Bits and bases

ACompressed recap

🧾

Binary places are powers of two; decimal → binary by repeated division (read up). Base r: Σ digit × r^position; fractions by repeated multiplication (read down); p/q ends in base r only if q's prime factors divide r. Octal = 3-bit groups, hex = 4-bit groups from the point. Find-the-base: polynomials in b with b > every digit. Binary arithmetic carries at 2; n-bit sums need n + 1 bits, n × m products need n + m bits; shifts multiply or divide by powers of 2.

BMerged rule card

0.1.1 What "digital" means: bits, binary numbers and place value

• Binary places are powers of 2: 1, 2, 4, 8, 16, …; value = sum of the places holding 1.
• Decimal → binary: divide by 2 repeatedly, read remainders bottom-up.
• n bits: 2ⁿ patterns; unsigned range 0 … 2ⁿ − 1; N needs ⌈log₂(N + 1)⌉ bits.
• LSB rightmost, MSB leftmost; nibble 4 bits, byte 8 bits.

0.1.2 Octal, hexadecimal and converting between any two bases

• Base r: digits 0 … r − 1; value = Σ digit × r^position (negative positions after the point).
• Decimal → base r: integer part by repeated division (read up), fraction by repeated multiplication (read down).
• Octal = groups of 3 bits, hex = groups of 4, grouped from the point; octal ↔ hex via binary.
• p/q terminates in base r only if q's prime factors divide r (binary: q a power of 2).
• Find-the-base: write polynomials in b, solve, keep b > every digit.

0.1.3 Binary arithmetic by hand; how many values n bits can hold

• Binary column addition: carry when a column reaches 2; subtraction borrows 2.
• Multiplication: shift and add copies of the multiplicand.
• Left shift k = × 2ᵏ; right shift k = ÷ 2ᵏ (remainder dropped).
• Sum of two n-bit numbers needs up to n + 1 bits (carry out); product of n-bit and m-bit needs up to n + m bits.

CInterleaved GATE-style questions

Q1NAT1 markBinary to decimal

What is 10110.01₂ in decimal?

Numerical answer: ________

Q2NAT1 markBits needed

Minimum number of bits to represent the unsigned number 1024?

Numerical answer: ________

Q3MCQ1 markDecimal to hex

250 in hexadecimal is:

• (A) 15A
• (B) FA
• (C) AF
• (D) F5

Q4NAT2 marksFind the base

121_b = 144₈. What is b?

Numerical answer: ________

Q5MCQ1 markTerminating fractions

Which decimal fraction has a binary expansion that ends (terminates)?

• (A) 0.1
• (B) 0.3
• (C) 0.2
• (D) 0.375

Q6NAT1 markRegrouping

How many hex digits does the 22-bit binary number 1011011101011011101011 have?

Numerical answer: ________

Q7NAT1 markArithmetic

What is 101₂ × 11₂, written in decimal?

Numerical answer: ________

Q8NAT2 marksProduct size

Minimum number of bits that can hold the product of any two 16-bit unsigned numbers?

Numerical answer: ________

DAnswer Vault