Convert Base 5 to Base 7

Convert Base 5 (base 5) to Base 7 (base 7) instantly and accurately.

Selected Base 5 (base 5) - available characters: 0, 1, 2, 3, 4

Conversion Formula

Step-by-step example using the value 42 (decimal):

Convert Base 5 (base 5) → Base 7 (base 7)

Step 1: Expand each digit of 132 (base 5) by position:

        1 × 5^2 = 25
        3 × 5^1 = 15
        2 × 5^0 =  2
        ────────────
        Sum = 42 (decimal)

Step 2: Divide 42 by 7 repeatedly (read remainders upward):

        42 ÷ 7 = 6  r 0
         6 ÷ 7 = 0  r 6
        Read remainders upward: 60

────────────────────────────
Result:  132 (base 5) = 60 (base 7)

About Base 5

Quinary (base 5) uses digits 0-4. Each position is a power of 5: 5⁰=1, 5¹=5, 5²=25. One of the most historically widespread alternative bases, reflecting one-hand finger counting - Proto-Indo-European *penkwe = 'five/fist'. Tally marks group strokes in fives. Babylonian sexagesimal (base 60) used a 5×12 sub-structure. Fibonacci numbers mod 5 have Pisano period 20. Conversion: decimal 42 = 1×25+3×5+2 = 132₅.

About Base 7

Septenary (base 7) uses digits 0-6. Each position is a power of 7: 7⁰=1, 7¹=7, 7²=49. Seven is prime, so every fraction p/q (q not divisible by 7) produces a purely repeating expansion. Seven days of the week derive from Babylonian astronomy (the 7 naked-eye celestial objects). Reed-Solomon codes used in QR codes and Blu-ray draw on prime-field properties of GF(7). Conversion: decimal 42 = 6×7+0 = 60₇.

Quick Reference Table

→ Full Number Base Converter