5.5 Complex Numbers and Functions: Mathematics Communication
So far, you have focused on written mathematical language. In this section, you will practise understanding and taking part in spoken mathematical discussions and written email exchanges. You will read and analyse a dialogue between two students, and an email correspondence between a student and a lecturer addressing the whole class.
The goal of this section is not just to understand the dialogue and the email, but to help you feel confident in having similar discussions and writing emails yourself. You will learn how to have a discussion about mathematics, ask and answer questions, and explain your ideas clearly to someone else. This will help you use English for practical conversations about mathematics, both in class and out of class. To do this, you will need to:
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notice useful phrases and sentence structures in the dialogues,
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memorise them,
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practise saying them,
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and finally, use them in real conversations.
Each line in the dialogues is numbered to help you talk about specific parts easily. For example, if you have a question, you can say: What does the second word from the end of line 8 mean?
By working with these texts, you will improve your ability to discuss mathematics in English, ask and answer questions, and communicate your ideas clearly with others.
Dialogue 1: two students
An example dialogue between two university students, William and Chloe, discussing complex numbers. This dialogue occurs in a study space with the two students sitting close to each other. It is a relaxed study session between peers reviewing a lecture.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | William: | Hey Chloe. How are you doing? Shall we sit here? |
| (2) | Chloe: | Hi, William. Looks fine to me, let’s grab a seat. How are things going for you? It’s a busy part of the semester. |
| (3) | William: | It sure is! There’s a lot on. I've been going through my notes from today’s lecture on complex numbers, and I'm confused about the imaginary unit. Can you please help me a bit to review? |
| (4) | Chloe: | Of course. So, we started by defining the imaginary unit, which we usually denote by the letter i, using its special property: i squared is equal to minus one. |
| (5) | William: | But how does that make sense? The square of any real number is always non-negative, isn't it? How can i squared be equal to minus one? |
| (6) | Chloe: | You’re right, that's why it is called an imaginary unit. It's not a real number. Think of it as a mathematical object that was invented, or we might say ‘defined into existence’, to allow us to find square roots of negative numbers. |
| (7) | William: | OK…so, if i squared is minus one, then the square root of minus one is just i. So, extending that, the square root of minus four is the square root of four, times the square root of minus one, which is two i. |
| (8) | Chloe: | Perfect, you've got the idea. And using i, we can form complex numbers, which are mathematical constructs that extend the real number system to include solutions to equations that cannot be solved over the real numbers. A complex number is in the form a plus b i, where a and b are real numbers. |
| (9) | William: | So, an example of a complex number is three plus two i? |
| (10) | Chloe: | Yes, in this case, a is three, and b is two. We call a, the real part of the complex number, and we call b the imaginary part. And using these coefficients, we can even represent complex numbers on a diagram. |
| (11) | William: | The complex plane? I remember that vaguely from the lecture… |
| (12) | Chloe: | Yes, or the Argand diagram, as it is sometimes called. This is the two-dimensional plane where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. |
| (13) | William: | I see, so our complex number three plus two i, can be represented by the point three comma two. |
| (14) | Chloe: | Exactly, and if we apply Pythagoras’ theorem, the modulus of a complex number is just the distance from the origin. |
| (15) | William: | OK in our case, the modulus is the square root of, three squared plus two squared, which equals the square root of fourteen. |
| (16) | Chloe: | Careful, that’s the square root of thirteen. Nine plus four, thirteen. |
| (17) | William: | Aaah yes of course. And that’s where we stopped in the lecture, right? |
| (18) | Chloe: | In the last five minutes, the lecturer briefly introduced the argument and polar form of complex numbers. But she said we will cover it properly next time. |
| (19) | William: | OK I will wait for our next lecture then. This was really helpful, Chloe, thank you. I'm starting to get it now! |
| (20) | Chloe: | No worries, William, happy to help. |
| (21) | William: | Shall we meet the same time next week? |
| (22) | Chloe: | Yeah sure. See you then. Enjoy the rest of your day. |
| (23) | William: | And you, Chloe. See you next week. |
Activity 1: Global Understanding
From the options below, decide which summary you think most accurately describes the overall structure of the dialogue. Next, select the ‘Answer’ button below the options.
Option 1: Chloe works with William to develop his understanding of some key concepts in complex numbers.
Option 2: Chloe supports William in completing a series of calculations which require the understanding of complex numbers.
Option 3: Chloe supports William in recapping everything that was covered in the most recent lecture.
Option 4: Chloe draws the Argand diagram for William as part of the homework for the next seminar.
Answer
Option 1 is the correct answer. William is unsure about the imaginary unit and the related concept of complex numbers. Chloe understands this and helpfully and effectively helps William to develop his understanding.
Option 2 is incorrect as the discussion does not involve working through any calculations but rather understanding the key concepts in complex numbers.
Option 3 is incorrect as, although some reference is made to the most recent lecture (e.g. in (3), (11), (18) and (19)), this is not the central focus of the conversation.
Option 4 is incorrect as, although the Argand diagram is mentioned by Chloe, she does not draw this diagram and it is not part of any homework.
Activity 2: Identifying Useful Language
Question 1: In (8), Chloe says And using i, we can form complex numbers, which are mathematical constructs that extend the real number system to include solutions to equations that cannot be solved over the real numbers. This sentence has three parts:
- And using i, … - This part of the sentence states what mathematical construct we will use.
- … we can form complex numbers … - This part of the sentence states what mathematical object will emerge from a.
- … which are mathematical constructs … - This part of the sentence is a relative clause which provides further information about something already mentioned.
Using your knowledge of mathematics, match each sentence beginning on the left with its corresponding ending on the right. Next, select the ‘Answer’ button below the table.
| Parts 1 and 2 of the sentence | Part 3 of the sentence | |
|---|---|---|
| So by successively differentiating, we can determine higher-order derivatives, … | … which reveals the elegant connection between complex multiplication and rotation in the complex plane. | |
| By converting to polar form, we can express complex numbers in exponential notation, … | …which represents the distance from the origin in the complex plane. | |
| And using differentiation, we can determine the gradient of the tangent at a point, … | … which give us information about concavity, inflection points, and acceleration. | |
| Through the modulus, we can measure the magnitude of a complex number, … | … which gives us the instantaneous rate of change of the function. |
Answer
| Parts 1 and 2 of the sentence | Part 3 of the sentence |
|---|---|
| So by successively differentiating, we can determine higher-order derivatives, … | … which give us information about concavity, inflection points, and acceleration. |
| By converting to polar form, we can express complex numbers in exponential notation, … | … which reveals the elegant connection between complex multiplication and rotation in the complex plane. |
| And using differentiation, we can determine the gradient of the tangent at a point, … | … which gives us the instantaneous rate of change of the function. |
| Through the modulus we can measure the magnitude of a complex number, … | …which represents the distance from the origin in the complex plane. |
Question 2: In (3) William says:
I've been going through my notes from today’s lecture on complex numbers, and I'm confused about the imaginary unit. Can you please help me a bit to review?
In this sentence, the words which mean something like study are going through. This is a two-word verb, sometimes called a phrasal verb or multi-word verb.
The tense used is called the present perfect continuous. It is formed with a part of the verb have, the past participle of the verb be (i.e. been) and the present participle (the -ing form) of a verb. This gives us. This gives us I + have + been + going through which as a string is written I've been going through.
Now read the three sentences below (a, b, c) and answer the three questions that follow (1, 2, 3).
- I've been looking over what I noted down from the lecture today on complex numbers and the stuff about the imaginary unit is still some way from making complete sense. Any chance of a bit of help?
- I've been working through the lecture notes on complex numbers and I'm finding it all straightforward. Do you want a hand with this at all?
- I've been taking a look at the notes I made from the lecture earlier today on complex numbers and it's still a bit puzzling. Do you have a moment to help at all?
- Which two of the three sentences (a, b, c) have a similar meaning to line (3) in the dialogue?
- In each sentence (a, b, c), what words mean something similar to study?
- In which sentence(s) is the verb in the present perfect continuous tense?
Next, select the ‘Answer’ button below.
Answer
Question 1: The two sentences that have a similar meaning to (3) are a and c. Sentence b is in a certain sense the opposite of sentences a and b in that, in addition to making reference to the lecture notes (not the student's own notes), it uses the clause I'm finding it all straightforward, where straightforward has a similar meaning to easy. Furthermore, the second sentence in b (Do you want a hand with this, at all?) is an informal offer of help, not a request for help.
Question 2: In each sentence, the words that mean something similar to study are:
- look over
- work through
- taking a look at
Note that taking a look at (c) uses a noun phrase (a look) rather than a direct verb form, which makes it slightly different grammatically from look over and work through. However, for the purposes of this activity, it is sufficient to note that all three multi-word expressions are similar in meaning to study. Consult a good English–English dictionary for further examples of their use and to add them to your productive vocabulary.
Question 3: All three verbs are in the present continuous tense.
Language Focus
Observation 1
In Activity 2 Question 1, above, we note the word successively, which means one after the other without any breaks, similarly to consecutively. We should not confuse this with successfully, which means in a way that accomplishes a desired goal.
Observation 2
In (6), speaking about the imaginary unit, Chloe says Think of it as a mathematical object that was invented, or we might say ‘defined into existence’, to allow us to find square roots of negative numbers. The phrase defined into existence means that a mathematical object comes into existence through formal definition and axiomatic construction (where an axiom is a self-evident statement accepted without proof), rather than through physical discovery in nature.
This idea reflects a key discussion in philosophy of mathematics about whether mathematical objects are discovered (exist in nature independently), or invented (they exist because we define them). For example, one can argue that the golden ratio, [latex]\phi[/latex], and [latex]\pi[/latex] exist in nature independently and have been discovered by mathematicians. On the contrary, it can be argued that non-Euclidean geometry and abstract algebra were invented by mathematicians to develop theories to solve certain mathematical problems.
Discuss with a partner whether the following mathematical constructs have been defined into existence or not:
- negative integers
- zero
- irrational numbers
- symmetries
- infinity, calculus.
Feel free to come up with some of your own examples that fit clearly in one category or the other - or examples where it is unclear whether the concept has been discovered or invented.
Note that this is a discussion/debate, so there is no single right answer. Instead try to persuade your partner of your views. Some useful language for your conversations follows:
I think negative integers are an invention and a human mathematical construct. While positive integers can arguably be seen in nature, surely negative numbers - and furthermore the concept of zero - are invented.
Isn't symmetry something that we find in nature? Many organisms have a line of symmetry, don't they? For example animal faces and bodies are broadly symmetrical. And even some organs, for example the lungs. And don't snowflakes and flowers also have some version of rotational symmetry?
Activity 3: Your turn!
Goal
Practise having conversations about problems on the topic of complex numbers and functions in English. Focus on communication, not necessarily on solving new math problems.
Set-up
In pairs, take the following roles:
Speaker A is the student that has a question about a concept or set of concepts that they don't understand either in complex numbers or in any field of mathematics. Speaker A can either choose a concept that they are actually struggling with, or something that they do in fact understand and pretend they don't.
Speaker B is the student that has the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B.
Step 2: Decide what concept the conversation will be about. Student B at least should be able to explain the concept.
Step 3: Take 3 to 5 minutes individually or together to look over the dialogue above and note down any useful phrases.
Step 4: Run the dialogue. Speaker A should start, either with a variation on William's opening above, or with something else, for example, Hi [Speaker B]. Do you have a quick moment to help me out with [Problem P]. It shouldn't take more than a couple of minutes.
Step 5: Switch roles and practise again to experience both asking and answering.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced, how confident you felt, and if there are alternative ways of going through the dialogue.
Email Correspondence: A Module Announcement to the whole class
The text below is an email thread which begins with a module announcement from the lecturer, Dr Sophie Lu, to the whole class for a midterm university exam. A student, George Williams, responds with some questions, to which Dr Lu responds.
| Email Number | |
|---|---|
| (1) | From: s.lu@northbury.edu.ac.uk To: Mathematics Module Mailing List Subject: Midterm Semester 2 and lecture notes from this week Hello everyone, It was great seeing you again this week. I have now uploaded our lecture notes for this week onto our Virtual Learning Environment, Minerva. You can find them under the folders Semester 2-> Week 10. As I mentioned in today's lectures, you will soon have your midterm exam on Thursday, 27th November, and it will be one hour long. I have now created a new folder on Minerva called "Midterm and exam preparation". For now, you can find last year's midterm exam. Please go through it and see if you can answer it. Try and time yourself and aim to finish in one hour. In our lectures next week, we will then go through the solutions together, as well as any other questions that you may have. After that, I will then upload the solutions in the same folder on Minerva. The material that it will cover will be all of derivatives, integrals, sequences and series, and complex numbers and functions up to and including roots of complex numbers. See you all next week. Kind regards, Sophie |
| (2) | From: g.williams@northbury.edu.ac.uk To: s.lu@northbury.edu.ac.uk Subject: RE: Midterm Semester 2 and lecture notes from this week Dear Dr Lu, I hope this email finds you well. Thank you for letting us know about our Mathematics midterm exam. I am writing to ask about the specific time of the midterm exam on Thursday, 27th November. I have to catch a train leaving Northbury at 16:00 that day due to a family matter, and I would like to know if I will be able to take the exam before I leave for the train station. If the exam is scheduled in the morning or early afternoon, I should be able to sit it before my train. However, if it is later in the day, I may need to reschedule my travel to avoid missing the exam. Could you please confirm the exam time at your earliest convenience? This will help me decide whether I need to adjust my travel arrangements. Thank you for your help. Best regards, George Williams |
| (3) | From: s.lu@northbury.edu.ac.uk To: g.williams@northbury.edu.ac.uk Subject: RE: Midterm Semester 2 and lecture notes from this week Hello George, Thank you for reaching out, I should have specified the time in the original announcement. I understand the importance of your family matter, and I appreciate you letting me know about your travel plans. The midterm exam is scheduled for Thursday, 27th November at 09:00 AM, so you should have plenty of time to complete it before your 16:00 train. The exam is held in our usual venue (Building A, Room 214) and we should be finished by 10:15 AM at the latest. Given this timing, you should have no difficulty making your train. However, if there are any other complications with your travel arrangements or if you have further concerns, please do let me know. Best of luck with your exam preparation, and safe travels. Kind regards, Sophie |
| (4) | From: g.williams@northbury.edu.ac.uk To: s.lu@northbury.edu.ac.uk Subject: RE: Midterm Semester 2 and lecture notes from this week Dear Sophie, Thank you so much for the quick response and for confirming the exam time. I am relieved to hear it's scheduled for 09:00 AM; that gives me plenty of time before my train. I really appreciate your understanding regarding my situation. I will make sure to be well-prepared and arrive early on the day of the exam. Thanks again for your help! Best regards, George |
| (5) | From: s.lu@northbury.edu.ac.uk To: Mathematics Module Mailing List Subject: CLARIFICATION: Midterm Semester 2 and lecture notes from this week Hello everyone, I have just realised that I neglected to include the exam time in my earlier announcement about the midterm examination. To clarify: your midterm exam will be held on Thursday, 27th November at 09:00 AM in our usual venue: Building A, Room 214. The exam will last one hour and we should be done by 10:15 AM. If you have any questions or concerns about the exam logistics, please do not hesitate to get in touch. Kind regards, |
Activity 1: Global understanding
From the options below, decide which summary you think most accurately describes the email correspondence. Consider two key parameters: a) whether George's request is resolved or not, and b) whether the tone of the exchange is abrupt and urgent or polite and respectful. Next, select the ‘Answer’ button below the options.
Option 1: George's request is not resolved and the tone of the exchange is abrupt and urgent.
Option 2: George's request is resolved and the tone of the exchange is abrupt and urgent.
Option 3: George's request is not resolved and the tone of the exchange is polite and respectful.
Option 4: George's request is resolved and the tone of the exchange is polite and respectful.
Answer
Option 4 is the correct answer. George's request is to know the time of the exam (George's first email (2), paragraph 2) and this information is provided (Dr Lu's reply (2), paragraph 2). Moreover, the tone of the email communications is polite and respectful throughout. We consider specific examples of language that convey the polite and respectful tone in the immediately following activity, Activity 2.
Activity 2: Identifying useful language 1 - conveying politeness and respect
In Activity 1, the tone of the email thread was characterised as polite and respectful. Politeness and respect are essential in university communication. In this activity we explore the language that creates that tone.
In Dr Lu's post (1) on the Maths Module Mailing List, the following phrases indicate politeness and respect to the students:
- It was great seeing you again this week. This standard opening creates a positive tone to the text. It shows interest in students, and it creates a warm, friendly tone before asking for work.
- Please go through it and see if you can answer it. It would be shorter to write something like Look at it and answer it. However, the use of please and see if you can soften the communication and make it more respectful.
- Try and time yourself and aim to finish in one hour. Again, the Try helps to soften the tone of the email.
In addition to these words and phrases which create a tone of politeness and respect, the email is also well structured and clear. Clear, well-structured emails both show respect in the sense that they convey that care has been taken in writing them, and are easier to understand.
Re-read the email exchange and identify any more example of language that helps to create a tone of politeness and respect. It is important to note that the language used to convey politeness varies in different cultures and languages. This activity identifies language usage that conveys politeness and respect in contemporary British English politeness. This is, of course, not the only correct way to communicate politely. We encourage you to reflect on how politeness and respect might be communicated in your context. Next, select the ‘Answer’ button below.
Answer
George's first email (2):
Dear Dr Lu; I hope this email finds you well. Thank you for letting us know about our Mathematics midterm exam; I am writing to ask about ...; ... and I would like to know if ...; Could you please confirm the exam time at your earliest convenience? This will help me decide whether I need to adjust my travel arrangements; Thank you for your help; Best regards.
Dr Liu's reply (3):
The whole of paragraph 1: Thank you for reaching out, I should have specified the time in the original announcement. I understand the importance of your family matter, and I appreciate you letting me know about your travel plans.
Paragraph 2 offers further help: However, if there are any other complications with your travel arrangements or if you have further concerns, please do let me know.
In the end of the email: Best of luck with your exam preparation, and safe travels.
George's second email (4):
The first sentence recognises Dr Lu's efforts and says thank you: Thank you so much for the quick response and for confirming the exam time. I am relieved to hear it's scheduled for 09:00 AM; that gives me plenty of time before my train.
Paragraph 2 is an explicit expression of gratitude: I really appreciate your understanding regarding my situation.
Before the sign-off, George again expresses thanks: Thanks again for your help!
Dr Lu's second post to the Maths Module Mailing List (5):
In the first line of her announcement, Dr Lu writes I neglected to include the exam time in my earlier announcement. Dr Lu's use of neglected is an indirect apology through which she acknowledges the slight error that she made. It can be helpful to indirect apologies in writing if we make an slight mistake. Another way of expressing this is I have been made aware that there is an error in question 3 of the worksheet. It is of course acceptable to simply apologise directly as well.
In the final line of her announcement, Dr Lu writes If you have any questions or concerns about the exam logistics, please do not hesitate to get in touch. Once again, by explicitly inviting questions, Dr Lu maintains a tone of respectfulness and politeness in her writing.
Question for reflection and discussion: As we noted above, this activity on politeness and respect raises a set of interesting questions in cross-cultural language study. Specifically, politeness and respect are expressed in different ways in different cultures, countries and regions. You may wish to reflect on this and think about your own language and culture, if it is not British English.
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How do you show politeness in your language and culture?
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What words and phrases do you use?
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Are they similar to or different from British English?
This reflection should help you to understand politeness in different languages and cultures.
Activity 3: Identifying useful language 2 - formality and informality in sign-offs
You may have noticed in the fictionary email exchange between Dr Lu and George that above their typed signature, they respectively write, Kind regards and Best regards. This part of an email (or a letter) is called the sign-off. Both these examples of sign-offs are relatively informal and reflect British English norms when writing emails in a university context. There are various other sign-offs which vary in formality. Decide whether the sign-offs below are Formal, Neutral (like Kind regards and Best regards) or Informal. Next, select the ‘Answer’ button below the sign-offs.
- Yours sincerely, [name]
- Best, [name]
- Cheers, [name]
- Very best wishes, [name]
- See you soon, [name]
- Thanks, [name]
- With appreciation, [name]
Answer
Note that these answers reflect politeness norms in a contemporary UK university context. Different cultures, regions and countries have different assumptions about politeness and respect and the uses of language that go along with that.
- Yours sincerely, [name]: Formal - this is best for people you know less well or for people who are senior to you.
- Best, [name]: Informal - reserve this sign-off for fellow students and people you know well.
- Cheers, [name]: Highly Informal - reserve this sign-off for fellow students and people you know well.
- Very best wishes, [name]: Neutral - this positive, upbeat sign-off is probably OK in many contexts.
- See you soon, [name]: Informal - reserve this sign-off for fellow students and people you know well.
- Thanks, [name]: Neutral - this short sign-off is probably OK in many contexts.
- With appreciation, [name]: Formal - this sign-off is likely to be used when expressing thanks for something that has been done or is going to be done.
Language Focus
In addition to the sign-offs which we have looked at above, Dr Lu's and George's emails also include a line before the sign off which is related to the specific content of the email. We can call these pre-sign-off lines. These are, in order:
- Dr Lu's first announcement (1): See you next week.
- George's email (2): Thank you for your help.
- Dr Lu's email to George (3): Best of luck with your exam preparation, and safe travels.
- George's second email (4): Thanks again for your help.
- Dr Lu's second announcement (5): If you have any questions or concerns about the exam logistics, please do not hesitate to get in touch.
In this section we have discussed the practices of writing respectful and polite emails. These pre-sign-off lines contribute to a sense of interest and concern by the writer. You may wish to consider in your own university emails what pre-sign-off lines you could include.
Activity 4: Your turn!
You have read and studied the email correspondence between Dr Lu and George. Imagine you are a student in Dr Lu's class, studying a degree in Mathematics and Physics. Imagine that you have looked over your exam timetable and you noticed that you have a Physics exam starting at 10:00 AM on Thursday, 27th November. This is problematic as the mathematics mid-term will not finish until after 10:00 according to Dr Lu's announcement. Write an email to Dr Lu noting the problem you have identified and asking for advice on what to do next.
Write this email in which you do the following:
- Introduce yourself appropriately and begin the email in a standard way.
- State clearly what the problem is, i.e. there is an overlap between the end of the mathematics mid-term and the start of another physics exam that you have to sit.
- Politely and clearly ask Dr Lu for advice on what to do.
- End the email appropriately.
You may also wish to write a reply from Dr Lu. If you choose to do this, here is a framework for Dr Lu's reply:
- Recognise the importance of the problem
- State that you have contacted the exams office and that there is an error with the Physics exam timing: it starts at 11:00 instead of 10:00.
- Make any other relevant comments and end the email appropriately.
In this section, you practised reading and performing a dialogue (spoken communication) about mathematics, and explored written communication through email correspondence. In the next section, you will focus on longer written mathematical texts.
