I do hate sums

I do hate sums.
There is no greater mistake than to call arithmetic an exact science.
There are … hidden laws of number which it requires a mind like mine to perceive.
For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different.”

— Mrs. La Touche, Letters of a Noble Woman

The abacus is perhaps the oldest computational device humans used. Its primary purpose is to aid in addition. Computational devices at our disposal evolved from the first abacus (circa 2400 BC) to warehouses of thousands of computers capable of performing complex computations, simulations and data analysis. The most important operation performed by any computational device, however, is still addition. It is hard to find an operation more central to computing than repeated addition or summation. Sums are everywhere. No mean, no variance, no model can be computed without summing up some data points. When approximating irrational numbers like π or integrals of functions, we use summations.  Despite our technological advances, we haven’t created an error-free summation device. The reason for this deficiency is that computers are incapable of dealing with the continuity of real numbers. They live in a discrete world, while numbers live in a continuous one. I will describe the sources of error in a single addition, how these errors add-up with summation and how to compensate for these errors. Read the rest of this entry »