Binary Numbers — NDA Mathematics Formulas and Traps

Binary ↔ Decimal Conversion

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

Common traps

Place values grow leftward, from 2⁰ on the RIGHT

The rightmost bit is the

2^0 = 1

place, not the

2^1

place. Counting the powers from the left, or starting at

2^1

, shifts every weight and is the most common conversion slip.

No subscript means DECIMAL, not binary

A string like

1011

written with no base subscript is the decimal number one thousand eleven — only

(1011)_2

is binary (which equals

11_{10}

). When a question gives a plain number to convert TO binary, read it as base 10; don't treat its digits as bits.

A 0 bit contributes nothing — don't add its place value

Only positions holding a 1 are summed. The fastest error is to add every place value you wrote down. Cross out the 0-bit places before adding so only the active weights remain.

Read the division remainders from the BOTTOM up

Repeated division produces the least significant bit first. Reading the remainders top-to-bottom reverses the number. The first remainder you write is the rightmost bit of the answer.

Keep a 0 in every skipped power — don't drop empty places

With the greedy method, every power of 2 you pass over still needs a 0 in its column. Writing only the powers you used (e.g.

20 = 16 + 4

as

11

instead of

(10100)_2

) silently deletes the empty

8

,

2

and

1

places and gives the wrong number.

Binary Arithmetic — Addition, Division & Algebraic Identities

Learn this subtopic in the notes

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

Common traps

Every unknown is a BIT — only 0 or 1 is allowed

When you solve for

p, q, r, x, y

, the value must be 0 or 1. A solution like

p = 2

is impossible in base 2 — it means you mis-set the place values. Check each unknown lands in

\{0, 1\}

before choosing an option.

Convert the FINAL answer back to binary

If the question gives the numbers in binary, the options are usually binary too. Doing the addition in decimal is fine — but don't forget the last step of converting your decimal total back to base 2.

Quotient and remainder are usually asked in BINARY

After dividing in decimal you have two numbers to convert back — both the quotient and the remainder. Reading the remainder option in decimal (e.g. picking 4 instead of

(100)_2

) is the standard slip.

Spot the identity before cubing anything

Cubing two-digit numbers by hand is slow and error-prone. The questions are engineered so that, after converting, either

a = b + c

or a sum/difference-of-cubes factoring applies. Look for that structure first — the brute-force route is the trap.

(x − y)² + xy equals x² − xy + y²

Expanding,

(x-y)^2 + xy = x^2 - 2xy + y^2 + xy = x^2 - xy + y^2

. So a question asking for

(x-y)^2 + xy

is secretly asking for the cube-sum cofactor

x^2 - xy + y^2

— recognise them as the same target.

Binary Representation & Number Theory

Learn this subtopic in the notes

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

Common traps

Watch the digit COUNT in the options

Representation questions often offer distractors with the wrong number of bits (one too few or too many). Bracket

N

between consecutive powers of 2 first to know how many bits the correct answer must have, then convert.

Reduce the exponent by the CYCLE length, not the modulus

For

5^{99} \bmod 13

, the powers cycle with period 4 (not 13). Reduce

99

modulo 4, not modulo 13. Find the actual cycle length first by listing remainders until one repeats.

Sum of odd numbers is a PERFECT SQUARE

If a sum of

1 + 3 + 5 + \cdots

is given, the term count is

\sqrt{\text{sum}}

. Recognising a value like

12345678987654321

as

111111111^2

is the intended shortcut — don't try to add the series term by term.

Binary Numbers formulas

Binary ↔ Decimal Conversion

What Base 2 Means — Place Values Are Powers of 2

Converting Binary to Decimal

Converting Decimal to Binary

Common traps

Binary Arithmetic — Addition, Division & Algebraic Identities

Binary Addition, Subtraction & Unknown-Digit Puzzles

Binary Division — Quotient and Remainder

Algebraic Identities with Binary-Given Values

Common traps

Binary Representation & Number Theory

Representing a Number in Binary & Counting Its Bits

Number-Theory One-Liners — Remainder Cycles & Sum of Odd Numbers

Common traps

More NDA Mathematics formula sheets