Binary Number

Binary Number

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A binary number is written in base-2, using only the digits 0 and 1, where each position carries a weight of successive powers of 2. The rightmost digit has weight 2⁰, the next 2¹, then 2², and so on; the leftmost digit is the most significant bit (MSB) and the rightmost is the least significant bit (LSB).

Binary → decimal: multiply each bit by 2 raised to its position index and sum. For example, 1011₂ = 1·2³ + 0·2² + 1·2¹ + 1·2⁰ = 8 + 0 + 2 + 1 = 11.

Decimal → binary: repeatedly divide the number by 2 and record the remainders; read them bottom-up (reverse order).

Addition rule: 1 + 1 = 0 with carry 1. 1’s complement flips every bit; 2’s complement = 1’s complement + 1, used to represent negative numbers. NDA typically tests conversions and basic addition; expect 1 MCQ from this 2%-weighted topic.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Positional Value and Conversion

Binary is a positional numeral system with radix 2. An n-bit number written as b_{n−1} b_{n−2} … b₁ b₀ equals Σ (bᵢ · 2ⁱ). The smallest n-bit unsigned value is 0, and the largest is 2ⁿ − 1 (e.g., an 8-bit byte spans 0 to 255).

To convert a decimal integer N to binary, divide by 2 repeatedly until the quotient is 0; the remainders, read from last to first, give the binary digits. For fractions, multiply the fractional part by 2 repeatedly and record the integer parts — the process either terminates (denominator is a power of 2) or repeats.

Binary Arithmetic

Addition follows column-wise rules: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 10 (write 0, carry 1). Subtraction uses borrow at base 2, but in digital systems it is implemented by adding the 2’s complement of the subtrahend. The 1’s complement of a binary string simply inverts every bit; the 2’s complement adds 1 to that result, yielding the standard signed representation where the MSB acts as the sign bit (0 = positive, 1 = negative).

Quick Reference Table

Operation	Rule	Example
B → D	Σ bᵢ · 2ⁱ	1101₂ = 13
D → B	Divide by 2, reverse remainders	13 → 1101
1’s comp	Flip every bit	1101 → 0010
2’s comp	1’s comp + 1	1101 → 0011
Add 1+1	0, carry 1	—

Typical NDA Question Patterns

NDA Mathematics papers frame this topic either as direct conversion (decimal ↔ binary) or as a single arithmetic question involving 101 + 110 type additions, plus complement-based negation. Since weightage is only ~2%, mastery of four-bit conversions and two’s complement suffices.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Subtleties

A leading zero carries no value, so 01101₂ = 1101₂ = 13, but it matters when the bit-width is fixed (e.g., a signed 8-bit register). Fractions like 0.101₂ equal ½ + 0 + ⅛ = 0.625; binary fractions either terminate or recur, mirroring the way decimal fractions either terminate or recur depending on the prime factors of the denominator.

Worked Example

Convert 45 to binary: 45 ÷ 2 = 22 r 1; 22 ÷ 2 = 11 r 0; 11 ÷ 2 = 5 r 1; 5 ÷ 2 = 2 r 1; 2 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1. Reading remainders upward: 101101₂. Verify: 32 + 8 + 4 + 1 = 45. ✓

Now compute the 2’s complement of 45 in 8-bit form: 00101101 → flip bits → 11010010 → add 1 → 11010011₂, which represents −45 in signed arithmetic.

Common Mistakes

• Writing 10₂ as “ten” instead of two.
• Reading remainders top-to-bottom during decimal-to-binary conversion.
• Forgetting the carry when adding 1 + 1 + 1 = 11₂ (answer 1, carry 1).
• Confusing 1’s and 2’s complement when a question asks for the negative of a binary number.

Connection to Adjacent Topics

Binary arithmetic underpins Boolean algebra (AND, OR, XOR), digital logic gates, and hexadecimal (base-16) shorthand, where each hex digit maps to four binary bits. NDA occasionally links this to number systems in general (octal, hex) within the same 2% slice.

Practice Prompts

1. Convert 173 to binary and verify by expansion.
2. Find the 2’s complement of 01100100₂ and state the signed decimal value it represents.

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