notation love

# notation

## Definitions

### from The American Heritage® Dictionary of the English Language, 4th Edition

• n. A system of figures or symbols used in a specialized field to represent numbers, quantities, tones, or values: musical notation.
• n. The act or process of using such a system.
• n. A brief note; an annotation: marginal notations.

• n. The act, process, method, or an instance of representing by a system or set of marks, signs, figures, or characters.
• n. A system of characters, symbols, or abbreviated expressions used in an art or science or in mathematics or logic to express technical facts or quantities.
• n. A specific note or piece of information written in such a notation.

### from the GNU version of the Collaborative International Dictionary of English

• n. The act or practice of recording anything by marks, figures, or characters.
• n. Any particular system of characters, symbols, or abbreviated expressions used in art or science, to express briefly technical facts, quantities, etc.
• n. Literal or etymological signification.

### from The Century Dictionary and Cyclopedia

• n. The act of noting, in any sense.
• n. A system of written signs of things and relations (not of significant sounds or letters), used in place of language on account of its superior clearness and brevity. Notations are employed to advantage in every branch of mathematics, in logic, in astronomy, in chemistry, in music, in proofreading, etc. Thus, (1 /i) means that in the expression 1 / i all the whole numbers from 1 to n inclusive are to be successively substituted for i and the resulting values added together to give the quantity denoted by the expression. When the limits are not indicated, the lower one is to be understood as constant, and generally zero, and the upper one as one less than the actual value of the variable. For example, if we write ϲ (2 x + 1) = x, this signifies In like manner, δ is used to signify the difference, or the amount by which the quantity written after it would be increased by increasing the variable by unity. The variable may be indicated by a subjacent letter; thus, δxx = (x + 1) —x; but δyx = x + 1 —xy = (x— 1) xy. The product of two quantities is denoted by writing them in their order, either directly, or with an interposed cross or dot (.); thus, 3 x a = 3. a = 3 a. A quotient is usually denoted by one of the signs ÷ or: or /, with the dividend before it and the divisor after it, or by a horizontal line with the dividend above and the divisor below. A continued product is also written with 11, just as a summation is written with ϲ ; but when the limits are not indicated, the lower one is constant, and generally unity, and the upper one the actual value of the variable. A positive whole number with the mark of admiration (!) after it denotes the continued product of all numbers from 1 up to that number inclusive; thus, 4! = 24. Instead of the mark of admiration, a right-angled line beneath and at the left of the number is sometimes used: as, A power of a quantity is denoted by writing the exponent to the right and above the base; thus. x = x. x. x. This notation is extended to symbols of operation. Thus, δu = δδ u; and δ— u = ϲ u, because u = δδ— u = δ u. If the exponent is included in parentheses, the quantity denoted is the continued product of a number of factors equal to the exponent, one factor being the base, and the others the results of successive subtractions of 1 from the base; thus, x () = x (x— 1) (x—2). A root is denoted either by a fractional exponent, or by the sign √ written before the base, with the index above and to the left; thus, If the index is omitted, it is understood to be 2. One of the most important parts of algebraical notation is the use of parentheses, ( ), square brackets, [ ], braces, , and vincula or horizontal lines above the expressions, to signify that the symbols so included are to be treated as signifying one quantity. Thus, (3 + 2) x 5 = 25, but 3 + (2 x 5) = 13. Functions are usually denoted by operative symbols, especially f, F, φ, Φ, written before the variable, the latter being often inclosed in parentheses. If there are several variables, these are inclosed in one parenthesis and separated by commas, as F (x, y). Various special functions have special abbreviations, as log for logarithm, sin for sine, cos for cosine, tan for tangent, cot for cotangent, sec for secant, cosec for cosecant, vsin for versed sine, sinh for hyperbolic sine, am for amplitude, sn for sine of the amplitude, cn for cosine of the amplitude, etc. (For the special notation of matrices, determinants, graphs, and groups, see those words.) A differential is expressed by d before the function, and a partial differential is now generally written with δ instead of d; the variable is indicated, if necessary, by a subjacent letter. A variation is expressed by a δ before the varying quantity. A differential coefficient is most frequently expressed fractionally as a ratio of differentials, or by etc., written before the function. But the capital D is often used: thus, Dxxy = yx— 1, and Dxxy = log x. x. Differentiation relatively to the time is frequently expressed by accents: thus, s′ = Dts′ and s′ = Dts′ . Dots over the letters are also used instead of the accents, this being the original fluxional notation of Newton. The differential coefficients of a function are frequently denoted by accents attached to the operational symbols: thus, f″x = Dxfx. A number of other differential operations are indicated by special operational symbols, as for Laplace's operator. The integral of an expression is written with the sign f, introduced by Leibnitz, before the differential. The limits of a definite integral are written above and below this sign. Besides these notations, there are many others peculiar to different branches of mathematics.
• n. Etymological signification; etymology.
• n. In music, the act, process, or result of indicating musical facts by written or printed characters.