Theoretical Principles & Computational Physics Guide
Variables & Data Types: Dynamic Typing & Numerical Representation
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
State Representation: Variables store physical observables (mass, charge, velocity) and coordinates in computer memory.
IEEE 754 Floating-Point: Standard 64-bit double precision provides ~15-17 significant decimal digits with dynamic range $10^{-308}$ to $10^{308}$.
Complex Arithmetic: Quantum amplitudes and AC phasors are natively supported via the
j suffix (e.g. 3.0 + 4.0j).2. Mathematical Formulation & Algorithmic Pipeline
Machine precision $\epsilon_{\text{mach}}$ determines the smallest floating-point increment such that $1.0 + \epsilon \neq 1.0$:
$$ \epsilon_{\text{mach}} = 2^{-52} \approx 2.22 \times 10^{-16}, \quad z = x + iy \in \mathbb{C} \implies |z|^2 = x^2 + y^2 $$
3. Numerical Mechanics & Implementation Logic
- Type Conversion: Apply explicit typecasting (
float(),int(),complex()) to guarantee proper operator behavior. - Overflow & Underflow Guard: Monitor exponent scales in calculations involving extreme astronomical or quantum magnitudes.
- Immutability & Mutation: Distinguish between immutable scalars and mutable sequence references.
4. Relevant Computational Practice Exercises
Exercise 1
Numerically determine machine epsilon $\epsilon_{\text{mach}}$ by repeatedly halving $\epsilon$ until $1.0 + \epsilon == 1.0$.
Exercise 2
Define a normalized quantum state amplitude $z = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}$ and verify that $|z|^2 = 1.0$.
Exercise 3
Compare arbitrary-precision integer factorials $N!$ against standard 64-bit floating-point limits.
- David Goldberg, "What Every Computer Scientist Should Know About Floating-Point Arithmetic", ACM Computing Surveys (1991).
- Mark Newman, "Computational Physics", University of Michigan (2013).