1. Numbers and Quantization Errors

Human brains understand conceptually continuous quantities such as real numbers as well as discrete quantities such as integers and characters. On the other hand, if you are asked to write down a continuous number explicitly, you quickly realize that it is not possible. For example, we cannot write down the exact value of \(\pi\) as a string of numbers. We can write it only approximately like 3.14. When we calculate mathematical expressions symbolically, we don’t have to worry about errors caused by such approximation.. However, when we calculate numerically, we use such approximation unless the number is integer or exact fraction. The situation is similar to hand calculation we do regularly. When we carry out a lengthy calculation, we often write down intermediate values temporarily on the back of envelope with only finite figures and use them at a later time. They are not exact numbers but we hope accurate enough in practice. The current digital computers work essentially in the same manner. We must keep it in our mind that digital computers can understand only very limited numbers as we discuss in this chapter. We have to ask computers only what they can understand. When we force computers to do something beyond their capabilities, they often pretend that they understood it and give us an answer, but a wrong one. At present computers are not smart enough to tell what they can do and what they cannot do. Therefore, we first need to understand how the digital computers handle numerical quantities and their limitation.