Dual numbers: a+bε where ε²=0 (but ε≠0). Multiplication: (a+bε)(c+dε) = ac+(ad+bc)ε. The magic: f(a+ε) = f(a)+f'(a)ε — automatically computes derivatives! Foundation of forward-mode automatic differentiation in ML/AI.
How to Use
Enter two dual numbers:
- Operations: +, −, ×, ÷
- Auto-diff: f(a+ε) gives f'(a)
- Steps: Detailed computation
Auto Differentiation
Want f'(3) for f(x)=x²+2x? Compute f(3+ε) = (3+ε)²+2(3+ε) = 9+6ε+ε²+6+2ε = 15+8ε. The ε-coefficient 8 = f'(3) = 2(3)+2. No symbolic math needed!
ML/AI Connection
Deep learning frameworks (PyTorch, JAX) use automatic differentiation for backpropagation. Forward mode (dual numbers) computes Jv products. Reverse mode computes v^TJ. Both are essential for training neural networks.
Step-by-Step Instructions
- 1Enter real + dual parts.
- 2Choose operation.
- 3Compute result.
- 4Read derivative.
- 5Verify.