quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part. The new denominator is the same as the denominator of the improper fraction. In the example, it is 4. Thus, . Historical notions Egyptian fraction An Egyptian fraction is the sum of distinct positive unit fractions, for example . This definition derives from the fact that the ancient Egyptians expressed all fractions except , and in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, can be written as Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write are and . Complex and compound fractions In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, and are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example: If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations, e.g. as or as A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on multiplication). For example, of is a compound fraction, corresponding to . The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction is equivalent to the complex fraction .) Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts". Arithmetic with fractions Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero. Equivalent fractions