Vinculum Numbers

Vinculum numbers or bar numbers are numbers which can have both positive and negative digits in their representation. They are of great facility in mental computation and can significantly reduce the burden of carry operations both in multiplication and division.

Vinculum Form of a Number
In transforming a number from canonical to vinculum form, digits greater than 5 are transformed to digits less than 5. The general principle is based on the observation that a number can be considered as greater than a smaller whole number base, or less than a greater whole number base.



In this example we note that 47 is either 7 above 40, or 3 below 50. The use of the bar over the 3 greatly facilitates using the vinculum in computation.



The general rule for obtaining the vinculum form of a two-digit number, is to increase the first digit by 1, and obtain the ten-friend of the final digit, by subtracting it from 10. A simple extension of this rule allows us to find the vinculum form of a number with an arbitrary number of digits.



To get the canonical form of a two-digit number, simply transpose the operation by reducing the first digit by 1 and again finding the ten's complement, ten-friend of the last digit by once again subtracting it from 10.

Multiplication Using the Vinculum

The vinculum form of a number may be used in any way that the original, canonical, form may be used. This is most easily demonstrated in the case of multiplication.



In this example 48 is first transformed to it's vinculum form, then a simple multiplication ensues, following which the answer is restored to it's normal, canonical, form. It is worth noting that, in the answer the vinculum form has three digits, with the final one being a bar digit, hence to transform it to canonical form we simply reduce the second or, middle, digit by 1, and obtain the ten-friend of the last.



These examples are straightforward and the product 97 x 6 is particularly interesting. In that case when we multiply bar 3 by 6 we obtain bar 18, and we carry forward a bar 1. The second thing to note is that in transposing the answer to canonical form, the vinculum form has two bar digits at the end. The final answer is 18 below 600 i.e. 600 - 18 = 582, from which we can develop the general principle for transforming a vinculum number to canonical form,

• reduce the first digit by one (One Less than the One Before)

• find the nine friend, nines-complement, of all the remaining digits from the left, except for

• find the ten friend, tens-complement, of the last (right-most) digit

As is always the case in computation, the converse, transforming a number from canonical to vinculum form is nearly the same with a single change,

• increase the first digit by one (One More than the One Before)

• find the nine friend, nines-complement, of all the remaining digits from the left, except for

• find the ten friend, tens-complement, of the last (right-most) digit

The principle of transforming a number to vinculum and back is of such importance that it is encapsulated in the sutra All from 9 and the Last from 10, and it has many other uses.