Arithmetic with Infinite Decimal ExpansionsOperational Viewpoint:In brief, arithmetic involving numbers given or represented by infinite decimal expansions is done by approximation with the hope or assumption (justified by error control, continuity and convergence analysis in calculus or beyond) that better and better results required more and more decimals in the calculation. In practice, apart from theory and with the aid of calculators, the number of significant or exact decimal places in a result that depends on approximations is estimated from which digits are changed as approximations to operands in the calculation get more precise - involve more decimal places. Calculation TheoryThe addition, subtraction, multiplication and division of finite and infinite decimal expansion may be defined by a sequence of approximations: do the calculations with say n leading decimals from the expansion involved to get an approximation f(n). By continuity or error control analysis, we may show that the approximations f(n) form a Cauchy sequence and therefore define the result, another finite or infinite expansion that gives the result. Details may be found in the first chapter in the book Calculus by L. Bers (Holt, Rinehart and Winston 1969, SBN 03-065240-5). The chapter in question provides further background information on the decimal and decimal-free representation of real numbers. See too Error Control Inequalities in the calculus area of this site. In practice, we do not do the error control analysis (too much work for hand computation) and instead take as many decimal places as a calculator permits or we examine the effect on the result (our approximation to it) of taking fewer and then more and more decimal places to approximate the numbers (infinite decimal expansions) involved. The extension of our concepts of what is a number from whole numbers and fractions to include infinite decimal expansions allows us to represent square roots of prime number, square root of 2 included, and the number p used in circle based calculations - perimeter, area, trig functions. By continuity or error control analysis, we may show that addition and multiplication of unsigned reals are each commutative and associative; that multiplication distributes over addition (so the distributive property holds); that division by an unsigned number N is equivalent to multiplication by the quotient 1 The finite or infinite decimal expansion of the latter can be computed by long division continued to finitely or infinitely many places after the decimal point.
Ratio of Unsigned Real NumbersA ratio of unsigned reals is a compound fraction M in which the numerator and denominator are both unsigned real numbers. When M and N are unsigned real numbers, then the compound fraction M can be approximated using say k decimals in the decimal expansions of M and N. Better approximations can be found by taking k larger and larger. Some error control analysis is required to see how accurate the approximations will be. The equality
which holds for simple fractions and hence decimal approximations to M and N also holds via error control arguments for M and N. Exercises or food for thought
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