exactvalue

An exact value is a quantity described in a precise form without rounding or approximation. In mathematics, exact values are typically expressed with fractions, radicals, logarithmic or exponential expressions, or symbolic constants such as pi and e. For instance, 3/5, sqrt(2), sin(pi/6), and pi/3 are exact values in contexts that permit symbolic evaluation; decimals like 0.6 are exact only as representations of those fractions, not as numerical approximations.

In computations, exact arithmetic preserves these representations. Rational numbers and certain algebraic numbers are manipulated without rounding, often within computer algebra systems. By contrast, floating-point arithmetic stores approximate numeric values and introduces rounding errors that accumulate in calculations.

Not all quantities have easily describable exact values. Some constants (such as pi and e) are treated as symbolic objects rather than finite decimal numbers, and many expressions cannot be simplified to a closed exact form. In these cases researchers rely on symbolic computation to maintain exactness or on well-controlled approximations.

Exact values are important in proofs, algebraic manipulations, and analytical reasoning, where precise relationships must be preserved. They contrast with approximate values used in numerical simulations, engineering estimates, or graphical computations, where speed and practicality require rounding or discretization.