Binary System — Set 1

Correct Answer: A. Base 2

• **Base 2** = the binary number system uses exactly two symbols — 0 and 1 — making it base 2. Every digital circuit is built on transistors that switch between two physical states (off/on), which maps perfectly to 0 and 1. • **Only two digits** — unlike decimal (0–9) or octal (0–7), binary has no digit 2 or higher, so every quantity must be expressed as combinations of 0s and 1s. • All modern CPUs, RAM, and storage devices ultimately store and process every kind of data — text, images, audio — as streams of binary digits. • Option B (Base 10) is wrong because that is the familiar decimal system used daily; Option C (Base 8) is wrong because that describes the octal system; Option D (Base 16) is wrong because that describes hexadecimal, which uses digits 0–9 plus A–F.

Correct Answer: D. Gottfried Wilhelm Leibniz

• **Gottfried Wilhelm Leibniz** = the German mathematician who formalized the modern binary system in 1679 and published 'Explication de l'Arithmétique Binaire' in 1703. He showed how every number can be expressed with only 0 and 1, and his work became the theoretical seed for digital logic centuries later. • **Legacy** — George Boole and Claude Shannon later built Boolean algebra and switching theory directly upon Leibniz's binary framework, making modern computers possible. • Although ancient cultures used binary-like patterns, Leibniz was the first to give it a complete arithmetic framework with a clear positional notation. • Option A (Charles Babbage) is wrong because Babbage designed mechanical decimal calculators in the 1800s; Option B (Alan Turing) is wrong because Turing formalized computation theory in the 1930s but did not invent binary; Option C (Ada Lovelace) is wrong because she wrote the first algorithm for Babbage's engine and never worked on binary number theory.

Correct Answer: A. A Bit

• **A Bit** = the smallest unit of digital information, short for 'binary digit.' A bit holds exactly one of two values — 0 or 1 — representing the low or high voltage state of a transistor in an electronic circuit. • **Physical reality** — in CMOS circuits, logic 0 is near 0 V and logic 1 is near 3.3 V or 5 V; this two-state voltage duality makes binary the natural language of electronics. • Eight bits together form one byte, the standard unit for encoding a single character such as 'A' (01000001 in ASCII). • Option B (A Nibble) is wrong because a nibble is 4 bits grouped together, not a single binary symbol; Option C (A Byte) is wrong because a byte is 8 bits; Option D (A Word) is wrong because a word is 16, 32, or 64 bits depending on the processor architecture.

Correct Answer: D. 8 bits

• **8 bits** = one standard byte consists of exactly eight binary digits. This grouping became universal because it is large enough to encode all 128 ASCII characters and is a convenient power of 2 (2³). • **Memory addressability** — most computer architectures treat the byte as the smallest individually addressable memory unit, so every RAM location stores exactly one byte. • A byte holds 2⁸ = 256 unique values (0–255), which is why pixel color channels (R, G, B) each range from 0 to 255 in standard image formats. • Option A (4 bits) is wrong because 4 bits form a nibble, not a byte; Option B (32 bits) is wrong because 32 bits form a double-word or a 32-bit integer; Option C (16 bits) is wrong because 16 bits form a word in many processor architectures.

Correct Answer: A. 1010

• **1010** = the binary representation of decimal 10. Calculation: 10 = 8 + 2 = 1×2³ + 0×2² + 1×2¹ + 0×2⁰, producing the bit pattern 1-0-1-0. • **Positional weights** — binary positions from right to left carry weights 1, 2, 4, 8 (powers of 2). Only the positions for 8 and 2 are turned on, giving 8+2 = 10. • Verification: 1010 → (1×8)+(0×4)+(1×2)+(0×1) = 8+0+2+0 = 10. Correct. • Option B (1100) is wrong because 1100 = 8+4 = 12; Option C (1001) is wrong because 1001 = 8+1 = 9; Option D (1111) is wrong because 1111 = 8+4+2+1 = 15.

Correct Answer: A. Nibble

• **Nibble** = exactly four binary bits, half of a byte. The name is a computing pun on 'byte' (a bite vs. a nibble), and it is standard terminology in digital electronics and assembly language programming. • **Hexadecimal link** — one nibble maps exactly to one hex digit (0–F) because 2⁴ = 16, the same as hex's base. This is why one byte always equals two hex digits — convenient for reading memory dumps. • Nibbles are also the storage unit in BCD (Binary-Coded Decimal) encoding, where each decimal digit 0–9 occupies its own 4-bit nibble. • Option B (Packet) is wrong because a packet is a network data unit measured in bytes, not bits; Option C (Word) is wrong because a word is 16 or 32 bits depending on architecture; Option D (Segment) is wrong because a segment is a memory management concept, not a bit grouping.

Correct Answer: D. 15

• **15** = the maximum decimal value representable by a 4-bit binary number. When all four bits are 1 (pattern 1111), the value is 8+4+2+1 = 15. The total state count is 2⁴ = 16, but counting starts at 0, so the max is 16−1 = 15. • **Why not 16?** — 16 is how many different values are possible (0 through 15), but the highest value in that range is 15, not 16. • The 0–15 range is also why one nibble maps perfectly to one hexadecimal digit, since hex digits span 0–F (0–15). • Option A (8) is wrong because 8 is only the weight of the MSB position (2³), not the maximum 4-bit value; Option B (31) is wrong because 31 = 11111 in binary, requiring 5 bits; Option C (16) is wrong because 16 is the count of possible values, not the maximum value itself.

Correct Answer: A. AND Gate

• **AND Gate** = a logic gate that outputs 1 only when ALL inputs are 1. Its two-input truth table yields only one true output: (1,1)→1; inputs (0,0), (0,1), and (1,0) all produce 0. • **Hardware implementation** — AND gates are built with two transistors in series so current flows to the output only when both transistors conduct simultaneously, physically enforcing the AND rule. • In binary arithmetic, AND behaves like multiplication: 1×1=1, 1×0=0, 0×0=0, making it the key operation for bit-masking (isolating specific bits in a value). • Option B (NOT Gate) is wrong because NOT is a single-input gate that inverts its input, not a conjunction; Option C (OR Gate) is wrong because OR outputs 1 whenever at least one input is 1; Option D (XOR Gate) is wrong because XOR outputs 1 only when inputs differ, not when both are 1.

Correct Answer: D. 1

• **1** = the binary representation of decimal 1. In any positional system, the rightmost place has weight base⁰ = 1, so a single '1' in that position represents exactly one — identical notation in both binary and decimal. • **No conversion required** — since 1 is less than the binary base (2), it fits in the units column alone without generating a carry, making it the trivial binary number. • Binary and decimal share the symbols '0' and '1'; it is only from the number 2 onward that notation diverges (decimal writes '2' while binary writes '10'). • Option A (0) is wrong because 0 represents zero, not one; Option B (11) is wrong because binary 11 = 2+1 = 3 in decimal; Option C (10) is wrong because binary 10 = 2 in decimal.

Correct Answer: A. NOT

• **NOT** = a unary logic operation that inverts a single bit: NOT 0 = 1, NOT 1 = 0. It is the simplest gate — one input, one output — and performs the logical complement of whatever bit it receives. • **Hardware** — a NOT gate (inverter) uses a single transistor: a high input voltage (logic 1) saturates the transistor and pulls the output low (logic 0), and vice versa. • NOT is the foundation of ones' complement arithmetic and is embedded in NAND, NOR, and XNOR gates, making it central to all digital circuit design. • Option B (OR) is wrong because OR takes two inputs and outputs 1 if either is 1 — it does not invert a bit; Option C (AND) is wrong because AND outputs 1 only when both inputs are 1, which is not inversion; Option D (NAND) is wrong because NAND is a two-input gate (NOT-AND) and cannot invert a single bit on its own.