The digital sum in base b of a set of natural numbers is calculated as follows: express each of the numbers in base b, then take the sum of corresponding digits and discard all carry overs. That is, the digital sum is the same as the normal sum except that no carrying is used.
For example in decimal (base 10) arithmetic, the digital sum of 123 and 789 is 802:
- 3 + 9 = 12, discard the 10 leaving 2.
- 2 + 8 = 10, discard the 10 leaving 0.
- 1 + 7 = 8, there is no carry to discard.
More usually the digital sum is calculated in binary (base 2) where the result only depends upon whether there are an even or odd number of 1s in each column. This is the same function as parity or multiple exclusive ors.
For example:
011 (3) 100 (4) 101 (5) --- 010 (2) is the binary digital sum of 3, 4 and 5.The binary digital sum is crucial for the theory of the game of nim.
The digital sum in base b is an associative and commutative operation on the natural numbers; it has 0 as neutral element and every natural number has an inverse element under this operation. The natural numbers together with the base-b digital sum thus form an abelian group; this group is isomorphic to the direct sum of a countable number of copies of Z/bZ.
Famous quotes containing the words sum and/or base:
“To sum up:
1. The cosmos is a gigantic fly-wheel making 10,000 revolutions a minute.
2. Man is a sick fly taking a dizzy ride on it.
3. Religion is the theory that the wheel was designed and set spinning to give him the ride.”
—H.L. (Henry Lewis)
“I am sure my bones would not rest in an English grave, or my clay mix with the earth of that country. I believe the thought would drive me mad on my death-bed could I suppose that any of my friends would be base enough to convey my carcass back to her soil. I would not even feed her worms if I could help it.”
—George Gordon Noel Byron (1788–1824)