ε² = 0, ε ≠ 0
A dual number is a + bε, where a,b ∈ R and the new infinitesimal unitε is nonzero but squares to zero. They are the smallest useful algebraic model of first-order change: values ride in the real part, derivatives ride in the ε part.
The dual numbers form the commutative real algebra R[ε]/(ε²). Every element has the form a + bε.
| operation | rule | condition / note |
|---|---|---|
| equality | a+bε = c+dε ⇔ a=c and b=d | ε part is independent of real part |
| addition | (a+bε)+(c+dε)=(a+c)+(b+d)ε | componentwise |
| multiplication | (a+bε)(c+dε)=ac+(ad+bc)ε | the bdε² term vanishes |
| powers | (a+bε)ⁿ = aⁿ + n aⁿ⁻¹ bε | integer n; for negative n need a≠0 |
| inverse | (a+bε)⁻¹ = 1/a − b/a² ε | exists exactly when a≠0 |
| division | (a+bε)/(c+dε)=a/c + (bc-ad)/c² ε | requires c≠0 |
bε has no inverse.ε · ε = 0 although ε ≠ 0.a+bε is invertible iff its real part a is nonzero.(ε); the residue field is R.(ε) square to zero.¯(a+bε)=a-bε, but (a+bε)(a-bε)=a², not a norm that detects ε.If f is differentiable at a, the nilpotence of ε truncates the Taylor series after the linear term:
f(a+bε) = f(a) + b f′(a) ε
exp(a+bε)=eᵃ + b eᵃ εsin(a+bε)=sin a + b cos a εcos(a+bε)=cos a − b sin a εlog(a+bε)=log a + b/a ε for a>0sqrt(a+bε)=√a + b/(2√a) ε for a>0Substituting x + ε into a polynomial forces exactly the first derivative to appear:(x+ε)³ = x³ + 3x²ε + 3xε² + ε³ = x³ + 3x²ε. For smooth functions this is the same Taylor-series phenomenon. No limiting process, symbolic simplification, or finite-difference step size is needed.
function square(x) { return x * x }
square(3 + 1ε) = 9 + 6ε
// value = 9, derivative at 3 = 6A dual number can be read as a point plus a tangent vector. The real projection a+bε ↦ aforgets tangent information; the ε coefficient stores a direction and scale. In differential geometry, maps on manifolds send tangent vectors forward; dual numbers implement that idea algebraically in one dimension.
Forward-mode AD evaluates an ordinary program with dual-number arithmetic. Each elementary operation propagates both value and derivative by the chain rule. It gives machine-precision derivatives and is especially efficient for functions with few input variables and many outputs.
i²=-1 creates rotations; ε²=0 creates tangents.j²=+1 model hyperbolic geometry; dual numbers are the degenerate case between circular and hyperbolic behavior.R[ε]/(ε²) is the order-1 version; R[ε]/(εⁿ) carries higher derivatives up to order n-1.