Ternary (Base 3) to Vigesimal (Base 20) Converter
Convert ternary numbers to vigesimal.
100110 (base 3) = CF (base 20)
Digits in base 3: 0 to 2. Digits in base 20: 0 to 9 and A to J.
How to convert Ternary to Vigesimal
Every positional number system works the same way: a digit is worth its own value times the base raised to the power of its position. In base 3 the place values, from the right, are 30 = 1, 31 = 3, 32 = 9 and so on.
Most pairs have no direct trick, so go through decimal, the base we all calculate in. First turn the base 3 number into decimal, then divide by 20 and read the remainders from the bottom up. Two short steps, and you can check the number in the middle.
The method
decimal value = each digit × its place value, all added upplace values in base 3: 1, 3, 9, 27, 81, and so ondivide by 20, keep the remainder, repeat until the quotient is 0the answer is the remainders read from last to firstdigits in base 3: 0 to 2digits in base 20: 0 to 9 and A to J
Worked example 1: 1200 from ternary to vigesimal
Step 1: base 3 to decimal1200 (base 3) = 1 × 33 + 2 × 32= 27 + 18= 45Step 2: decimal to base 2045 ÷ 20 = 2 remainder 52 ÷ 20 = 0 remainder 2Read the remainders from the bottom up: 251200 (base 3) = 25 (base 20)
Letters stand for values from 10 up: A = 10, B = 11, C = 12 and so on. Remainders come out last digit first, which is why you read them from the bottom up. Digits that are 0 add nothing to the sum, so they are left out of the expansion.
Worked example 2: 100110 from ternary to vigesimal
Step 1: base 3 to decimal100110 (base 3) = 1 × 35 + 1 × 32 + 1 × 31= 243 + 9 + 3= 255Step 2: decimal to base 20255 ÷ 20 = 12 remainder 15 (F)12 ÷ 20 = 0 remainder 12 (C)Read the remainders from the bottom up: CF100110 (base 3) = CF (base 20)
Worked example 3: 1101001 from ternary to vigesimal
Step 1: base 3 to decimal1101001 (base 3) = 1 × 36 + 1 × 35 + 1 × 33 + 1 × 30= 729 + 243 + 27 + 1= 1,000Step 2: decimal to base 201,000 ÷ 20 = 50 remainder 050 ÷ 20 = 2 remainder 10 (A)2 ÷ 20 = 0 remainder 2Read the remainders from the bottom up: 2A01101001 (base 3) = 2A0 (base 20)
Need the other direction? Open the Vigesimal to Ternary converter.
Ternary to Vigesimal examples
The decimal number 255 is written 100110 in ternary and CF in vigesimal.
| Ternary | Vigesimal | Decimal |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 21 | 7 | 7 |
| 22 | 8 | 8 |
| 101 | A | 10 |
| 120 | F | 15 |
| 121 | G | 16 |
| 1012 | 1C | 32 |
| 2101 | 34 | 64 |
| 10201 | 50 | 100 |
| 100110 | CF | 255 |
| 1101001 | 2A0 | 1000 |
Where Ternary to Vigesimal comes up
Ternary uses the digits 0, 1 and 2, and each ternary digit (a trit) carries about 1.58 bits of information. Ternary is rare in real hardware, though the Soviet Setun computer of 1958 used a balanced ternary design. Today it shows up mostly in computer science courses and puzzles.
Vigesimal counts in twenties, with the digits 0 to 9 and A to J. The Maya wrote numbers in a base 20 system, and French still says quatre-vingts, four twenties, for 80.
Going to a larger base packs more value into each digit, so the result is shorter. The number 255 needs 6 digits in ternary and 2 in vigesimal.
Ternary to Vigesimal FAQ
How do I convert ternary to vigesimal?
Convert to decimal first by adding up each digit times its place value, then divide that decimal number by 20 repeatedly and read the remainders from last to first.
Which digits does ternary use?
Base 3 uses the digits 0 to 2. For example, 3 is not a valid digit in base 3.
Will the vigesimal number be longer or shorter?
Shorter. Going to a larger base packs more value into each digit, so the result is shorter. Roughly 0.37 vigesimal digits for every ternary digit. The number 255 needs 6 digits in ternary and 2 in vigesimal.
What is 100110 in vigesimal?
100110 in ternary is CF in vigesimal. Both are the decimal number 255.
Can I convert negative numbers or fractions?
The converter handles whole numbers, and a leading minus sign works for negatives. Fractions use a different method, where you repeatedly multiply the fractional part by the new base, and they are not supported here.