Lecture 2: More Python, and numerical precision

As usual, material covered in class is in the notebook. Some lecture notes on numerical precision are included below. These notes reference the official Python documentation and some material from Chapter 4 of Computational Physics by Mark Newman.

Numerical precision

While integers in Python are represented with arbitrary precision, the precision of real or floating point numbers is limited. Typically, floating point numbers are represented by 64 bits of information: 1 bit for the sign, 11 bits for the exponent, and 52 for the significant digits.

Finite precision for real numbers has several consequences. First, it means that there is a limit to how large or how small of a number can be represented in Python. The largest number is around $10^308$, and the smallest around $10^{-308}$.

Second, not all decimals can be represented exactly in this binary form. For example, we know that it's impossible to write the fraction $1/3$ exactly as a decimal number in base 10. Similarly, the fraction $1/10 = 0.1$ is infinitely repeating in base 2. This can have strange consequences when we attempt to add such numbers together, if we expect infinitely precise results (see the notebook for more).

Third, precision will be lost when we try to compare numbers that differ dramatically in size, or numbers that almost perfectly cancel out. As a general rule we can assume that real numbers have about 16 significant figures -- anything beyond this is lost.