Uses
The Iverson bracket converts a Boolean value to an integer value through the natural map, which allows counting to be represented as summation. For instance, the Euler phi function that counts the number of integers less than n and coprime to it can be expressed by
More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example
In the first sum, the index is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where i is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions.
Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula
which is valid for n > 1 but which is off by 1/2 for n = 1. To get an identity valid for all positive integers n (i.e., all values for which is defined), a correction term involving the Iverson bracket may be added:
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