Finite differences let computers do calculus without ever reaching zero
Calculus defines a slope by shrinking a step until it vanishes, something no computer can literally do. Finite differences keep the step small but real: subtract two nearby function values, divide by the gap, and you get a workable estimate of the derivative, the trick behind countless numerical simulations.
The idea is older than it looks. It traces back to an algorithm by the Swiss clockmaker-mathematician Jost Burgi around 1592 and to work by Isaac Newton, and Brook Taylor formally introduced finite differences in 1715. Later writers, including George Boole in 1860, treated them as mathematical objects in their own right, a discrete counterpart to the calculus of infinitely small quantities.
Three flavours are standard. A forward difference compares the value at a point with the value one step ahead; a backward difference looks one step behind; a central difference straddles the point, taking half a step each way. Dividing any of them by the step size estimates the slope, and when no step size is stated, it is taken to be 1.
Accuracy differs. For a smooth function, the error of the forward and backward versions shrinks roughly in proportion to the step, while the central version shrinks with the square of the step, so halving the gap cuts its error about fourfold. The central scheme has a blind spot, though. A sequence that simply flips between 1 and 2 at each step will be judged to have zero slope everywhere, a real danger when data come on a discrete grid.
Applying the recipe twice gives second derivatives; the familiar central estimate adds the values on either side, subtracts twice the middle one, and divides by the step squared. Higher orders follow the same pattern, with coefficients read straight from rows of Pascal's triangle. These approximations sit at the heart of finite difference methods for solving differential equations, especially boundary value problems, and a difference equation plays the same role for them that a differential equation plays for derivatives.
Source: Finite difference
