Binary Numbers — NDA Maths

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 (5)

• Place values grow leftward, from 2⁰ on the RIGHT
  → What Base 2 Means — Place Values Are Powers of 2
• No subscript means DECIMAL, not binary
  → 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
• Keep a 0 in every skipped power — don't drop empty places
  → Converting Decimal to Binary

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

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