Numerator in the context of Rational function

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠${\displaystyle {\tfrac {p}{q}}}$⁠ of two integers, a numerator p and a non-zero denominator q. For example, ⁠${\displaystyle {\tfrac {3}{7}}}$⁠ is a rational number, as is every integer (for example, ${\displaystyle -5={\tfrac {-5}{1}}}$). The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also calledthe field of rationals or the field of rational numbers. It is usually denoted by boldface Q, or blackboard bold ⁠${\displaystyle \mathbb {Q} .}$⁠

A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).

An Egyptian fraction is a finite sum of distinct unit fractions, such as$${\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.}$$That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number ${\displaystyle {\tfrac {a}{b}}}$; for instance the Egyptian fraction above sums to ${\displaystyle {\tfrac {43}{48}}}$. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including ${\displaystyle {\tfrac {2}{3}}}$ and ${\displaystyle {\tfrac {3}{4}}}$ as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.

A unit fraction is a positive fraction with one as its numerator, 1/n. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole.

Multiplying two unit fractions produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into equivalent whole numbers, allowing modular division to be transformed into multiplication. Every rational number can be represented as a sum of distinct unit fractions; these representations are called Egyptian fractions based on their use in ancient Egyptian mathematics. Many infinite sums of unit fractions are meaningful mathematically.