2.1 Reality

The real world is full of real numbers. Quantities such as distances, velocities, masses, angles, and other quantities are all real numbers. In high performance computing we often simulate the real world, so it is somewhat ironic that we use simulated real numbers (floating-point) in those simulations of the real world. A wonderful property of real numbers is that they have unlimited accuracy. For example, when considering the ratio of the circumference of a circle to its diameter, we arrive at a value of 3.141592.... The decimal value for pi does not terminate. Because real numbers have unlimited accuracy, even though we can’t write it down, pi is still a real number. Some real numbers are rational numbers because they can be represented as the ratio of two integers, such as 1/3. Not all real numbers are rational numbers. Not surprisingly, those real numbers that aren’t rational numbers are called irrational. You probably would not want to start an argument with an irrational number unless you have a lot of free time on your hands.

Unfortunately, on a piece of paper, or in a computer, we don’t have enough space to keep writing the digits of pi . So what do we do? We decide that we only need so much accuracy and round real numbers to a certain number of digits. For example, if we decide on four digits of accuracy, our approximation of pi is 3.142. Some state legislature attempted to pass a law that pi was to be three. While this is often cited as evidence for the IQ of governmental entities, perhaps the legislature was just suggesting that we only need one digit of accuracy for pi . Perhaps they foresaw the need to save precious memory space on computers when representing real numbers.