Vigesimal (Base 20) — Definition, Formula & Examples

Vigesimal (base 20) is a number system that uses 20 distinct digits (0 through 19) instead of the 10 digits we use in everyday counting. Each place value represents a power of 20 rather than a power of 10.

A positional numeral system with radix 20, in which a number is expressed as a sum of terms $d_n \times 20^n + d_{n-1} \times 20^{n-1} + \cdots + d_1 \times 20^1 + d_0 \times 20^0$ , where each digit $d_i$ satisfies $0 \le d_i \le 19$ .

Key Formula

N = d_n \times 20^n + d_{n-1} \times 20^{n-1} + \cdots + d_1 \times 20^1 + d_0 \times 20^0

Where:

$N$ = The value of the number in base 10
$d_i$ = The digit in position i (ranging from 0 to 19)
$n$ = The position of the leftmost digit (starting from 0 on the right)

How It Works

In base 20, the rightmost digit represents

20^0 = 1

, the next digit left represents

20^1 = 20

, the next represents

20^2 = 400

, and so on. Since we need 20 distinct symbols, digits beyond 9 are often written as letters: A = 10, B = 11, up to J = 19. To convert a base-20 number to base 10, multiply each digit by its place value and add the results. The Maya civilization and some other cultures historically used base 20, likely because humans have 20 fingers and toes combined.

Worked Example

Problem: Convert the vigesimal (base 20) number 3A5₂₀ to base 10, where A represents 10.

Identify digits and place values: The digits from right to left are 5, A (which is 10), and 3. Their positions are 0, 1, and 2.

Multiply each digit by its power of 20: Compute each term using the place value formula.

3 \times 20^2 + 10 \times 20^1 + 5 \times 20^0

Calculate and add: Evaluate each power of 20 and sum the results.

3 \times 400 + 10 \times 20 + 5 \times 1 = 1200 + 200 + 5 = 1405

Answer: 3A5₂₀ = 1405 in base 10.

Why It Matters

Studying base 20 deepens your understanding of how positional number systems work, which is a key topic when learning about binary, hexadecimal, and other bases in computer science. The Maya numeral system used base 20, making vigesimal relevant in history and archaeology courses as well.

Common Mistakes

Mistake: Treating letters like A as their base-10 face value (1) instead of the assigned value (10).

Correction: In vigesimal, A = 10, B = 11, C = 12, and so on up to J = 19. Always substitute the correct numeric value before calculating.