Data representation and basic types

Number Systems & Bases

Computers represent all data as binary (base 2). Understanding how numbers are encoded prevents bugs in scientific computing.

Conversion formula: $1101_2 = 1\cdot2^{3} + 1\cdot2^{2} + 0\cdot2^{1} + 1\cdot2^{0} = 13_{10}$.

Basic Python types:

• int: exact integer (42)
• float: IEEE 754 double, $\approx15$ digits (3.14)
• str: text ("physics")
• bool: True / False

Floats, Conversion & Pitfalls

Floating-point caution: Not all decimals are exact in binary representation.

>>> 0.1 + 0.2
0.30000000000000004

Common pitfall: Floats are not real numbers. Compare numerical results using a tolerance (abs(a - b) < 1e-9), never with ==, or your simulation will fail due to pure representation artifacts.

Type conversion:

• int("42") converts str $\to$ int
• float("3.14") converts str $\to$ float
• str(99) converts int $\to$ str

Physics link: Watch for integer division (5 // 2 = 2) versus float division (5 / 2 = 2.5) to avoid off-by-one errors.