Binary Representation of Decimals + Floating Point Errors Explained | Problem 3
In this video we solve Problem 3 from Computation and Measurement,
covering binary (base-two) representation of decimal numbers and
the errors introduced by floating point storage in computers.
✏️ What's covered in this video:
📌 Part (a) – Binary representations of:
• 625 → 1001110001 (base 2)
• 6.25 → 110.01 (base 2)
• 0.0625 → 0.0001 (base 2)
📌 Part (b) – Why 1/5 = 0.2 CANNOT be represented
exactly in binary — it produces an infinitely
repeating binary fraction (like 1/3 in decimal).
📌 Part (c) – Floating point binary representation:
Numbers stored as 0.d₁d₂d₃...dₙ × 2ᵐ
📌 Part (d) – 8-digit floating point storage:
• 0.2 is stored as 0.11001100 × 2⁻² = 51/256
• Absolute error = |1/5 − 51/256| = 1/1280
• Relative error = |Xᴛ − X| / Xᴛ
📐 Key concepts explained:
• Base-2 vs base-10 numeration
• Powers of 2 decomposition
• Why some decimals repeat infinitely in binary
• Floating point representation (mantissa & exponent)
• Absolute error vs relative error
• Error propagation in computation
💡 This is essential knowledge for:
Numerical methods | Scientific computing |
Computer science | Engineering mathematics
⏱️ Timestamps:
0:00 – Introduction & Problem Statement
[add your timestamps here]
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