Positional Base Notation
In base b (b ≥ 2), a number (aₖ...a₁a₀)b represents the base-10 value: aₖ·bᵏ + ... + a₁·b¹ + a₀·b⁰ using digits {0, 1, ..., b-1}.
Grouping & Carry Rules
Decimal (Base 10) groups numbers by 10s. Base b groups numbers by powers of b (b⁰, b¹, b², ...):
- • Base 2 (Binary): Digits {0, 1}
- • Base 8 (Octal): Digits {0 to 7}
- • Base 16 (Hexadecimal): Digits {0-9, A=10, B=11, C=12, D=13, E=14, F=15}
Base System questions test algebra substitution, number range counting, and non-decimal arithmetic in CAT, XAT, and IIFT:
Decimal → Base b
Read Remainders Bottom-to-Top
Example: 53 to Base 5 → 53 ÷ 5 = 10 rem 3, 10 ÷ 5 = 2 rem 0, 2 ÷ 5 = 0 rem 2 → (203)₅.
Base b → Decimal
∑ (Digitᵢ × bⁱ)
Example: (342)₇ to Base 10 → 3×7² + 4×7¹ + 2×7⁰ = 147 + 28 + 2 = 177.
Fractional Base Conversion
Read Integers Top-to-Bottom
Example: 0.625 to Base 2 → 0.625×2 = 1.25 (1), 0.25×2 = 0.5 (0), 0.5×2 = 1.0 (1) → (0.101)₂.
Convert (432)₅ to Base 8.
• Step 1: Convert (432)₅ to Base 10:
4 × 5² + 3 × 5¹ + 2 × 5⁰ = 100 + 15 + 2 = 117.
• Step 2: Convert 117 to Base 8:
117 ÷ 8 = 14 rem 5
14 ÷ 8 = 1 rem 6
1 ÷ 8 = 0 rem 1
• Answer: Read bottom-to-top → (165)₈.
If (34)b + (45)b = (112)b, find base b.
• Convert to Base 10:
(3b + 4) + (4b + 5) = b² + b + 2
• Simplify: 7b + 9 = b² + b + 2 → b² - 6b - 7 = 0
• Factorize: (b - 7)(b + 1) = 0 → b = 7 (since base > 5).
Perform Base 6 Multiplication: (43)₆ × (25)₆.
• Method 1 (Direct Base 6):
3 × 5 = 15 = (23)₆ → write 3, carry 2
4 × 5 + 2 = 22 = (34)₆ → write 343₆
43₆ × 20₆ = 1300₆ → 343₆ + 1300₆ = (2043)₆.
• Method 2 (Via Decimal):
(43)₆ = 27, (25)₆ = 17 → 27 × 17 = 459.
Convert 459 to Base 6: 459 ÷ 6 = 76 r 3, 76 ÷ 6 = 12 r 4, 12 ÷ 6 = 2 r 0, 2 ÷ 6 = 0 r 2 → (2043)₆.
Mistake 1: Invalid Digit for Base
In Base b, digits can ONLY be 0 to (b - 1). For example, a number in Base 5 can NEVER contain digits 5, 6, 7, 8, or 9!
Mistake 2: Top-to-Bottom Remainder Reading
When converting integer decimal to Base b via division, remainders MUST be read from bottom to top!
Mistake 3: Subtraction Borrow Value
In decimal subtraction, borrowing 1 brings 10. In Base b subtraction, borrowing 1 brings a value of b to the lower column!
Q1: How many 3-digit numbers in base 7 retain 3 digits when converted to base 5?
View Solution
• 3-digit range in Base 5: 5² = 25 to 5³ - 1 = 124.
• Overlap in decimal: 49 to 124 → 124 - 49 + 1 = 76.
Q2: If N = (1331)b = 512 in Base 10, find the base b.
View Solution
• Notice that (1331)b = b³ + 3b² + 3b + 1 = (b + 1)³.
• (b + 1)³ = 512 = 8³.
• b + 1 = 8 → b = 7.
Q3: Convert decimal fraction 0.4375 to Base 4.
View Solution
• 0.4375 × 4 = 1.75 → integer part 1
• 0.75 × 4 = 3.0 → integer part 3
• Read integers top-to-bottom → (0.13)₄.
How do you convert a decimal number to base n?
Successively divide the decimal number by base n, record the remainders at each step from bottom to top (last remainder becomes the most significant digit).
How do you convert a number from base n to decimal (base 10)?
Multiply each digit by nk, where k is the positional index starting from 0 at the unit digit, and sum all terms.
How do you convert a fractional decimal number to base n?
Multiply the fractional part by n. The integer part becomes the first digit after the radix point. Repeat with the remaining fractional part from top to bottom.
How do you calculate the number of digits of N in base b?
The number of digits of N in base b is given by ⌊log_b N⌋ + 1.
What is the divisibility rule for (b - 1) in base b?
Just like a number is divisible by 9 (which is 10 - 1) in base 10 if the sum of digits is divisible by 9, a number in base b is divisible by (b - 1) if the sum of its digits in base b is divisible by (b - 1).