Uppercase Δ: change or difference
Read Δx as “delta x” or “the change in x”. It is always final minus initial:
Δx = x₂ − x₁
If a car's odometer reads 120 miles at the start and 185 miles at the end, Δ(distance) = 185 − 120 = 65 miles. If a temperature falls from 30° to 22°, ΔT = 22 − 30 = −8°; the sign records the direction of the change.
Slope is Δy over Δx
The slope formula most students meet in Algebra 1 is delta notation written out:
m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)
Between (1, 2) and (4, 8), Δy = 6 and Δx = 3, so m = 2. This is exactly the calculation DeltaMath's Slope From Two Points skill practices; the graphical version counts Δy as rise and Δx as run.
Average rate of change
For any function, Δf/Δx over an interval is the average rate of change, the slope of the line joining two points on the curve. For f(x) = x² between x = 1 and x = 3: Δf = 9 − 1 = 8, Δx = 2, so the average rate of change is 4.
Physics and everyday use
- Δt: elapsed time. Δv: change in velocity, so acceleration a = Δv/Δt.
- ΔH in chemistry: change in enthalpy. ΔG: change in free energy.
- Finance and data: Δ often labels a period-over-period change column.
Δ as the discriminant
In algebra, many textbooks (especially outside the U.S.) use Δ for the discriminant of a quadratic ax² + bx + c = 0:
Δ = b² − 4ac
- Δ > 0: two distinct real roots.
- Δ = 0: one repeated real root.
- Δ < 0: no real roots (two complex roots).
For x² − 6x + 2, Δ = 36 − 8 = 28, so two real roots. The full solution is worked on the quadratic formula guide.
Lowercase δ: small quantities and special functions
| Symbol | Name | Meaning | Where you meet it |
|---|---|---|---|
| ε, δ | Epsilon-delta | For every ε > 0 there is a δ > 0 such that |x − a| < δ implies |f(x) − L| < ε | Formal definition of a limit (Calculus, Real Analysis) |
| δᵢⱼ | Kronecker delta | 1 if i = j, otherwise 0 | Linear algebra, identity matrices, tensor notation |
| δ(x) | Dirac delta | A “function” that is zero everywhere except at 0 and integrates to 1 | Physics, signal processing, differential equations |
| δ | Small change / variation | A tiny increment, as in δx, or a variation in the calculus of variations | Physics, optimization |
Other uses of Δ
- Laplacian: Δf = ∂²f/∂x² + ∂²f/∂y² + … in multivariable calculus and PDEs (also written ∇²f).
- Symmetric difference: A Δ B is the set of elements in A or B but not both.
- Triangles: geometry texts write ΔABC for the triangle with vertices A, B, C.
- Finite differences: the forward difference operator Δf(n) = f(n + 1) − f(n) in sequences and numerical analysis.