Question 1

Addition terminology: In an addition operation, the input operands are called the “augend” and the “addend,” whose sum produces the result.

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Introduction / Context: Clear language helps when documenting arithmetic units, verifying HDL, or teaching fundamentals. The standard names for the two inputs in addition are “augend” and “addend.” Given Data / Assumptions: Binary or decimal does not change the operand naming. The operation is x + y = z. We consider integer addition here, but the terminology is general. Concept / Approach: The augend is the first operand; the addend is the second operand being added to the augend. Their order has no effect on the sum due to commutativity, but the names are widely used in arithmetic descriptions and adder block diagrams. Step-by-Step Solution: Define roles: x is the augend, y is the addend.Compute sum: z = x + y using a half adder/full adder network as needed.Confirm terminology independent of hardware implementation. Verification / Alternative check: Textbook diagrams typically label inputs to an adder as augend and addend, with carry chains for multi-bit adders. Why Other Options Are Wrong: Claiming dependence on floating-point or word length is incorrect; names are conceptual, not implementation-specific. Common Pitfalls: Using “operand A/B” instead is fine in engineering drawings, but knowing formal terms helps in academic contexts. Final Answer: Correct

Question 2

Compute the two’s complement of the binary number 110110₂. Choose the correct 6-bit result.

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Introduction / Context: Two’s complement is obtained by inverting all bits (one’s complement) and then adding 1. This operation is central to implementing subtraction using a binary adder and to representing negative integers in hardware. Given Data / Assumptions: Original number: 110110₂ (6 bits). We want the 6-bit two’s complement. No overflow beyond 6 bits is retained. Concept / Approach: The algorithm is: two’s complement(x) = invert_bits(x) + 1 (within the same bit width). The invert operation flips each 1 to 0 and each 0 to 1. The final addition is performed modulo 2^n, where n is the word size (here n = 6). Step-by-Step Solution: Start with x = 1 1 0 1 1 0.One’s complement: invert bits → 0 0 1 0 0 1.Add 1: 001001 + 000001 = 001010.Therefore, two’s complement(110110₂) = 001010₂. Verification / Alternative check: Check by addition: 110110₂ + 001010₂ = 1000000₂ (a 7-bit result). Discarding the carry beyond 6 bits yields 000000₂, confirming they are complements within 6 bits. Why Other Options Are Wrong: 110100₂ and 101010₂ are not obtained by invert-plus-one. 001011₂ is off by 1 from the correct complement (it is the one’s complement plus 0). Common Pitfalls: Forgetting the final +1 after inversion, or accidentally extending the width during addition. Final Answer: 001010₂

Question 3

Definition of a full adder: A full adder stage adds two single-bit operands along with an incoming carry (Cin) and produces a Sum and a carry-out (Cout). What does a full adder add?

Accepted Answer

Introduction / Context: A full adder is the fundamental building block for multi-bit ripple-carry adders. Understanding its inputs clarifies how larger adders are constructed by cascading stages. Given Data / Assumptions: Single full-adder cell, not a 4-bit adder IC. Inputs: A (1 bit), B (1 bit), Cin (1 bit). Outputs: Sum (1 bit), Cout (1 bit). Concept / Approach: The full adder solves a three-input addition problem at the bit level. The logic implements the equations Sum = A XOR B XOR Cin and Cout = majority(A, B, Cin), enabling carry propagation from less significant bits to more significant bits in a ripple chain. Step-by-Step Solution: Accept A, B, Cin as inputs.Compute Sum using cascaded XOR operations.Compute Cout using combinations of pairwise products (A*B + A*Cin + B*Cin) or a majority gate equivalence. Verification / Alternative check: Truth tables confirm that for all 8 combinations of A, B, Cin, the Sum/Cout match binary addition rules. Why Other Options Are Wrong: Options referring to “2-bit” or “4-bit” numbers confuse a single full-adder cell with multi-bit adder ICs that contain several full-adder stages. Common Pitfalls: Mixing up half adder vs full adder; a half adder lacks Cin and cannot be directly cascaded for accurate multi-bit addition with carry propagation. Final Answer: two single bits and one carry bit

Question 4

Carry propagation delay in ripple adders: In a 4-bit ripple-carry adder built from full-adder stages, how does carry propagation delay affect overall speed?

Accepted Answer

Introduction / Context: Ripple-carry adders (RCAs) are simple but slow because each full-adder stage must wait for the carry from the previous stage. Understanding carry propagation is key to appreciating faster adder architectures like carry look-ahead. Given Data / Assumptions: A 4-bit adder made of four cascaded full adders. Each full adder has a finite propagation delay from inputs to Sum/Cout. Worst-case input change requires the carry to ripple through all stages. Concept / Approach: The total worst-case delay is approximately the sum of the per-stage carry propagation delays plus the final Sum delay at the most significant stage. Therefore, as you add more bits, the delay grows linearly, directly limiting maximum clock frequency or throughput for arithmetic operations. Step-by-Step Solution: Model one stage delay as tFA (carry propagation).For 4 bits, worst-case carry path traverses 4 stages → roughly 4 * tFA (+ Sum output delay at MSB).Maximum operating speed is bounded by this accumulated delay. Verification / Alternative check: Compare to carry look-ahead adders where generate/propagate logic reduces the dependency chain, dramatically cutting the worst-case delay. Why Other Options Are Wrong: “Not a consideration” is incorrect; even nanosecond delays matter at high frequencies. Claims that delay decreases with more stages are physically wrong. Saying it “never limits speed” ignores real timing constraints. Common Pitfalls: Neglecting the Sum output delay at the MSB and ignoring wiring/capacitive loading, which further increases the effective delay. Final Answer: is cumulative for each stage and limits the speed at which arithmetic operations are performed

Question 5

Digital arithmetic systems: what must be indicated for numbers?  In digital electronics and computer arithmetic, any system that performs addition, subtraction, multiplication, or division must represent numbers unambiguously. Which pair of attribut…

Accepted Answer

Introduction / Context: Digital systems execute arithmetic on binary-encoded data. For any computation to be meaningful, the representation must clearly convey whether the value is positive or negative and how large it is. This question checks your understanding of the two essential attributes that must be specified for numbers used in arithmetic operations inside processors, digital signal processors, and arithmetic-logic units. Given Data / Assumptions: Numbers are represented in a positional binary format inside digital hardware. Arithmetic operations require unambiguous interpretation of the operands. The context could be signed magnitude, 2’s complement, 1’s complement, or other signed schemes. Concept / Approach: Two independent properties define a signed numeric quantity: the sign (positive or negative) and the magnitude (the absolute size). Binary formats encode these in different ways. Signed-magnitude uses a dedicated sign bit plus magnitude bits. Two’s complement embeds the sign within the most significant bit and defines a wrap-around coding for negatives, but the underlying conceptual attributes remain sign and magnitude. The radix (base) of the system is also important for interpretation, but arithmetic hardware is already designed for a fixed base (almost always base-2), so the critical per-number attributes are sign and magnitude. Step-by-Step Solution: Identify what arithmetic needs: direction (sign) and amount (magnitude).Check representations: signed-magnitude and two’s complement both encode sign and magnitude (albeit differently).Eliminate distractors: base is fixed for a given machine and not part of each number’s per-value attributes.Conclude: the required pair is sign and magnitude. Verification / Alternative check: Consider subtraction A − B implemented as A + (−B). To form −B you need to know the sign rule and the magnitude of B; two’s complement does this by inversion plus 1, but conceptually still uses sign and magnitude. Why Other Options Are Wrong: Size, base: “size” is vague; base is fixed by the system, not per number.Sign, base: sign is relevant, but base is not the second per-number attribute needed.Magnitude, base: omits sign, which is critical for negative values. Common Pitfalls: Confusing format (e.g., two’s complement) with the conceptual attributes; assuming base must be labeled per value rather than per system. Final Answer: sign, magnitude

Question 6

ALU configuration: what does the mode pin select?  An arithmetic-logic unit (ALU) includes a “mode” control input. What does this mode input determine about the operation being performed?

Accepted Answer

Introduction / Context: An arithmetic-logic unit (ALU) is the computational heart of a processor. It typically supports two broad classes of operations: arithmetic (add, subtract, increment, etc.) and logical (AND, OR, XOR, NOT, etc.). Many classic ALUs expose a “mode” pin to select which class of function is active. Given Data / Assumptions: The device is an ALU from a standard family (e.g., 74xx181) where a mode input exists. Control lines further select a specific function within the chosen mode. Carry inputs and complements may be controlled by other pins, not the primary mode line. Concept / Approach: Mode lines are high-level selectors. In many ALUs, a single mode input chooses between arithmetic and logic “blocks.” Additional select inputs then choose a particular arithmetic (e.g., A + B, A − 1) or logic (e.g., A XOR B) function. Thus, the mode pin toggles the operational domain rather than the sign, one’s complementing, or carry handling directly. Step-by-Step Solution: Identify role of mode: top-level function class selection.Check common ALUs: datasheets show M=0 for arithmetic, M=1 for logic (or vice versa).Notice that carry and complement are managed by separate inputs (C_in, function selects).Therefore, mode determines arithmetic vs logic operation. Verification / Alternative check: Review truth tables from standard ALUs: the same function select codes produce different results depending on the mode state. Why Other Options Are Wrong: One’s complemented: inversion is typically a separate control or part of logic functions.Positive or negative: sign is a property of operands/results, not a mode selection.With or without carry: carry enable/propagate is a separate signal. Common Pitfalls: Assuming “mode” equals “subtract mode”; subtraction usually uses addition plus two’s complement, controlled by carry and invert signals. Final Answer: arithmetic or logic

Question 7

How is the sign indicated in binary number systems?  In common signed binary representations used by digital systems, how is the sign of a number conveyed within its bit pattern?

Accepted Answer

Introduction / Context: Hardware works with fixed-size binary words. To represent negative values, digital systems employ signed encodings such as signed magnitude, one’s complement, and two’s complement. Each scheme embeds sign information within bits of the word rather than attaching a printed minus sign like in human-readable notation. Given Data / Assumptions: The question refers to how the sign is indicated inside the binary encoding itself. We consider standard signed schemes used in computers. Magnitude bits indicate size; a special position usually conveys sign. Concept / Approach: Signed formats dedicate or effectively use the most significant bit (MSB) to encode sign. In signed magnitude, the MSB is 0 for positive and 1 for negative, with remaining bits as magnitude. In two’s complement, the MSB still indicates negativity (1 means negative), but the encoding of magnitude is transformed to support easy arithmetic. Regardless of the exact scheme, the presence of a sign-designating bit alongside magnitude-related bits is the unifying concept. Step-by-Step Solution: Identify that the sign must be part of the bit pattern.Recall that the MSB functions as a sign indicator in signed encodings.Confirm that human symbols like “−” are not stored; machines use bits.Select: “including a sign bit along with the magnitude bits.” Verification / Alternative check: Arithmetic units detect overflow by examining MSB and carry behavior in two’s complement; this relies on sign being encoded in the MSB. Why Other Options Are Wrong: Leading 0 for negative: opposite of convention; 0 usually indicates non-negative.Inverting bits if negative: that is part of one’s complement formation, not the universal sign indication.Placing a negative sign in front: that is textual, not binary storage. Common Pitfalls: Confusing signed magnitude with two’s complement; both still have a sign-determining MSB. Final Answer: including a sign bit along with the magnitude bits

Question 8

Cascading adders: result of two 4-bit blocks  If you cascade two standard 4-bit binary adders (properly connecting carry-out to carry-in), what width of a parallel adder can you construct?

Accepted Answer

Introduction / Context: Modular design allows wider arithmetic units to be built from smaller ICs. A classic example is using two 4-bit adder ICs (such as 74xx83) with the carry chain properly linked to realize an 8-bit adder. Understanding cascading is fundamental to building scalable datapaths. Given Data / Assumptions: Each IC implements a 4-bit parallel adder with carry-in and carry-out. Carry-out of the lower nibble feeds carry-in of the upper nibble. Inputs and sums are arranged to form a continuous 8-bit addition. Concept / Approach: Parallel adders add all bits of each operand simultaneously, with the carries rippling from least significant to most significant positions. By chaining two 4-bit blocks, you extend the operand width to 8 bits, preserving the same arithmetic behavior with a longer carry path. Step-by-Step Solution: Take two 4-bit adders: lower handles bits 3..0, upper handles bits 7..4.Connect CO (lower) → CI (upper) to propagate carries.Wire operand bits accordingly and collect an 8-bit sum plus a final carry-out.Resulting function: an 8-bit parallel-adder circuit. Verification / Alternative check: Timing analysis shows ripple-carry delay accumulates across both devices; this matches expectations for an 8-bit adder built from two 4-bit blocks. Why Other Options Are Wrong: 16-bit adder: would require four 4-bit adders.Full-adder circuit: each 4-bit IC already contains multiple full adders; cascading yields a wider parallel adder, not a single full adder.Arithmetic-logic unit: an ALU includes more than addition (logic ops, shifts); two adders alone are insufficient. Common Pitfalls: Forgetting the carry chain; without CO→CI linkage the upper nibble ignores carries. Final Answer: 8-bit parallel-adder circuit

Question 9

Why use two's complement for signed integers?  Select the primary advantage of the 2's complement number system for representing signed binary integers in digital hardware.

Accepted Answer

Introduction / Context: Two’s complement is the dominant signed-integer representation in general-purpose processors. Its popularity stems from the way it simplifies hardware design, especially for subtraction, overflow detection, and representation of zero and negative values without dual “+0/−0.” Given Data / Assumptions: We are comparing signed encodings (signed magnitude, one’s complement, two’s complement). Goal: identify a key computational advantage in hardware. Standard binary adders are available. Concept / Approach: In two’s complement, subtraction A − B is implemented as A + (two’s complement of B). The same adder circuit performs both addition and subtraction by conditionally inverting B and adding 1 via the carry-in. This unifies the hardware path and eliminates the need for a separate subtractor. Additional benefits include a unique zero and straightforward overflow rules. Step-by-Step Solution: Form −B by inverting B then adding 1.Compute A − B = A + (−B) using the existing adder.Use carry-in = 1 to add the +1 needed for two’s complement.Hardware re-use reduces complexity and speeds arithmetic units. Verification / Alternative check: Examine an ALU control table: addition and subtraction share the adder core, differing only by operand inversion and carry-in setting. Why Other Options Are Wrong: Addition by performing subtraction: not a typical or advantageous approach.Division through repeated subtraction: possible in principle but not a special benefit of the encoding.None of the above: incorrect because the core advantage is well known. Common Pitfalls: Confusing one’s complement with two’s complement (one’s complement has +0 and −0 representations). Final Answer: can perform subtraction by performing addition

Question 10

Compute the 1's complement of a binary word  Find the 1's complement (bitwise inversion) of the binary number 101010₂. Select the correct result.

Accepted Answer

Introduction / Context: Complements are foundational in digital arithmetic. The 1’s complement of a binary number is produced by flipping every bit: 1 becomes 0 and 0 becomes 1. This operation is used in some arithmetic schemes and in checksum calculations, so fluency with complements is essential. Given Data / Assumptions: Original number: 101010₂ (six bits). 1’s complement means bitwise inversion only (no +1 as in 2’s complement). We preserve the bit-width. Concept / Approach: Apply the transformation b_i → 1 − b_i to each bit. This is equivalent to XOR with 1 for every position. Perform it left to right or right to left—order does not matter as long as each bit is inverted once. Step-by-Step Solution: Write the original: 1 0 1 0 1 0Invert each bit: 0 1 0 1 0 1Collect the result: 010101₂Check parity of pattern: alternating bits remain alternating after inversion. Verification / Alternative check: Use a quick mental check: the pattern 101010 becomes 010101, confirming that each position flipped. Why Other Options Are Wrong: 010110₂: differs in the fifth bit; not a pure inversion.110111₂: multiple bits do not match an inversion of the original.101011₂: differs only in LSB; that is a +1 change, not inversion. Common Pitfalls: Confusing 1’s complement (invert) with 2’s complement (invert then add 1). Final Answer: 010101₂

Question 11

Terminology: what do we call the bits that convey “how large” a number is?  In signed binary representations, which term refers specifically to the bits that indicate the size (absolute value) of the number, independent of its sign?

Accepted Answer

Introduction / Context: Signed number encodings separate the idea of sign (positive/negative) from size (absolute value). The bits that reflect the size are essential for understanding range, resolution, and arithmetic behavior. This question tests your familiarity with the standard terminology used in digital design. Given Data / Assumptions: We are considering signed formats (e.g., signed magnitude, two’s complement). The sign information is represented (explicitly or implicitly) by specific bit positions (often the MSB). Other bits describe the absolute quantity. Concept / Approach: In any signed encoding, we conceptually parse the word into sign information plus the remaining bits that specify the absolute size. Those remaining bits are called magnitude bits. Even in two’s complement, where interpretation is more involved, the term “magnitude bits” is useful when discussing the absolute value of the representation. Step-by-Step Solution: Identify the role of each field in the word: sign vs. size.Recognize the dedicated terminology: “magnitude bits” = bits representing size.Eliminate unrelated terms (carry, MSB-only) that do not capture the concept.Select “magnitude bits.” Verification / Alternative check: Consider signed-magnitude encoding: the MSB is the sign bit; the remaining bits are literally the magnitude bits. Why Other Options Are Wrong: Carry bits: refer to carry signals between adders, not size representation.Sign bits: indicate polarity, not size.Most significant bits: a positional descriptor; MSBs can include sign in signed formats, but “MSB” is not synonymous with “magnitude bits.” Common Pitfalls: Equating “most significant bits” with magnitude bits without considering that the MSB may be a sign bit in signed formats. Final Answer: magnitude bits

Arithmetic Operations and Circuits Questions