1.6 General Terminology: Exploring Mathematics Through Text

In Activity 2, you looked at the dual purpose of this text: the text both informs and invites, i.e. it has informational and invitational purposes. When making an invitation, it is common to use language that is encouraging, warm, positive. We might call this 'upbeat language': language that has a positive and encouraging purpose to it. Other sections that consider positive/upbeat language include Activity 2 in Section 3.6 and Language Focus 1 in Section 7.6.

Upbeat language might be stronger or weaker, more explicit or more implicit. If we say to someone It would be really great if you joined our team for the maths quiz because you're a great team player and really knowledgeable about mathematics, this is clearly a greater quantity of more explicitly 'upbeat' language than Would you like to join our team? 

Question 1: Stronger and weaker expressions of invitational language in paragraph 1

Look again at paragraph 1, reproduced below for convenience, and identify some stronger and weaker forms of upbeat language. Next, select the ‘Answer’ button below.

Welcome back, and a warm welcome to those of you joining the School for the first time. As we begin a new academic year, I wanted to share some news from the School, outline what is coming up over the next few months, and flag several opportunities I hope you will consider taking up.

Question 2: Upbeat language in the remainder of the text

Look over the remainder of the text and note down any further examples of upbeat language. Decide whether you think they are stronger/more explicit or weaker/more implicit. Next, select the ‘Answer’ button below.

Follow-up activities: Her are some activities that build on the activities above

1. Make a list of other examples of upbeat language that might occur in certain academic contexts.
2. Mathematics is a serious discipline. The everyday image of mathematics is not, perhaps, one in which upbeat language frequently occurs. To what extent and in what communicative contexts could or should mathematics communication try to be upbeat at least some of the time?
3. To what extent is your communication in English or any other languages 'upbeat'?

This Language Focus is designed to raise your awareness of and skill in noticing significant features of language in the text. To do this, we draw on a single sentence and notice aspects of the lexis (words and word combinations), the morphology (the structure of words) and the syntax (the structure of phrases and clauses).

In the final section of the text above ('Mathematical quiz'), we encounter the following sentence: The quiz is run by students for students and has proven a genuinely effective way of building confidence with the language of mathematics.

There is a great deal of linguistic interest in this sentence.

Some collocations: collocation are collections of words that occur together. The following collocations occur in this sentence: the verb + noun collocation to run a quiz (in its passive form, the quiz is run); the adverb plus adjective collocation genuinely effective; the verb + object noun collocation to build confidence.

A phrase which can be used more broadly: to prove a genuinely effective way of doing something. This can be adapted to multiple other contexts e.g. memorising equations has been consistently proven to be a genuinely - and powerfully - effective way of enhancing one's ability to solve problems using said equations. 

Noteworthy grammar: the passive form of to run a quiz; the irregular past participle of prove as proven.

Clausal structure in relation to meaning: There are two clauses to the sentence linked by and. The meaning of this and is arguably slightly closer to and therefore than to simply a linking and as in I'm strongest at number theory and geometry.

Follow-up activities: Clearly this activity is very distinct from the first activity with which we began this unit which focussed on the overall meaning and purpose of the text. Working with a single sentence has a greater focus on form, although meaning of course is not excluded. You can work with activities like this which focus on the lexico-grammar of a short of text in various ways. These include:

1. Simply memorising the language and running it through your mind while reflecting on the grammatical structure of the language. This may be similar to how you study and learn equations or mathematical definitions.
2. Locating other 'noteworthy sentences' in the text and starting to notice significant features of them in terms of any of the following aspects:

• • individual words
  • particular word combinations such as collocations
  • noteworthy phrases
  • aspects of the grammar
  • how the structure contributes to the form

In the section titled 'School Mathematics Prize', we saw the UG essay title rhombitrihexagonal tessellations. The adjective rhombitrihexagonal contains the elements -tri- and hexagonal. Tri- refers to the idea of three and hexagonal refers to a polygon with six sides. The focus of this exercise is on expressions of numbers such as this.

As mathematicians, we are well aware of the positive integers: 1, 2, 3, 4 … 27 … We can talk about any given integer (say 27) in various ways: for example 27, the 27th item in the list, minus 27, 27 squared, the square root of 27, log 27. However, some of the smaller numbers, in particular 1, 2, 3 and 4, can be expressed in many different ways through language. As you will see in this exercise, there are various ways of forming words related to 1, 2, 3 and 4: sometimes whole new word stems will be used (for example one, first, monad and unit); alongside this however, you will see that prefixes and suffixes are also used to create new kinds of words (for example the prefix bi- relates to the core meaning of '2' in the words binary and bisect). Other activities that look at the use of prefixes and suffixes include the Mathematics Focus in Section 7.3 and the Mathematics Focus in Section 8.5.

Question 1: Consider the unordered list of mathematical terms below and group them according to whether they refer to 0-ness, 1-ness, 2-ness, 3-ness, 4-ness, many-ness or infinity. Then, select the ‘Answer’ button below. As noted above, prefixes and suffixes are used in some of these words; aim to notice some of the prefixes and suffixes that occur.

quadruple, tetrad, dichotomy, plurality, singleton, unit, dyad, triad, raised to the power of four, tripartite, cubed, uncountable, four-way division, empty set, squared, individual, quadripartite, duo, trichotomy, trio, double, solo, binary, quaternary, bipartite, ternary, unary, triple, monad, single, pair, group of four, quadruplet, triplet, null, void set.

Question 2: We now turn to the meanings and uses of these various forms. Complete the table below for the words for 1-ness, 2-ness, 3-ness and 4-ness. Two columns of the table and a number of individual cells have been completed. Next, select the ‘Answer’ button below.

Question 3: While the groupings above are useful, full knowledge of these terms involves understanding the differences in meaning and use between them. One way to explore this is to take two or three terms from the same row and use an English dictionary to understand differences in grammar, meaning and use. For example, take the two terms ternary and triple.

• Ternary: Ternary branching in a tree structure is when a node in a tree branches into three other nodes. Ternary logic allows three values: True, False and a third value (e.g. Unknown). In both these uses, ternary is an adjective modifying the nouns branching and logic.
• Triple: A Pythagorean triple is a set of three numbers such that the square of the largest is equal to the sum of the squares of the other two. The first integer triple is [latex](3, 4, 5)[/latex]; another is [latex](5, 12, 13)[/latex]. Here, triple is used as a noun.

Given this pair of words, we might say that ternary often suggests a structural pattern involving three distinct options or branches, whereas triple emphasises the grouping of three items itself. That said, ultimately the best way to understand these terms is through specific examples in context.

Your task: With this in mind, consider choosing any two of the terms from a same row or column and investigating individual uses of the terms in a mathematics or science and engineering context.

Extension Activity: Number Concepts in Mathematical Contexts
This section shows how number-based language appears across different mathematical areas. You do not need to read all of it; choose the topics most relevant to your studies or interests.

• Coordinates and tuples

• • 2-ness: [latex](x,y)[/latex] is an ordered pair (or 2-tuple), used for coordinates in two dimensions.
  • 3-ness: [latex](x,y,z)[/latex] is an ordered triple (or 3-tuple), used for coordinates in three dimensions.
  • 4-ness: [latex](x,y,z,t)[/latex] is an ordered quadruple (or 4-tuple), often used in spacetime.
  • n-ness: [latex]n[/latex]-tuple (a general ordered list of [latex]n[/latex] elements).

• Polynomials (Number of Terms)

Algebra uses a specific naming convention based on how many distinct terms represent an expression.

• • 1-ness: Monomial (an expression with one term, e.g., [latex]3x^2[/latex]).
  • 2-ness: Binomial (an expression with two terms, e.g., [latex]x + y[/latex]).
  • 3-ness: Trinomial (an expression with three terms, e.g., [latex]x^2 + 2x + 1[/latex]).
  • Many-ness: Polynomial (an expression with many terms).
• Degree of Equations

 The quad trap: In general, quad means 4. However, in algebraic equations, quadratic means 2 (squared). The suffix -ic is used to form the words in this set (cubic, quadratic, quintic).

• • 1-ness: Linear (degree 1, e.g., [latex]y = mx + c[/latex]).
  • 2-ness: Quadratic (degree 2, e.g., [latex]y = ax^2 + bx +c[/latex]). Note: From the Latin 'quadratus' (square). A square has 4 sides, but an area of side-length [latex]\times[/latex] side-length (2 dimensions).
  • 3-ness: Cubic (degree 3).
  • 4-ness: Quartic (degree 4, e.g., [latex]y = ax^4 + bx^3 +cx^2+dx+e[/latex]). A biquadratic is a special type of quartic with only even powers (e.g., [latex]y = ax^4 + bx^2+c[/latex]), which can be solved by substituting [latex]u=x^2[/latex].
  • 5-ness: Quintic (degree 5, e.g., [latex]y = ax^5 + bx^4 +cx^3+dx^2+ex+f[/latex]).
• Geometry / Division: the prefixes bi- and tri- are used to form words in this set related to '2' and '3'.
  • 2-ness: Bisect (to cut something into two equal parts).
  • 2-ness: Bilateral symmetry (symmetry across two sides, common in biology and geometry).
  • 3-ness: Trisect (to cut something into three equal parts).
  • 4-ness: Quadrants (the four regions of the Cartesian plane divided by the axes).
  • 4-ness: Tetrahedron (a three-dimensional shape with four triangular faces).
  • 4-ness: Quaternion (a number system that extends complex numbers, used in three-dimensional rotations and mechanics).
  • many-ness: Polyhedron (three-dimensional object with flat polygonal faces, straight edges and sharp corners or vertices).
• Sets and Elements
  • 0-ness: Empty set or void set (a set containing no elements, denoted [latex]\varnothing[/latex] or [latex]\emptyset[/latex] or [latex]\{\}[/latex].
  • 1-ness: Singleton (a set containing exactly one element, e.g., [latex]\{0\}[/latex]).
  • 1-ness: Identity (the element that leaves others unchanged, like 1 in multiplication or 0 in addition).
  • 1-ness: Unique (existing as only one, as in 'Existence and Uniqueness theorems').
• Probability and Logic
  • 2-ness: Boolean logic / Binary logic (using two values: True/False or 0/1).
  • 3-ness: Three-valued logic (True, False, and a third value such as Unknown or Undefined).
• Infinity and set size:

In mathematics, we distinguish between different 'sizes' of infinity.

• • Countably infinite: A set that can be put into one-to-one correspondence with the natural numbers [latex]\mathbb{N}[/latex], e.g. the set of integers [latex]\mathbb{Z}[/latex].
  • Uncountably infinite: A set that is larger than the natural numbers (e.g. the set of real numbers [latex]\mathbb{R}[/latex]).
  • Cardinality: A measure of the 'size' of a set.
• Special Cases: Unit Circle and Unit Vector

We use the terms unit circle and unit vector to denote a circle or vector with magnitude 1. Here, unit emphasises the standard measure of 1-ness in geometry and linear algebra.

• • Unit circle: A circle with radius 1, centred at the origin.
  • Unit vector: A vector with magnitude (length) 1, often used to indicate direction.