NDA Maths · Teaching notes

Binary Numbers — NDA Maths

Binary Numbers is a small but reliable chapter: 13 PYQs span 2017–2025, and almost every one rewards the same first move — translate the binary strings into ordinary decimal, do the easy arithmetic there, and (if asked) translate the answer back. The marks are rarely in the binary itself; they are in spotting that a question dressed up in base 2 is really a one-line place-value conversion, a simple division, or a familiar algebra identity. The notes teach in three movements, foundations first: (1) Binary to Decimal Conversion — what base 2 means, why place values are powers of 2, converting a binary string to decimal, and converting decimal back to binary by repeated division; (2) Binary Arithmetic — adding, subtracting and dividing in binary (and the unknown-digit puzzles that hide an addition), plus the recurring cube identities where the numbers just happen to be given in binary; (3) Binary Representation and Number Theory — counting/representing numbers and the few modular-arithmetic and perfect-square recall items the chapter files here. Convert-first is the chapter's centre of gravity: master decimal ↔ binary, and the rest is arithmetic you already know. Every PYQ is tagged.

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Subtopic notes

PYQ weightage by concept

8 concepts · 13 PYQs — where the marks actually sit, so you know what to drill first

Binary ↔ Decimal Conversion3 PYQs · 23%

Concept	PYQs	Share
Converting Binary to Decimal	2	15%
Converting Decimal to Binary	1	8%
What Base 2 Means — Place Values Are Powers of 2foundation	—	—

Binary Arithmetic — Addition, Division & Algebraic Identities7 PYQs · 54%

Concept	PYQs	Share
Binary Addition, Subtraction & Unknown-Digit Puzzles	3	23%
Binary Division — Quotient and Remainder	2	15%
Algebraic Identities with Binary-Given Values	2	15%

Binary Representation & Number Theory3 PYQs · 23%

Concept	PYQs	Share
Number-Theory One-Liners — Remainder Cycles & Sum of Odd Numbers	2	15%
Representing a Number in Binary & Counting Its Bits	1	8%

Formula & revision sheet

8 formulas · 11 gotchas across all subtopics — the exam-eve cheat-sheet

Binary ↔ Decimal Conversion

Formulas (3)

What Base 2 Means — Place Values Are Powers of 2 · Powers of 2 (place values) $\begin{array}{c|ccccccccc} 2^n & 2^9 & 2^8 & 2^7 & 2^6 & 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 \\ \hline \text{value} & 512 & 256 & 128 & 64 & 32 & 16 & 8 & 4 & 2 & 1 \end{array}$
Converting Binary to Decimal · Binary → decimal $(b_n\ldots b_1 b_0)_2 = \sum_{i=0}^{n} b_i\, 2^i$
Converting Decimal to Binary · Repeated division by 2 $N \xrightarrow{\div 2} (q_1, r_1) \xrightarrow{\div 2} (q_2, r_2) \to \cdots \to 0;\quad N = (\,r_k \ldots r_2\, r_1)_2$

Watch out for (3)

Place values grow leftward, from 2⁰ on the RIGHT
→ What Base 2 Means — Place Values Are Powers of 2
A 0 bit contributes nothing — don't add its place value
→ Converting Binary to Decimal
Read the division remainders from the BOTTOM up
→ Converting Decimal to Binary

Binary Arithmetic — Addition, Division & Algebraic Identities

Formulas (3)

Binary Addition, Subtraction & Unknown-Digit Puzzles · Binary addition carry rule $1 + 1 = (10)_2,\qquad 1 + 1 + 1 = (11)_2$
Binary Division — Quotient and Remainder · Division identity $A = B\,Q + R,\qquad 0 \le R < B$
Algebraic Identities with Binary-Given Values · Key cube identities $x^3 + y^3 = (x+y)(x^2 - xy + y^2);\qquad a = b + c \ \Rightarrow\ a^3 - b^3 - c^3 - 3abc = 0$

Watch out for (5)

Every unknown is a BIT — only 0 or 1 is allowed
→ Binary Addition, Subtraction & Unknown-Digit Puzzles
Convert the FINAL answer back to binary
→ Binary Addition, Subtraction & Unknown-Digit Puzzles
Quotient and remainder are usually asked in BINARY
→ Binary Division — Quotient and Remainder
Spot the identity before cubing anything
→ Algebraic Identities with Binary-Given Values
(x − y)² + xy equals x² − xy + y²
→ Algebraic Identities with Binary-Given Values

Binary Representation & Number Theory

Formulas (2)

Representing a Number in Binary & Counting Its Bits · Bit count of N $2^{k-1} \le N \le 2^k - 1 \ \Longrightarrow\ N \text{ uses } k \text{ bits}$
Number-Theory One-Liners — Remainder Cycles & Sum of Odd Numbers · Sum of first n odd numbers $1 + 3 + 5 + \cdots + (2n - 1) = n^2$

Watch out for (3)

Watch the digit COUNT in the options
→ Representing a Number in Binary & Counting Its Bits
Reduce the exponent by the CYCLE length, not the modulus
→ Number-Theory One-Liners — Remainder Cycles & Sum of Odd Numbers
Sum of odd numbers is a PERFECT SQUARE
→ Number-Theory One-Liners — Remainder Cycles & Sum of Odd Numbers