Binary⇄Decimal Converter – Bases 2, 8, 10, 16

Every positional numeral stores digits times powers of its base. The binary string 1101 breaks into 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8+4+0+1 = 13 in decimal. Going the other way, repeated division by 2 collects remainders bottom-up: 13 ÷ 2 = 6 r1, 6 ÷ 2 = 3 r0, 3 ÷ 2 = 1 r1, 1 ÷ 2 = 0 r1, which reads as 1101. The converter performs this for bases 2, 8, 10 and 16 simultaneously, so 13 also shows as 15₈ and D₁₆. Hex digit values beyond 9 map as A=10 through F=15, making 2F₃₁₆ = 2×16+15 = 755. Bit grouping shortcuts appear too: three binary digits per octal digit (11 011 → 33₈... for 27) and four per hex digit (1010 1111 → AF₁₆), which is why programmers read hex as compressed binary.

1. 1

   Enter a value and base

   Type a number in any of the four bases

2. 2

   Expand by place value

   Each digit multiplies its base power, e.g. 1×2³ for the leading bit

3. 3

   Divide for the reverse trip

   Repeated division collects remainders as new digits

4. 4

   Read all four bases

   Binary, octal, decimal and hex update together

• 1

  Hex is binary in groups of 4: F₁₆ = 1111₂

• 2

  Octal groups binary in 3s: 777₈ = 111 111 111₂ = 511

• 3

  Fractional parts convert by repeated multiplication, not division

• Reading binary left-to-right as ones-place first; weights grow right-to-left

• Treating hex letters casually: F = 15, so 2F = 47 not 2+6

• Confusing 1000₂ with one thousand — it equals 8

• Forgetting octal digits only run 0–7