Options Are Wrong: 011100111 and 111010101 rearrange groups, not matching the digit-wise mapping. 343 is neither a binary string nor a valid base-2 representation; it is a decimal-looking number. Common Pitfalls: Dropping leading zeros in a 3-bit group (e.g., 2 should be 010, not 10), or reversing digit order. Keep exactly three bits per octal digit to avoid ambiguity. Final Answer: 101010111

Question 16

Binary to decimal — evaluate positional weights Convert the binary number 01011 (base 2) into its decimal (base 10) value.

Accepted Answer

Introduction / Context: Binary-to-decimal conversion uses positional weights: each bit contributes a power of 2 if set. Reading from right to left, the weights are 1, 2, 4, 8, 16, and so on. We will apply this directly to 01011_2 and verify by alternative grouping intuition if desired. Given Data / Assumptions: Binary input: 01011. Unsigned interpretation. No fixed width needed beyond the given bits. Concept / Approach: Sum the weights corresponding to 1-bits. A 0-bit contributes nothing for its position. The leading zero does not change the value; it only reflects formatting width. Step-by-Step Solution: Write weights for each position (left→right): 16, 8, 4, 2, 1.Match to bits: 0·16 + 1·8 + 0·4 + 1·2 + 1·1.Compute: 0 + 8 + 0 + 2 + 1 = 11.Therefore, 01011_2 = 11_10. Verification / Alternative check: Binary 01011 equals 1011 if leading zero is dropped; 1011 is a familiar pattern (8 + 2 + 1) → 11. Both views agree. Why Other Options Are Wrong: 35 and 15: far larger than the sum of these five positions with only three ones set; 15 would require 1111. 10: one less than the correct value; it omits the LSB contribution of 1. Common Pitfalls: Misaligning positional weights or dropping the final 1-bit contribution. Always label powers of 2 before summing to avoid mistakes. Final Answer: 11

Question 17

Binary to octal — group bits by threes Convert the binary number 010111100 (base 2) into its octal (base 8) representation.

Accepted Answer

Introduction / Context: Octal and binary interconvert by grouping binary digits into sets of three, starting from the right. Each 3-bit group corresponds to one octal digit (0–7). This method is fast and avoids arithmetic division when the binary string is already available. Given Data / Assumptions: Binary input: 010111100. We may pad the most significant side with zeros to form complete 3-bit groups. Octal digits map from 000→0 up to 111→7. Concept / Approach: Partition the binary string into 3-bit chunks from right to left, then translate each chunk to its octal digit. Keep the left-to-right order of the resulting octal digits. Step-by-Step Solution: Create groups: 010 111 100.Map to octal: 010→2, 111→7, 100→4.Concatenate octal digits: 274 (base 8).Match option formatting with trailing 8: 2748. Verification / Alternative check: Round-trip: 2→010, 7→111, 4→100; concatenation recreates 010111100. The mapping is consistent. Why Other Options Are Wrong: 1728 and 1748: first digit 1 would require leading bits 001 (not present here). 2728: middle digit 7 is correct, but the final digit 2 would require last group 010, whereas we have 100 → 4. Common Pitfalls: Grouping from the left instead of the right, or forgetting to pad on the leftmost side. Always ensure the total bit count is a multiple of three before conversion. Final Answer: 2748

Question 18

Digital number systems — terminology check. What is a single hexadecimal digit (one 4-bit value) commonly called in digital electronics?

Accepted Answer

Introduction / Context: A core idea in digital systems is that 4 bits conveniently map to one hexadecimal digit. Knowing the standard names for common bit-widths makes documentation, debugging, and memory/encoding discussions precise and fast. Given Data / Assumptions: One hexadecimal digit ranges from 0 to F (0–15 in decimal). A hexadecimal digit corresponds to exactly 4 binary bits. Standard computer-architecture terminology is used (bit, nibble, byte, word). Concept / Approach: Each hex digit expands to a 4-bit pattern (0000 through 1111). The well-accepted term for a 4-bit quantity is “nibble” (also spelled “nybble”). Eight bits form a “byte,” and a “word” is an architecture-dependent wider unit (often 16, 32, or 64 bits). Step-by-Step Solution: Map: 1 hex digit ⇔ 4 bits.Standard terminology: 4 bits = nibble.Therefore, one hex digit is a nibble. Verification / Alternative check: Hex is popular because 4 bits align perfectly with one digit, letting engineers compress long binary strings. For example, 0xAB maps to 1010 1011; each 4-bit group (1010 and 1011) is one “nibble.” Documentation and instruction sets frequently refer to high nibble (upper 4 bits) and low nibble (lower 4 bits) of a byte. Why Other Options Are Wrong: Byte: 8 bits, not 4. Grouping: a vague term, not a standard size. Instruction: an operation encoding; width varies. Word: platform-dependent width, larger than a nibble. Common Pitfalls: Confusing nibble with byte due to casual usage. Assuming “word” is universally 16 bits; it varies by CPU. Final Answer: nibble

Question 19

Base conversion — convert the hexadecimal number to binary. Task: Convert C0B₁₆ into its binary equivalent (expand each hex digit into 4 bits).

Accepted Answer

Introduction / Context: Hexadecimal and binary interconvert cleanly because one hex digit equals a 4-bit group. Engineers convert hex to binary to inspect bitfields, flags, and masks precisely during low-level programming and hardware work. Given Data / Assumptions: Hex number: C0B₁₆ Digit values: C = 12, 0 = 0, B = 11 No sign or fractional part; simple integer conversion Concept / Approach: Replace each hex digit with its 4-bit binary nibble: C → 1100, 0 → 0000, B → 1011. Concatenate the nibbles in order to obtain the final binary string with no leading suppression beyond the natural nibbles. Step-by-Step Solution: C → 11000 → 0000B → 1011Concatenate: 1100 0000 1011 ⇒ 110000001011 Verification / Alternative check: Convert back to hex by regrouping: 1100 (C), 0000 (0), 1011 (B). The round-trip confirms correctness. As a decimal sense-check: C0B₁₆ = 1216^2 + 016 + 11 = 3072 + 11 = 3083; binary 110000001011₂ also equals 3083₁₀. Why Other Options Are Wrong: 110000001001 / 110000001100 / 110100001011 / 111000001011: each has at least one nibble mismatched to the original hex digits. Common Pitfalls: Mistyping B as 1010 (A) or 1101 (D). Dropping or adding zeros in the middle nibble for 0. Final Answer: 110000001011

Question 20

Digital vs. analog — evaluate the statement. Claim: “The voltages in digital electronics are continuously variable.” Decide whether this claim is true or false.

Accepted Answer

Introduction / Context: Digital electronics represent information using discrete logic levels (commonly two: LOW and HIGH). Although underlying voltages are physical and can vary, digital design defines threshold regions to interpret signals as one of a small set of stable symbols, not a continuum of values for data encoding. Given Data / Assumptions: Standard 2-level digital logic (TTL/CMOS) is assumed. Logic thresholds partition voltages into LOW, undefined, and HIGH regions. Noise margins ensure robust binary interpretation. Concept / Approach: In digital systems, a wire carries symbols (0 or 1) determined by thresholding, not precise analog magnitudes. While signals physically transition continuously during edges, the information model uses discrete states. Multilevel schemes (e.g., PAM-4) still use a finite set of levels, not continuous values. Step-by-Step Reasoning: Define logic levels: e.g., for 5 V CMOS, VIL and VIH set LOW/HIGH thresholds.Interpretation: any voltage below VIL is a logic 0; above VIH is logic 1.Transient behavior is analog, but the data abstraction is discrete. Verification / Alternative check: Look at datasheet noise margins: as long as signals stay within guaranteed ranges, the receiver decodes a clean 0 or 1 regardless of small analog variation, proving information is not “continuously variable.” Why Other Options Are Wrong: True / conditional variants: they confuse physical voltages with the symbolic digital abstraction; even multilevel signaling uses discrete sets. Common Pitfalls: Equating analog transitions during edges with analog information encoding. Assuming ADC inputs define digital logic; ADCs convert analog to discrete codes. Final Answer: False

Number Systems and Codes Questions