FractionA fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: and ) consists of an integer numerator, displayed above a line (or before a slash like ), and a non-zero integer denominator, displayed below (or after) that line.
Egyptian fractionAn Egyptian fraction is a finite sum of distinct unit fractions, such as 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 ; for instance the Egyptian fraction above sums to . Every positive rational number can be represented by an Egyptian fraction.
Generalized continued fractionIn complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.
Nitrogen cycleThe nitrogen cycle is the biogeochemical cycle by which nitrogen is converted into multiple chemical forms as it circulates among atmospheric, terrestrial, and marine ecosystems. The conversion of nitrogen can be carried out through both biological and physical processes. Important processes in the nitrogen cycle include fixation, ammonification, nitrification, and denitrification. The majority of Earth's atmosphere (78%) is atmospheric nitrogen, making it the largest source of nitrogen.
HydrolysisHydrolysis (haɪˈdrɒlɪsɪs; ) is any chemical reaction in which a molecule of water breaks one or more chemical bonds. The term is used broadly for substitution, elimination, and solvation reactions in which water is the nucleophile. Biological hydrolysis is the cleavage of biomolecules where a water molecule is consumed to effect the separation of a larger molecule into component parts. When a carbohydrate is broken into its component sugar molecules by hydrolysis (e.g.
Partial fraction decompositionIn algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms.
Continued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression.
Unit fractionA 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.
Chloroacetic acidChloroacetic acid, industrially known as monochloroacetic acid (MCA), is the organochlorine compound with the formula ClCH2CO2H. This carboxylic acid is a useful building block in organic synthesis. It is a colorless solid. Related compounds are dichloroacetic acid and trichloroacetic acid. Chloroacetic acid was first prepared (in impure form) by the French chemist Félix LeBlanc (1813–1886) in 1843 by chlorinating acetic acid in the presence of sunlight, and in 1857 (in pure form) by the German chemist Reinhold Hoffmann (1831–1919) by refluxing glacial acetic acid in the presence of chlorine and sunlight, and then by the French chemist Charles Adolphe Wurtz by hydrolysis of chloroacetyl chloride (ClCH2COCl), also in 1857.
Irreducible fractionAn irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). In other words, a fraction a/b is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials.
Acetic acidAcetic acid əˈsiːtᵻk, systematically named ethanoic acid ˌɛθəˈnoʊᵻk, is an acidic, colourless liquid and organic compound with the chemical formula (also written as , , or ). Vinegar is at least 4% acetic acid by volume, making acetic acid the main component of vinegar apart from water and other trace elements. Acetic acid is the second simplest carboxylic acid (after formic acid). It is an important chemical reagent and industrial chemical, used primarily in the production of cellulose acetate for photographic film, polyvinyl acetate for wood glue, and synthetic fibres and fabrics.
Trichloroacetic acidTrichloroacetic acid (TCA; TCAA; also known as trichloroethanoic acid) is an analogue of acetic acid in which the three hydrogen atoms of the methyl group have all been replaced by chlorine atoms. Salts and esters of trichloroacetic acid are called trichloroacetates. It is prepared by the reaction of chlorine with acetic acid in the presence of a suitable catalyst such as red phosphorus. This reaction is Hell–Volhard–Zelinsky halogenation. CH3COOH + 3 Cl2 → CCl3COOH + 3 HCl Another route to trichloroacetic acid is the oxidation of trichloroacetaldehyde.
FertilizerA fertilizer (American English) or fertiliser (British English) is any material of natural or synthetic origin that is applied to soil or to plant tissues to supply plant nutrients. Fertilizers may be distinct from liming materials or other non-nutrient soil amendments. Many sources of fertilizer exist, both natural and industrially produced. For most modern agricultural practices, fertilization focuses on three main macro nutrients: nitrogen (N), phosphorus (P), and potassium (K) with occasional addition of supplements like rock flour for micronutrients.
ProteinProteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, responding to stimuli, providing structure to cells and organisms, and transporting molecules from one location to another. Proteins differ from one another primarily in their sequence of amino acids, which is dictated by the nucleotide sequence of their genes, and which usually results in protein folding into a specific 3D structure that determines its activity.
Dichloroacetic acidDichloroacetic acid (DCA), sometimes called bichloroacetic acid (BCA), is the chemical compound with formula C H Cl2COOH. It is an acid, an analogue of acetic acid, in which 2 of the 3 hydrogen atoms of the methyl group have been replaced by chlorine atoms. Like the other chloroacetic acids, it has various practical applications. The salts and esters of dichloroacetic acid are called dichloroacetates. Salts of DCA have been studied as potential drugs because they inhibit the enzyme pyruvate dehydrogenase kinase.
Nitrogen fixationNitrogen fixation is a chemical process by which molecular nitrogen (N2), which has a strong triple covalent bond, is converted into ammonia (NH3) or related nitrogenous compounds, typically in soil or aquatic systems but also in industry. The nitrogen in air is molecular dinitrogen, a relatively nonreactive molecule that is metabolically useless to all but a few microorganisms. Biological nitrogen fixation or diazotrophy is an important microbe-mediated process that converts dinitrogen (N2) gas to ammonia (NH3) using the nitrogenase protein complex (Nif).
Well-defined expressionIn mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if takes real numbers as input, and if does not equal then is not well defined (and thus not a function).
Algebraic fractionIn algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions. A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials. Thus is a rational fraction, but not because the numerator contains a square root function. In the algebraic fraction , the dividend a is called the numerator and the divisor b is called the denominator.
Greedy algorithm for Egyptian fractionsIn mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as 5/6 = 1/2 + 1/3. As the name indicates, these representations have been used as long ago as ancient Egypt, but the first published systematic method for constructing such expansions was described in 1202 in the Liber Abaci of Leonardo of Pisa (Fibonacci).
Protein (nutrient)Proteins are essential nutrients for the human body. They are one of the building blocks of body tissue and can also serve as a fuel source. As a fuel, proteins provide as much energy density as carbohydrates: 4 kcal (17 kJ) per gram; in contrast, lipids provide 9 kcal (37 kJ) per gram. The most important aspect and defining characteristic of protein from a nutritional standpoint is its amino acid composition. Proteins are polymer chains made of amino acids linked together by peptide bonds.
