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Pure mathematics

Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on. Another insightful view is that of pure mathematics as not necessarily applied mathematics.

History

Ancient Greece

Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being." Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must needs make gain of what he learns." The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted, They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason. And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that the subject is one of those that "...seem worthy of study for their own sake."

19th century

The term itself is enshrined in the full title of the Sadleirian Chair, founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

20th century

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.

In fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training.

Generality and abstraction

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality.

  • Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
  • Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
  • One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.
  • Generality can facilitate connections between different branches of mathematics Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more abstract mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. Undeniably, though

Activity (chemistry)

In chemical thermodynamics, activity (symbol: a) is a measure of the “effective concentration� of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.

By convention, activity is treated as a dimensionless quantity, although its actual value depends on customary choices of standard state for the species. The activity of pure substances in condensed phases (solid or liquids) is normally taken as unity. Activity depends on temperature, pressure and composition of the mixture, among other things. For gases, the effective partial pressure is usually referred to as fugacity.

The difference between activity and other measures of composition arises because molecules in non-ideal gases or solutions interact with each other, either to attract or to repel each other. The activity of an ion is particularly influenced by its surroundings.

Activities should be used to define equilibrium constants but, in practice, concentrations are often used instead. The same is often true of equations for reaction rates. However, there are circumstances where the activity and the concentration are significantly different and, as such, it is not valid to approximate with concentrations where activities are required. Two examples serve to illustrate this point:

  • In a solution of potassium hydrogen iodate at 0.02 M the activity is 40% lower than the calculated hydrogen ion concentration, resulting in a much higher pH than expected.
  • When a 0.1&nbsp;M hydrochloric acid solution containing methyl greenindicator is added to a 5&nbsp;M solution of magnesium chloride, the color of the indicator changes from green to yellow—indicating increasing acidity—when in fact the acid has been diluted. Although at low ionic strength (<0.1&nbsp;M) the activity coefficient decreases with increasing ionic strength, this coefficient can actually increase with ionic strength in a high ionic strength regime. For hydrochloric acid solutions, the minimum is around 0.4&nbsp;M.

Definition

The activity of a species i, denoted ai, is defined as:

a_i = \exp\left (\frac{\mu_i - \mu^{\ominus}_i}{RT}\right )

where μiis thechemical potential of the species under the conditions of interest, μoi is the chemical potential of that species in the chosen standard state, R is the gas constant and T is the thermodynamic temperature. This definition can also be written in terms of the chemical potential:

\mu_i = \mu_i^{\ominus} + RT\ln{a_i}

Hence the activity will depend on any factor that alters the chemical potential. These include temperature, pressure, chemical environment etc. In specialised cases, other factors may have to be considered, such as the presence of an electric or magnetic field or the position in a gravitational field. However the most common use of activity is to describe the variation in chemical potential with the composition of a mixture.

The activity also depends on the choice of standard state, as it describes the difference between an actual chemical potential and a standard chemical potential. In principle, the choice of standard state is arbitrary, although there are certain conventional standard states which are usually used in different situations.

Activity coefficient

The activity coefficient γ, which is also a dimensionless quantity, relates the activity to a measured amount fractionxi,molalitymioramount concentrationci:

a_i = \gamma_{x,i} x_i\,
a_i = \gamma_{m,i} m_i/m^{\ominus}\,
a_i = \gamma_{c,i} c_i/c^{\ominus}\,

The division by the standard molality mo or the standard amount concentration co is necessary to ensure that both the activity and the activity coefficient are dimensionless, as is conventional.

When the activity coefficient is close to one, the substance shows almost ideal behaviour according to Henry's law. In these cases, the activity can be substituted with the appropriate dimensionless measure of composition xi, mi/mo or ci/co. It is also possible to define an activity coefficient in terms of Raoult's law: the International Union of Pure and Applied Chemistry (IUPAC) recommends the symbol Æ’ for this activity coefficient, although this should not be confused with fugacity.

a_i = f_i x_i\,. Solution can also become too diluted when necessary.

Standard states

Gases

In most laboratory situations, the difference in behaviour between a real gas and an ideal gas is dependent only on the pressure and the temperature, not on the presence of any other gases. At a given temperature, the "effective" pressure of a gas i is given by its fugacityÆ’i: this may be higher or lower than its mechanical pressure. By historical convention, fugacities have the dimension of pressure, so the dimens


From Yahoo Answers

Question:I have this question on a home work paper for chemistry and I only have a short time to do it so I need the answer by Monday and quick.

Answers:By definition an element is composed of only one type of atom, where a compound is 2 or more.

Question:In my chemistry book, a pure substance is defined as "matter with distinct properties and composition that doesn't vary from sample to sample." A homogenous mixture is "a mixture uniform throughout." It later says that if matter is uniform throughout it's homogenous, if it DOES have variable composition, it is a homogenous mixture (or a solution), and if it DOES NOT have variable composition, it's a pure substance. I'm not seeing how this makes sense... help?!

Answers:Think of it as a pure substance is like an element, the element is always the same. Every time, Carbon is going to have the same properties. A homogeneous mixture is when two things are combined and you can not see any difference. The example i like is water and sand. When you put the sand in the water, it doesn't mix so the mixture is heterogeneous. If you mix water and sugar; however, the end product is still aqueous and appears uniform. It will be this way until you get into over saturation.

Question:Chem booklet for school and I need a little help please:) also the definitions for 1. a compound 2.a mixture

Answers:I'm in college chemistry and we're learning about this too (but I learned it first in HS) A substance cannot be separated by physical means like distillation. It's chemically bound and can only be broken my chemical means. Also no matter where it came from the ratio is always the same. Water always has a 2:1 ratio of Hydrogen to Oxygen. An element IS a substance, but it's in it's purest form because it CAN'T be broken by chemical means into a simple substance. It's pure. Like with water, H2O is a substance, but it can be broken down into into it's elements--Hydrogen and Oxygen. But that takes a chemical reaction. Salt in water is can be separated by physical means (like distillation) so it is not a substance. This is probably confusing... but read it slowly and try and understand it. lol. and I'm not answering the rest cuz... it's your homework. I just know these two can be confusing.

Question:Hey people! I'm moving back to Canada this year, but I have never been to a Canadian School, the new school's confusing, it has Applied, and Academics mathematics... But for American school, they have separate courses like Algebra, Geometry, Algebra II/ Trig,etc and PS: if im taking geometry summerschool, what kind of "course name" do they have for Geometry? Please help! Thanks!!

Answers:This sounds similar to Australia, where most Universities distinguish between "Pure Mathematics" and "Applied Mathematics". The names mean what you would expect: maths with particular applications in mind. Algebra, Trig and Geometry would all be "Pure Mathematics", as would be Number Theory and the analysis part of Calculus. Applied maths would be things like Differential Equations, Waves, Optimisation, and sometimes some overlap with computing and statistics. As an example, check out these Senior (3rd year University) Pure and Applied subjects at the University of Sydney. http://www.maths.usyd.edu.au/u/UG/SM/

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