calculations

Calculations are processes of determining numerical results from inputs by applying rules of arithmetic, algebra, or more general mathematical methods. They can be performed by humans using mental or written techniques, or by machines and software, from simple calculators to high-powered computers. Calculations underpin science, engineering, finance, and everyday decision making.

Categories include arithmetic calculations (addition, subtraction, multiplication, division), algebraic calculations (solving equations and manipulating symbols), geometric calculations (areas, volumes, lengths), and statistical calculations (averages, variances, probabilities). More advanced calculations involve numerical methods for solving equations, optimization, integration, and differential equations, as well as symbolic computation that manipulates expressions exactly.

Accuracy and precision are important aspects of calculations. Results may be exact or approximate, and are affected by input quality, rounding, and representation limits in finite systems. Significant figures, units, and error analysis help convey reliability.

Historically, calculations were performed with tallying, counting boards, or the abacus; later devices such as Napier's bones and the slide rule aided complex computations. The advent of mechanical and electronic calculators, followed by computers, greatly expanded the scale and speed of calculations, enabling modern science and finance.

In summary, calculations are fundamental methods for extracting quantitative information from data, guided by conventions, standards, and tools that suit the task at hand.