In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, a formula (plural: formulas or formulae [1]) is an entity constructed using the symbols and formation rules of a given logical language.
In science Science is a systematic enterprise of gathering knowledge about nature and organizing and condensing that knowledge into testable laws and theories. As knowledge has increased, some methods have proved more reliable than others, and today the scientific method is the standard for science. It includes the use of careful observation, experimentation,, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula A chemical formula or molecular formula is a way of expressing information about the atoms that constitute a particular chemical compound), or a general relationship between quantities. Colloquial use of the term in mathematics often refers to a similar construct.
Such formulas are the key to solving an equation An equation is a mathematical statement that asserts the equality of two expressions. Equations consist of the expressions that have to be equal on opposite sides of an equal sign, as in with variables. For example, determining the volume Volume is how much three-dimensional space a substance or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of of a sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The requires a significant amount of integral calculus Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral; but, having done this once, mathematicians can produce a formula to describe the volume in terms of some other parameter (the radius In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter for example). This particular formula is:
Having obtained this result, and knowing the radius of the sphere in question, we can quickly and easily determine its volume. Note that the quantities V, the volume, and r the radius are expressed as single letters. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate larger and more complex formulae.
Expressions In mathematics, an expression is a finite combination of symbols that are well-formed according to the rules applicable in the context at hand. Symbols can designate values , variables, operations, relations, or can constitute punctuation or other syntactic entities. The use of expressions can range from simple arithmetic operations like are distinct from formulas in that they cannot contain an equals sign; whereas formulas are comparable to sentences, expressions are more like phrases.
In a general context, formulae are applied to provide a mathematical solution for real world problems. Some may be general: F = ma, which is one expression of Newton's second law Newton's laws of motion are three physical laws that form the basis for classical mechanics. They have been expressed in several different ways over nearly three centuries, and can be summarised as follows:, is applicable to a wide range of physical situations. Other formulae may be specially created to solve a particular problem; for example, using the equation of a sine curve The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields. Its most basic form as a function of time is: to model the movement of the tides in a bay. In all cases however, formulae form the basis for all calculations.
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In computing
In computing Computing, also known as computer science, is usually defined as the activity of using and improving computer technology, computer hardware and software. It is the computer-specific part of information technology. Computer science is the study and the science of the theoretical foundations of information and computation and their implementation, a formula typically describes a calculation The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition or calculating the chance of a successful relationship between two people, such as addition, to be performed on one or more variables. A formula is often implicitly provided in the form of a computer A computer is a programmable machine that receives input, stores and manipulates data//information, and provides output in a useful format instruction In computer science, an instruction is a single operation of a processor defined by an instruction set architecture. In a broader sense, an "instruction" may be any representation of an element of an executable program, such as a bytecode such as
- Total fruit = number of Apples + number of Oranges.
- Degrees Celsius = (5/9)*(Degrees Fahrenheit -32)
In computer spreadsheet A spreadsheet is a computer application that simulates a paper, accounting worksheet. It displays multiple cells that together make up a grid consisting of rows and columns, each cell containing alphanumeric text, numeric values or formulas. A formula defines how the content of that cell is to be calculated from the contents of any other cell each terminology, a formula is usually a text string In mathematical logic, more precisely in the theory of formal languages, and in computer science, a string is a sequence of symbols that are chosen from a set or alphabet.[citation needed] containing cell references A spreadsheet is a computer application that simulates a paper, accounting worksheet. It displays multiple cells that together make up a grid consisting of rows and columns, each cell containing either alphanumeric text or numeric values. A spreadsheet cell may alternatively contain a formula that defines how the content of that cell is to be, e.g.
- =A1+A2
where both A1 and A2 describe "cells" (column A, row 1 or 2) within the spreadsheet. The result appears within the cell containing the formula itself (possibly A3, at end of values in column A). The = sign precedes the right hand side of the formula indicating the cell contains a formula rather than data. The left hand side of the formula is, by convention, omitted because the result is always stored in the cell itself and would be redundant.
Formulae with prescribed units
A physical quantity Physical quantity is the numerical value of a measurable property that describes a physical system's state at a moment in time. For that reason, the changes in the physical quantities of a system describe its transformation can be expressed as the product of a number and a physical unit A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement. A formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid is that all terms have the same dimension In mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature, . For example, speed, which may be measured in meters per second (m/s), miles per hour (mi/, meaning every term in the formula could be potentially converted to contain the identical unit (or product of identical units).
In the example above, for the volume of a sphere, we may wish to compute with r = 2.0 cm, which yields
There is vast educational training about retaining units in computations, and converting units to a desirable form, such as in units conversion by factor-label Many, if not most, parameters and measurements in the physical sciences and engineering are expressed as a numerical quantity and a corresponding dimensional unit; for example: 1000 kg/m³, 100 kPa/bar, 50 miles per hour, 1000 Btu/lb. Converting from one dimensional unit to another is often somewhat complex and being able to perform such.
However, the vast majority of computations with measurements is done in computer programs with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation. This requires that the universal formula be converted to a formula that is intended to be used only with prescribed units, meaning the numerical quantity is implicitly assumed to be multiplying a particular unit. The requirements about the prescribed units must be given to users of the input and the output of the formula.
For example suppose the formula is to require that , where tbsp is the U.S. tablespoon (as seen in conversion of units Conversion of units refers to conversion factors between different units of measurement for the same quantity) and VOL is the name for the number used by the computer. Similarly, the formula is to require . The derivation of the formula proceeds as:
Given that , the formula with prescribed units is
The formula is not complete without words such as: "VOL is volume in tbsp and RAD is radius in cm". Other possible words are "VOL is the ratio of V to tbsp and RAD is the ratio of r to cm."
The formula with prescribed units could also appear with simple symbols, perhaps even the identical symbols as in the original dimensional formula:
and the accompanying words could be: "where V is volume (tbsp) and r is radius (cm)".
If the physical formula is not dimensionally homogeneous, and therefore erroneous, the falsehood becomes apparent in the impossibility to derive a formula with prescribed units. It would not be possible to derive a formula consisting only of numbers and dimensionless ratios.
References
See also
- Mathematical notation Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as numbers 1 and 2, function symbols sin and +; conceptual symbols, such
- Formula (mathematical logic) In mathematical logic, a well-formed formula is a word (i.e. a finite sequence of symbols from a given alphabet) which is part of a formal language. It is a syntactic object that can be given a semantic meaning. A formal language can be considered to be identical to the set containing all and only its wffs
- Spreadsheet A spreadsheet is a computer application that simulates a paper, accounting worksheet. It displays multiple cells that together make up a grid consisting of rows and columns, each cell containing alphanumeric text, numeric values or formulas. A formula defines how the content of that cell is to be calculated from the contents of any other cell each
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