symbolic vs numerical computation

Suppose you ask a computer for one third. A numerical answer is 0.3333333333333333 — a fast, finite, slightly-wrong decimal. A symbolic answer is the fraction 1/3, held as a pair of integers and exactly equal to one third forever. These are two whole styles of computing with mathematics: one juggles approximate numbers as quickly as the hardware allows, the other manipulates exact symbols and formulas the way you would by hand. Knowing which one a task wants is half the battle.

Numerical computation works with floating-point numbers — a fixed budget of about sixteen decimal digits — and trades a small, controlled error for enormous speed and scale. Symbolic computation (computer algebra) instead represents mathematical objects exactly: the integer 1/3, the symbol pi, the unevaluated expression x^2 + 2x + 1, even sqrt(2) as a thing that squares to 2. It can simplify, differentiate, factor, and solve in closed form, returning answers that are not merely close but provably correct. The price is that exact intermediate results can grow explosively in size and time (expression swell), so symbolic methods rarely scale to the large problems numerics shrug off. A rough rule: if you need a formula, a proof, or perfect cancellation, go symbolic; if you need a number from a big or messy problem, go numerical.

Neither is universally better, and the most powerful tools are hybrid: derive a formula symbolically, then evaluate it numerically; or use exact arithmetic only at the spots where round-off would be catastrophic. The honest framing is that symbolic computation buys exactness at the cost of scalability, while numerical computation buys speed and scale at the cost of a (usually tiny, usually understood) error. Treating one as a strict upgrade of the other is the classic beginner mistake.