5.6 Complex Numbers and Functions: Exploring Mathematics Through Text

This exercise looks at instruction verbs. These are verbs used to convey what a question or problem is asking you to do, and they appear frequently in exams, problem sheets, and coursework. Common examples in mathematics include calculate, demonstrate, find, and convert. Understanding the precise meaning of these verbs will help you both to answer exam and problem sheet questions accurately, and to discuss them with your peers and lecturers.

In Section 2 of the Assessment Brief, the LOs are stated:

• LO1: Perform arithmetic operations (addition, subtraction, multiplication, division) on complex numbers in Cartesian form
• LO2: Convert between Cartesian and polar forms and apply De Moivre's theorem to compute powers
• LO3: Find complex roots of polynomial equations and apply the Fundamental Theorem of Algebra
• LO4: Represent complex numbers on an Argand diagram and interpret geometric transformations (addition, multiplication, conjugation)
• LO5: Apply complex numbers to solve problems in mathematics, science and engineering contexts
• LO6: Construct clear mathematical arguments, show all working, and justify solution methods
• LO7: Reflect on your own learning progress and provide constructive feedback to peers

The grammatical form of these LOs is broadly consistent: each begins with an instruction verb (sometimes called an imperative verb or command verb), followed by a direct object, and often includes additional information such as prepositional phrases or purpose clauses.

The table below shows the grammatical structure of each LO:

Question 1 - meaning: Study the table above and look up each verb in a good English dictionary. Note the meanings of each one and how these instruction verbs combine with objects and prepositions. When you feel you have an understanding of the meaning of these instruction verbs, work through the activity below.

Below, you will see nine of the instruction verbs from the table above, arranged into three groups of three. For each group, there are three corresponding sentences, each containing one gap. Your task is to complete each gap with an appropriate form of one of the three verbs in that group. Use both meaning and grammar clues to decide, referring back to the table above for guidance on how each verb is typically used. Next, select the ‘Answer’ button below.

Note that for any given verb, several forms are possible. For example, for the verb show, the possible forms include: the base (or bare infinitive) form show; the -ing form showing (identical to the present participle); and the past participle form shown.

Group A: perform, convert, apply

1. Before you can add or multiply complex numbers in polar form, it often helps to             them into Cartesian form first.

2. De Moivre's theorem can be             to find the nth roots of any non-zero complex number.

3. To divide one complex number by another, you should             the operation by multiplying both the numerator and denominator by the complex conjugate of the denominator.

Group B: find, represent, interpret

1. Using the quadratic formula,             the two complex roots of the equation [latex]z^2+z+1=0[/latex].
2. Geometrically, we can             multiplication by [latex]i[/latex] as a rotation of [latex]90[/latex] degrees anticlockwise about the origin in the complex plane.
3. An Argand diagram is the standard way of             a complex number as a point, or a position vector, in a two-dimensional plane.

Group C: construct, show, justify

1.             an Argand diagram to illustrate the complex number [latex]z=3+4i[/latex] and its complex conjugate [latex]\bar{z} =3-4i[/latex].
2. To             that [latex]z \bar{z}=|z|^2[/latex] for any complex number [latex]z[/latex], begin by writing [latex]z[/latex] in the form [latex]a+bi[/latex].
3. Before applying De Moivre's theorem to simplify a power of a complex number, you should             why the formula holds for any integer exponent [latex]n[/latex].

Question 2 - grammatical properties: The table below gives a breakdown of the verbs listed above in two columns:

• Column 1: Examples showing how each instruction verb is used, with the verb and any associated prepositions underlined.
• Column 2: The noun form of each instruction verb, with the endings underlined.

Note that two verbs (find and show) typically use the noun form -ing (see the grammar comments in the table).

Your task: Using your knowledge of mathematics and the verb-noun patterns explored in this activity, match the two parts of the sentences below. Some sentences use the verbal form, while others use the noun form. Next, select the ‘Answer’ button below the table.

In Language Focus 1, you encountered instruction verbs alongside their noun forms (e.g. perform → performance). Two other sections of the Assessment Brief that contain instruction verbs are Section 5 and Section 7.

Your Task: Work in pairs, students A and B: student A takes Section 5, and student B takes Section 7. Each student should read their section, attempt the tasks below and then deliver a mini teaching session:

1. Read the whole section and produce a short summary (2-3 sentences) of what it instructs students to do. State what the section is basically about as well as any additional relevant details.
2. Working with the grammatical ideas in Language Focus 1, identify some other examples of instruction words in the section and note aspects of their surrounding grammar. In Section 5, the first instruction verb is Record. The object of this verb is a 1–2 minute video. In Section 7, the first instruction verb is attempt in the sentence Systematically attempt all problem sheet questions. The instruction verb occurs with the adverb Systematically and its object is all problem sheet questions. 
3. Make any other comments on useful, significant or noteworthy language. For example in Section 5, we see the sentence The system will randomly pair you with another student. This may be considered useful, significant or noteworthy language as we have the word pair used as as verb with the adverb randomly. This is a noteworthy multi-word string and also has relevance to mathematical language. As a second example in Section 7, we see two uses of the adverb thoroughly. This is noteworthy as it reflects the tone of the section as it advises students to be systematic and careful in their preparation.

Here is some useful language to help you with the discussion:

• Language to get the conversation started:

1. Are we ready to start?
2. Would you like to go first, or should I?
3. I'll go first. So I've looked at Section 5 and I'll first give you a summary with some other bits of detail. Then I'll cover some of the instruction verbs and make a couple of other comments. Here we go! 

• Language to summarise the main point with some hedging or softening language included:

1. The main point of Section 5/7 is/seems to be …
2. It seems to me that Section 5/7 is basically about …
3. I'd say that the central idea in Section 5 /7 is …

• Language for supplying additional details:

1. Alongside this central message, Section 5 /7 also gives some details on …
2. The additional details that are covered in Section 5/7 are firstly …, secondly … finally …
3. I think there are essentially three other noteworthy details in Section 5 /7 which, are firstly … 

This chapter is about Complex Numbers. In this two-word term, complex is an adjective and number is a noun. Mathematical language contains many adjectives, and understanding their form and pronunciation is important for clear communication. Each of the LOs in the Assessment Brief contains at least one adjective. The adjectives are underlined below, with the stressed syllable in bold:

• LO1: Perform arithmetic operations (addition, subtraction, multiplication, division) on complex numbers in Cartesian form
• LO2: Convert between Cartesian and polar forms and apply De Moivre's theorem to compute powers
• LO3: Find complex roots of polynomial equations and apply the Fundamental Theorem of Algebra
• LO4: Represent complex numbers on an Argand diagram and interpret geometric transformations (addition, multiplication, conjugation)
• LO5: Apply complex numbers to solve problems in mathematics, science and engineering contexts
• LO6: Construct clear mathematical arguments, show all working, and justify solution methods
• LO7: Reflect on your own learning progress and provide constructive feedback to peers

Note that all the underlined items in the LOs above are adjectives, not nouns. In English, nouns can also modify other nouns. Some examples from the same texts above are:

• Argand diagram  (Argand is a noun used as modifier; diagram is the main noun)
• mathematics contexts (mathematics is a noun; contexts is the main noun)
• solution methods (solution is a noun; methods is the main noun)
• learning progress (learning is a noun; progress is the main noun).

We are not focusing on these noun-noun combinations in this exercise. However, it is important to recognise that these are not adjective-noun combinations. If you wish to revise adjective + noun combinations in relation to noun + noun combinations (and other relevant patterns), see Language Focus 3 in Section 5.3.

Returning to adjectives, another important aspect of adjective grammar is whether adjectives have an ending, and if so, what that ending is. The endings of the adjectives identified above are, with the stress in bold:

1. no ending: complex, clear
2. -ic: arithmetic, geometric (from geometry)
3. -ian: Cartesian from the name Descartes
4. -ar: polar, from pole
5. -(i)al: polynomial and mathematical (from mathematics)
6. -ive: constructive (from construct)

Your task: Go through the text and identify as many adjectives as you can. For each one, consider the following:

1. Does it have an adjective ending and what is that ending?
2. If it has an ending, is there a similar word without that ending?
3. Where is the stress? You may need to consult a good English dictionary for this.
4. What noun follows the adjective?

For example, consider Section 3, repeated below for convenience:

Weekly problem sheets are a key learning tool. They help you practise the material covered in lectures. Problem sheets are set weekly in Weeks 1-4 and 6-8 (Week 5 is the mid-term test, and Week 9 is revision week). No mark is awarded (they count 0% towards your final mark), but they are essential for learning. There is no strict deadline.

The adjectives are the following, with the stressed syllable in bold: weekly; learning; essential; strict. The first three have an ending, the last one does not.