Mathematics - Fractions
Fractions, Common and Decimal
There are many ways to make change for a dollar: two half-dollars, four quarters, ten dimes, 20 nickels, or 100 pennies. No matter how the change is made, the dollar is broken up - "fractured" - into several pieces. These pieces are called fractions, from the same Latin word (fractus, meaning "broken") that fracture comes from.
All fractions represent parts of a whole. It has long been convenient and customary to divide things into segments. Hours are divided into 60 minutes each. Days are divided into 24 hours and years into 12 months. Miles are divided into feet and kilometers into meters. Each of these segments can be expressed as a fraction. One inch is one twelve part, or one-twelfth, of a foot. Fractions are very helpful because they make possible measurements in other than whole numbers such as 1, 2, or 5. Measurements with fractions can often be more precise: it is more exact to say "four and one-tenth gallons" than "a little more than four gallons."
Types of Fractions
In everyday mathematics there are two types of fractions, common and decimal. The only difference between the two is in how they are written. All fractions are written using the same symbols used to write whole numbers, but the symbols are used in a different way. Common fractions are written as 4/10 or 7/100: four over ten and seven over one hundred. The same numbers, when appearing as decimal fractions, would be 0.4 and 0.07. These would usually be read as "point four" and "point zero seven." They express the same amounts.
In a common fraction, the number below the line is the denominator, and the number above the line is the numerator. In reading a common fraction, the numerator is stated first. Thus, 2/3 is read as two-thirds. Any number except zero can be either a denominator or a numerator. What is being expressed in a common fraction is not only a quantity but also a ratio: the relationship of one quantity to another. For example, the fraction 1/2 expresses the ratio of one to two: The relationship of one to two is that one is half of two. The uses of a ratio are quite common. When a baker makes a cake, he may use two cups of sugar for every three cups of flour: The ratio is two to three, and it may be expressed as a fraction, 2/3.
Decimal fractions are so called because they are based on the decimal, or base-ten, numbering system (see Numeration Systems and Numbers). Sometimes referred to simply as "decimals," all decimal fractions consist of one or more numbers preceded by a dot called the decimal point: 0.4, for example, is read as four-tenths. If there is only one figure to the right of the decimal point, the fraction is always read as "tenths." If there are two figures, the fraction is read as "hundredths," and if there are three, it is read as "thousandths." In other words, decimal fractions follow the same progression as do whole numbers, where the first digit is in the "tens" column, the second in the "hundreds," and so forth (see Arithmetic). As examples, the decimal 0.075 is read as "seventy-five thousandths," and the fraction 0.3852 as "three thousand, eight hundred fifty-two ten-thousandths."
In common fractions, any number may be a denominator. But in decimals, the unwritten denominator is always 10, or some power of 10 such as 100, 1,000, 10,000, and so on. This means that it is simple to change a decimal fraction into a common fraction by putting the proper denominator under the number to the right of the decimal point. Thus, 0.85 becomes the common fraction 85/100.
To change a common fraction into a decimal, one must divide the numerator by the denominator. In this way, 3/4 can be changed into the decimal 0.75. Not all common fractions, however, can be changed into such precise decimals: 2/3 as a decimal is an endless series of sixes to the right of the decimal point.
Common fractions. There are four kinds of common fractions: proper, improper, mixed, and complex. A proper fraction has a numerator smaller than the denominator, such as 3/4. Therefore the value of a proper fraction is always less than one. In improper fractions, the numerator is equal to, or larger than, the denominator, as 4/4 or 6/5. Thus all improper fractions are equal to, or larger than, one.
A mixed fraction, also called a mixed number, consists of a whole number and a fraction, such as 2 1/3. Any mixed fraction can be changed into an improper fraction by multiplying the whole number by the denominator, adding the result to the numerator, and placing the total over the original denominator. In this way 2 1/3 can be changed into 7/3. Complex fractions, used in higher mathematics, do not consist of natural numbers. For instance, the square root of two over the square root of five is considered a complex fraction:
Similarly, the use of mixed fractions for both numerator and denominator would create a complex fraction:
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