denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction" means "a piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook The Ground of Arts. In general, a common fraction is said to be a proper fraction, if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3. Reciprocals and the "invisible denominator" The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of , for instance, is . The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction. When the numerator and denominator of a fraction are equal (for example, ), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer, except for zero, has a reciprocal. For example. the reciprocal of 17 is . Ratios A ratio is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group n". For example, if a car lot had 12 vehicles, of which 2 are white, 6 are red, and 4 are yellow, then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow