Decimal Expansion
Decimal expansion is a numerical system widely utilized in everyday life, characterized by its use of base 10. In this system, numbers are represented with digits that hold specific place values, where each digit's worth is determined by its position relative to the decimal point. The part of the number to the left of the decimal point is known as the integral part, while the part to the right is the fractional part. Decimal expansions can either terminate, like 8.75, or be nonterminating, such as 24.4411333..., with the latter sometimes repeating indefinitely. Rational numbers can be expressed as either terminating or repeating decimals, while nonterminating, nonrepeating decimals correspond to irrational numbers. Although decimal expansion simplifies calculations and is familiar to most people, it is less precise than the actual rational numbers it represents, as slight inaccuracies may occur due to rounding. This imprecision is usually negligible in manual calculations, but it can become significant in computer applications, leading some experts to question its continued use in computing contexts. Understanding decimal expansion is essential for grasping basic mathematical concepts and performing everyday calculations effectively.
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Decimal Expansion
Decimal expansion is the form of numerical expression most people are accustomed to working with in school and at work. Decimal expansion uses base 10 to express numbers. Each digit in a number has a place value in decimal notation, which means that it is equivalent to the number multiplied by a power of ten; a 3 in the tens place means 3 × 10, or 30, while a 2 in the hundreds place means 2 × 100, or two hundred. The place values decline as one moves from left to right.
Because decimal expansion is so familiar, it can be difficult to think about in abstract terms because many people consider it not so much a form of mathematical expression but as just the way numbers are. Decimal expansion is best understood by contrasting it with other forms of numerical notation, such as expanded notation. For example, the number 8,432.71, which is written in decimal expansion, would be written in expanded notation as 8 × 103 + 4 × 102 + 3 × 101 + 2 × 100 + 7 × 10 – 1 + 1 × 10 – 2. Decimal expansion notation makes it much easier to perform calculations by hand, which explains the average person’s greater familiarity with it.
Overview
Decimal expansions have two different parts, which are separated by the decimal point. (When all digits to the right of the decimal point are zero, the decimal point is often omitted, but presumed to be there.) The number to the left of the decimal point in a decimal expansion is called the integral part, while the number to the right of the decimal point is called the fractional part. In some cases the decimal point is referred to as the radix point. The integral part of the decimal expansion is composed of integers—hence its name—and the fractional part takes its name from its value being less than one.
Some numbers written in decimal expansion terminate, while others do not. A nonterminating number, such as 24.4411333…, continues on forever; a nonterminating number also periodically repeats in an unending sequence, such as 8981.67676…. As it turns out, nonterminating, nonrepeating decimal expansions are irrational numbers. Decimal expansions that terminate or repeat periodically are rational numbers. So-called terminating decimal expansions can be seen instead as nonterminating expansions with the rightmost, repeating digit equal to zero: 8,432.71 could thus be written as 8,432.71000000….
A significant characteristic of numbers expressed in decimal expansion is the following: They are less precise than the rational numbers they represent. This is because some degree of accuracy has been sacrificed in exchange for how much easier calculations are to perform on numbers in decimal expansion. Often the discrepancies in calculations are so small that they are difficult to detect. With the invention of computers, however, these discrepancies have become more problematic. Some computer scientists would prefer that decimal expansion not be used because of its imprecision.
Bibliography
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