Making you own custom number system: Base-5

After learning about Binary System, you might be wondering how to create your own number system.

Here we will create a 5-digit number system, Base-5. We will call this number system Pental Number.

Rules #

The rules are similar to binary, but instead of two numbers (0 and 1), we use 0 to 4 as the numbers.

0, 1, 2, 3, 4 = Base-5

Conversion #

We will also use modulo, but \(5\) as the divisor.

$$ 45_{10} = n_5 $$

$$ 45 \space{mod}\space 5 = 0 \qquad{quot.} = 9\\ 9 \space{mod}\space 5 = 4 \qquad{quot.} = 1\\ 1 \space{mod}\space 5 = 1 \qquad{quot.} = 0\\ $$

$$ 45_{10} = 140_5 $$

Another example:

$$ 173_{10} = n_5 $$

$$ 173 \space{mod}\space 5 = 3 \qquad{quot.} = 34\\ 34 \space{mod}\space 5 = 4 \qquad{quot.} = 6\\ 6 \space{mod}\space 5 = 1 \qquad{quot.} = 1\\ 1 \space{mod}\space 5 = 1 \qquad{quot.} = 0\\ $$

$$ 173_{10} = 1143_5 $$

Decimal to Pental Conversion #

To convert to decimal, we must note how many digits there are in \(n_5\).

For example \(140_5\), there are 3 digits:

Then what we do is add \({dn}*5^d\) for each digit. Starting from the rightmost digit is to add \(5^0\)

$$ (1\ast5^2) + (4\ast5^1) + (0\ast5^0) \\\ 25 + 20 + 0 = 45 $$

Another example:

$$ 1143_5 = n_{10} $$

$$ (1\ast5^3) + (1\ast5^2) + (4\ast5^1) + (3\ast5^0)\\ 125 + 25 + 20 + 3 = 173 $$