Denominator in the context of "Egyptian fractions"

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).

The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as ${\displaystyle {\sqrt {2}}}$ or ${\displaystyle 2^{1/2}}$. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction ⁠99/70⁠ (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.

In chemistry and fluid mechanics, the volume fraction ${\displaystyle \varphi _{i}}$ is defined as the volume of a constituent V_i divided by the volume of all constituents of the mixture V prior to mixing:

Being dimensionless, its unit is 1; it is expressed as a number, e.g., 0.18. It is the same concept as volume percent (vol%) except that the latter is expressed with a denominator of 100, e.g., 18%.

A continued fraction is a mathematical expression written as a fraction whose denominator contains a sum involving another fraction, which may itself be a simple or a continued fraction. If this iteration (repetitive process) terminates with a simple fraction, the result is a finite continued fraction; if it continues indefinitely, the result is an infinite continued fraction. The special case in which all numerators are equal to one is referred to as a simple (or regular) continued fraction. Any rational number can be expressed as a finite simple continued fraction, and any irrational number can be expressed as an infinite simple continued fraction.

Different areas of mathematics use different terminology and notation for continued fractions. In number theory, the unqualified term continued fraction usually refers to simple continued fractions, whereas the general case is referred to as generalized continued fractions. In complex analysis and numerical analysis, the general case is usually referred to by the unqualified term continued fraction.

The highest averages, divisor, or divide-and-round methods are a family of apportionment rules, i.e. algorithms for fair division of seats in a legislature between several groups (like political parties or states). More generally, divisor methods are used to round shares of a total to a fraction with a fixed denominator (e.g. percentage points, which must add up to 100).

The methods aim to treat voters equally by ensuring legislators represent an equal number of voters by ensuring every party has the same seats-to-votes ratio (or divisor). Such methods divide the number of votes by the number of votes per seat to get the final apportionment. By doing so, the method maintains proportional representation, as a party with e.g. twice as many votes will win about twice as many seats.