Mathematics is often described as the academic discipline that uses simple words to label complicated ideas. Whereas Chemists might refer to Tetrakis[3,5bis(trifluoromethyl)phenyl]borate ^{[1]}, Biologists to Macracanthorhynchus hirudinaceus ^{[2]} and Physicists to Dissipative Decoherence ^{[3]}, Mathematicians talk about Rings for arguably just as complicated concepts. This brief article provides an inexhaustive list of examples of this phenomena; one to which I may add over time.
Notes:
All “normal” definitions are taken from Dictionary.com. Where there are multiple parts to a definition, I have in general only reproduced the first few of these. However, links to the full Dictionary.com definitions are provided, just click on the relevant picture of a link. In some cases, the simple everyday word relates to a Mathematical concept which is even more complicated than on average. In these cases, I merely provide a handwavy picture of the Mathematics, lest I fill this article with thousands of words of background material. In these cases, I will generally provide a link to more information, typically on Wikipedia. 
Analysis  
Everyday Usage
Mathematical Usage Analysis is a broad and fundamental part of Mathematics which is concerned with all aspects of limits ^{[4]}. In turn, limits relate to what happens to expressions, shapes, sequences, series and other concepts when some parameter becomes infinitely big or infinitely small; for example the sum of an infinite number of terms. Both elements of The Calculus, namely Differentiation ^{[5]} and Integration, are based on the concept of limits and thus are part of Analysis. The study of infinite sequences and the sum or product of infinite series (particularly power series) are similarly part of this area. Analysis is typically split into studies relating to Real Numbers (Real Analysis) and relating to Complex Numbers (unsurprisingly Complex Analysis).


Bundle  
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Cage  
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Category  
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Complex  
Everyday Usage
Mathematical Usage A set of numbers, denoted by , whose members are of the form , where both and are Real Numbers and . The Complex Numbers form each of a Group (under both addition and multiplication), a Field, a Space and a Division Algebra (a set of numbers that behave well with respect to division, save for when the denominator is zero of course). They are closed under algebraic operations (addition/subtraction, multiplication/division ^{[6]}, raising to powers/taking roots and so on). This means that if you perform any of these operations on Complex Numbers, you always end up with more Complex Numbers. In fact the Complex Numbers are what you get if you extend the Real Numbers ^{[7]} to include the result of any algebraic operation which is performed on them.


Derivative  
Everyday Usage
Mathematical Usage The derivative is essentially the slope of a curve (more properly a function ^{[8]}) at a point. However, where such a slope varies from point to point, the derivative captures the slope at any point in another expression (using the prefered language of functions, the derivative of a function, is another function, denoted by or . So is the slope of function at the point ). As an example, if we have , then ^{[9]} Derivatives can also be calculated for twodimensional surfaces (or higher dimensional objects). Often the approach is to fix one or more of the dimensions. So for a two dimensional surface, if we fix one dimension, we reduce the problem to a one dimensional curve. Such derivatives are called partial derivatives with the slightly different notation, .


Dual  
Everyday Usage
Mathematical Usage
A Polyhedron is a three dimensional solid object whose faces consist of Polygons. If all of the faces are identical, then the shape is a Regular Polyhedron. Two obvious examples of Regular Polyhedra are the Tetrahedron and the Cube. The former has four faces, each of which is an equilateral triangle. The latter has six faces, each of which is a square. The above image also shows some examples of vertices, points where three or more faces meet. We are now ready to offer our definition of the Dual of a Polyhedron. It is simply the Polyhedron which we obtain by joining up the midpoints of all of the faces of our original Polyhedron. A diagram is probably called for – let’s look at the Cube: If we identify the midpoints of each Cube face in the lefthand diagram and link these, we create a shape inside the Cube which is two square pyramids glued basetobase; this is called an Octahedron. Moving to the righthand diagram, if we carry out the same process on an Octahedron, we recover a Cube again. Alternatively, you can start the process by constructing a shape outside of the cube and with the midpoints of its faces touching the Cube’s vertices; again the result is an Octahedron. This is probably obvious to see by thinking about the two exhibits above from right to left instead of left to right. Some Polyhedra, like the Tetrahedron above, are their own Dual. We have covered three of the five Platonic Solids above. The remaining ones are the Dodecahedron and Icosahedron. These two are also each other’s Duals in the same way that the Cube and Octahedron are. 

Expectation  
Everyday Usage
Mathematical Usage Expectation (or Expected Value) is a concept in Probability Theory. Consider a finite set of outcomes, , which occur with probabilities (noting of course that because one of the outcomes must happen), then we have: The canonical example is the longterm average value thrown by a standard sixsided die (one with the number of dots on each face ranging from one to six. The outcomes here are: and the probabilities of these are: So the Expectation is: Here we can see that (given the likelihood of throwing any number with the die is the same for all cases) the Expectation is the same as the Mean of the outcomes. So for a Icosahedral die labelled one to twenty, the Expectation is the mean of one to twenty, or . If some outcomes are more likely than others, then this does not hold of course. One might view the Expectation as a sum of the values of the outcomes, weighted by their probability of occuring. The above definition generalises for countably infinite sets of outcomes and their probabilities, with replacing as the upper bound of the sum. For continuous (i.e. uncountable) distributions, the concept generalises to an integral rather than a sum. 

Field  
Everyday Usage
Mathematical Usage
A Field is essentially a Mathematical structure that behaves the same way as the Rational Numbers with respect to addition, multiplication and division. There are some quite recherché examples, but if you think of the Rational or the Real Numbers as the prototype, then you won’t go far wrong. The set of formal properties that a Field, , exhibits is as follows: Closure For all : and Commutativity For all : and Identities There exist and , such that for all : and Inverses For all , there exists , such that: For all where , there exists , such that: Associativity For all : and Distributivity For all : From the above we can see that a Field F is effectively two for the price of one with respect to Groups, is a Group and so is . What is more both Groups are Abelian. 

Function  
Everyday Usage
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Graph  
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Group  
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Identity  
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Imaginary  
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Integral  
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Irrational  
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Manifold  
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Map  
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Measure  
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Module  
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Normal  
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Perfect  
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Rational  
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Real  
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Ring  
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Simple  
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Smooth  
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Space  
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Tree  
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Variety  
Everyday Usage
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<<  The Irrational Ratio  
Part of the peterjamesthomas.com Maths and Science archive. 
Notes
^{ [1]}  [{3,5(CF_{3})2C_{6}H_{3}}_{4}B]^{−}, or more colloquially BARF. 
^{ [2]}  A type of thornyheaded worm, or Acanthocephala. 
^{ [3]}  You most likely don’t want to know. It involves stuff like: 
^{ [4]}  See a section of Euler’s Number for more background on limits. 
^{ [5]}  See another section of Euler’s Number for more background on differentiation. 
^{ [6]}  If we exclude dividing by zero of course. 
^{ [7]}  The Real Numbers are not algebraically closed, not least as is not a Real Number. 
^{ [8]}  A curve implies a onedimensional line. Derivatives may be taken of twodimensional surfaces, or indeed higherdimensional shapes. 
^{ [9]}  A section of Euler’s Number shows how to calculate the derivative of a function. 
Text & Images: © Peter James Thomas 2018. 
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