Several scales (radix) have been used by mathematicians. Binary (2), Ternary (3), Quaternary (4), Quinary (5), Senary (6), Septenary (7), Octenary(8), Nonary (9), Denary (10/Decimal), Undenary(11), Duodenary (12) and of course, hexadecimal (16). Note that in any scale the base radix is “10”. Thus, “10” stands for 2 in “binary”, “ten” in “decimal”, 8 for “octal” radix respectively, etc.

Let the digits used in a proposed scale(radix r) be . Let us express an integer in this scale. Let be unit’s digits. Analagous to the place value system (in decimal):

Now, let us say we want to express this number N in terms of these digits s.

Dividing N by , we get the unit’s digit as the remainder; and the quotient is:

.

Dividing the above quotient by r, we get as the remainder and the quotient as:

, and so on.

Example: Express the denary number 5213 in the scale of seven.

Solution: gives 5 as remainder and as quotient.

gives 2 as remainder and as remainder.

Continuing this way, we are able to express:

. That is . You can check the equivalence by converting both to decimal values.

Cheers,

Nalin Pithwa.