Simple Words, Complicated Ideas

Part of the peterjamesthomas.com Maths and Science archive

Mathematics is often described as the academic discipline that uses simple words to label complicated ideas. Whereas Chemists might refer to Tetrakis[3,5-bis(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 hand-wavy 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

The separating of any material or abstract entity into its constituent elements (opposed to synthesis).
This process as a method of studying the nature of something or of determining its essential features and their relations: the grammatical analysis of a sentence.

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).

Further reading:

Several sections of Euler’s Number

Several sections of The Equation

Bundle

Everyday Usage

Several objects or a quantity of material gathered or bound together: a bundle of hay.
An item, group, or quantity wrapped for carrying; package.
A number of things considered together: a bundle of ideas.

Mathematical Usage

XXX

Cage

Everyday Usage

A boxlike enclosure having wires, bars, or the like, for confining and displaying birds or animals.
Anything that confines or imprisons; prison.
Something resembling a cage in structure, as for a cashier or bank teller.

Mathematical Usage

XXX

Category

Everyday Usage

Any general or comprehensive division; a class.
A classificatory division in any field of knowledge, as a phylum or any of its subdivisions in biology.

Mathematical Usage

XXX

Complex

Everyday Usage

Composed of many interconnected parts; compound; composite: a complex highway system.

Characterized by a very complicated or involved arrangement of parts, units, etc.: complex machinery.

So complicated or intricate as to be hard to understand or deal with: a complex problem.

Mathematical Usage

A set of numbers, denoted by $\mathbb{C}$ , whose members are of the form $a+ib$ , where both $a$ and $b$ are Real Numbers and $i^2=-1$ . 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.

Further reading:

The relevant section of A Brief Taxonomy of Numbers, this provides a brief overview

Glimpses of Symmetry, Chapter 7 – Imaginary Battleships, this is a gentler, more discursive introduction

Glimpses of Symmetry, Chapter 11 – Root of the Problem, this is a little more advanced and demonstrates a surprising link between certain types of Complex Numbers and geometry

Derivative

Everyday Usage

Derived
Not original; secondary.

Mathematical Usage

The differential as a limit

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, $f(x)$ is another function, denoted by $f^{\prime}(x)$ or $\frac{d}{dx}\hspace{1mm}f(x)$ . So $f^{\prime}(1)$ is the slope of function $f(x)$ at the point $1$ ).

As an example, if we have $f(x)=x^2+x+1$ , then $f^{\prime}(x)=2x+1$ ^[9]

Derivatives can also be calculated for two-dimensional 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, $\frac{\partial}{\partial x}\hspace{1mm}f(x,y)$ .

Further reading:

Euler’s Number, which explains derivatives from first principles

Dual

Everyday Usage

Of, relating to, or noting two.
Composed or consisting of two people, items, parts, etc., together; twofold; double: dual ownership; dual controls on a plane.
Having a twofold, or double, character or nature.

Mathematical Usage

Note:

The word Dual actually has many different meanings in Mathematics and one could argue that – looking for what these have in common – the overall sense is not a million miles from quotidian usage. The general concept is to be able to regard one mathematical object as the equivalent of (or at least strongly related to) another via some way of translating between them. Here I will concentrate just one one of best known Dual concepts, that which applies to Regular Polyhedra.

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.

Tetrahedron and Cube

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:

Cube and Octahedron

If we identify the mid-points of each Cube face in the left-hand diagram and link these, we create a shape inside the Cube which is two square pyramids glued base-to-base; this is called an Octahedron. Moving to the right-hand 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 mid-points 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

The act or the state of expecting: to wait in expectation.
The act or state of looking forward or anticipating.
Something expected; a thing looked forward to.

Mathematical Usage

Expectation (or Expected Value) is a concept in Probability Theory. Consider a finite set of outcomes, $O=\{o_1,o_2,\ldots,o_n\}$ , which occur with probabilities $P=\{p_1,p_2,\ldots,p_n\}$ (noting of course that $p_1+p_2+\cdots+p_n=1$ because one of the outcomes must happen), then we have:

$E(O)=\displaystyle\sum_{i=1}^{n}o_ip_i$

The canonical example is the long-term average value thrown by a standard six-sided die (one with the number of dots on each face ranging from one to six. The outcomes here are:

$O=\{1,2,3,4,5,6\}$

and the probabilities of these are:

$P=\bigg\{\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{6}\bigg\}$

So the Expectation is:

$\begin{aligned}E(O)&=1\cdot\dfrac{1}{6}+2\cdot\dfrac{1}{6}+3\cdot\dfrac{1}{6}+4\cdot\dfrac{1}{6}+5\cdot\dfrac{1}{6}+6\cdot\dfrac{1}{6}\\&\\&=3\frac{1}{2}\end{aligned}$

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 $10.5$ . 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 $\infty$ replacing $n$ 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

An expanse of open or cleared ground, especially a piece of land suitable or used for pasture or tillage
Sports. A piece of ground devoted to sports or contests; playing field.
A sphere of activity, interest, etc., especially within a particular business or profession: the field of teaching; the field of Shakespearean scholarship.

Mathematical Usage

Note:

This definition is adapted from the first section of Glimpses of Symmetry, Chapter 16 – …But not as we know it.

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, $F$ , exhibits is as follows:

Closure

For all $a, b \in F$ :

$a + b \in F$ and

$ab \in F$

Commutativity

For all $a, b \in F$ :

$a + b = b + a$ and

$ab = ba$

Identities

There exist $0$ and $1 \in F$ , such that for all $a \in F$ :

$a + 0 = a$ and

$1a = a$

Inverses

For all $a \in F$ , there exists $-a \in F$ , such that:

$a + -a = 0$

For all $a \in F$ where $a \ne 0$ , there exists $a^{-1} \in F$ , such that:

$aa^{-1} = 1$

Associativity

For all $a, b, c \in F$ :

$a + (b + c) = (a + b) + c$ and

$a(bc) = (ab)c$

Distributivity

For all $a, b, c \in F$ :

$a(b + c) = ab + ac$

From the above we can see that a Field F is effectively two for the price of one with respect to Groups, $(F,+)$ is a Group and so is $(F\backslash0,\times)$ . What is more both Groups are Abelian.

Function

Everyday Usage

Mathematical Usage

XXX

Graph

Everyday Usage

Mathematical Usage

XXX

Group

Everyday Usage

Mathematical Usage

XXX

Identity

Everyday Usage

Mathematical Usage

XXX

Imaginary

Everyday Usage

Mathematical Usage

XXX

Integral

Everyday Usage

Mathematical Usage

XXX

Irrational

Everyday Usage

Mathematical Usage

XXX

Manifold

Everyday Usage

Mathematical Usage

XXX

Map

Everyday Usage

Mathematical Usage

XXX

Measure

Everyday Usage

Mathematical Usage

XXX

Module

Everyday Usage

Mathematical Usage

XXX

Normal

Everyday Usage

Mathematical Usage

XXX

Perfect

Everyday Usage

Mathematical Usage

XXX

Rational

Everyday Usage

Mathematical Usage

XXX

Real

Everyday Usage

Mathematical Usage

XXX

Ring

Everyday Usage

Mathematical Usage

XXX

Simple

Everyday Usage

Mathematical Usage

XXX

Smooth

Everyday Usage

Mathematical Usage

XXX

Space

Everyday Usage

Mathematical Usage

XXX

Tree

Everyday Usage

Mathematical Usage

XXX

Variety

Everyday Usage

Mathematical Usage

XXX

<<	The Irrational Ratio
Part of the peterjamesthomas.com Maths and Science archive.

Notes

^[1]	[{3,5-(CF₃)2C₆H₃}₄B]⁻, or more colloquially BARF.
^[2]	A type of thorny-headed 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 $\sqrt{-1}$ is not a Real Number.
^[8]	A curve implies a one-dimensional line. Derivatives may be taken of two-dimensional surfaces, or indeed higher-dimensional shapes.
^[9]	A section of Euler’s Number shows how to calculate the derivative of a function.

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Simple Words, Complicated Ideas

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