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FRACTION (from the Latin frangere, to break) signifies, in arithmetic and algebra, a combination of numbers representing one or more parts of a unit or integer: thus four fifths is a fraction, formed by dividing a unit into five equal parts, and taking one part four times. Fractions are divided into vulgar and decimal. Vulgar fractions are expressed by two numbers with a line between them. The lower, the denominator, indicates into how many equal parts the unit is divided ; and the number above the line, called the numerator, indicates how many of such parts are taken; as, in &, 8 is the denominator, 7 the numerator. Vulgar fractions have been divided, though not very accurately, into proper, improper, simple, compound and mixed, viz.:A proper fraction is when the numerator is less than the denominator, as f 5 § i h T9T? fo TJ &C" An improper fraction is when the numerator is equal to or greater than the denominator, as J, J, j§, Jf2-, &c. A simple fraction is that which consists of a single numerator and single denominator ; and is either proper or improper, as f, JT, |f, &c. A compound fraction is a fraction consisting of two or more other fractions connected by the word of; thus § of f, or § of T7T of f, &c, are compound fractions. A complex fraction is that whose numerator and denominator are both fractions ; thus sris a complex fraction. These twf distinctions, though frequently made by authors on arithmetic, are certainly improper, the former indicating an operation in multiplication, and the latter an operation in division. It is, therefore, improper to apply to them the denomination of fractions. An integer and fraction together is called a mixed number; that is, 7f, 9£, &c, are mixed numbers. The theory of vulgar fractions is one of the most important in algebra, but is rarely, we think, developed in a clear, simple and easy manner in books on arithmetic. A correct understanding of them is of great importance for the proper prosecution of arithmetical and mathematical studies.Decimal fractions include every fraction, the denominator of which is 10 or a power of it; as T5^, Ww &c* Our beautiful system of writing numbers enables us to write decimal fractions without expressing the denominators, just as we are enabled to write the whole number without mentioning whether they are hundreds, thousands, &c. The following scheme will explain it. _____540432.47230 7____On the left of the point are the whole numbers; and just as every place in that series in proceeding to the left increases in value ten times, so every place to the right from the stop decreases in value ten times. Writing decimal fractions is therefore only an extension of our system of writing whole numbers. Yet, though it is as simple as it is important, the system was unknown to the ancients, and was first discovered by the German mathematician Regiomontanus in 1464. All calculations in decimal fractions are very easy and simple.