Questions tagged [real-numbers]

For questions on teaching and introducing real numbers and properties of real numbers.

Filter by
Sorted by
Tagged with
6
votes
3answers
302 views

Cardinal vs. ordinal: When learned? When needed?

Is the distinction between cardinal numbers and ordinal numbers taught as part of mathematics (as opposed to part of learning the language distinction between "one" and "first") in pre-college or ...
13
votes
8answers
3k views

Should high school teachers say “real numbers” before teaching complex numbers?

Before complex numbers are introduced in senior high school courses, should we emphasise that solutions (e.g. to quadratic equations) are real solutions? If we do, then when non-real numbers finally ...
20
votes
14answers
7k views

Why is it possible to teach real numbers before even rigorously defining them?

In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach ...
20
votes
5answers
841 views

Inability to work with an arbitrary mathematical object

This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to ...
5
votes
2answers
316 views

How to teach real analysis?

I am recently going to make a series of videos about real analysis and measure theory. I wonder if anyone can give me some suggestions on how to arrange the material of the course. Should I introduce ...
7
votes
4answers
485 views

What number is the sum of two roots

This is one of these questions that students ask and for which I have never found an answer that students would accept as convincing. Here is an instance: Student: 2•3 is a number, namely 6. 2+3 is a ...
8
votes
1answer
254 views

Why many people believe that: $\displaystyle c>0\implies \frac{1}{c}<0$?

I came across many people who believe the below false implication. I don't know why people believe it true in high school and middle school and also students in university level. Really I would like ...
6
votes
3answers
141 views

Specific Intervention(s) for Middle School 'Place Value' confusion

I'm working with a middle school student (grade 8) who recently displayed a misunderstanding of place value in decimal numbers. The student believes, for example, that $0.125$ is bigger than $0.12$ in ...
8
votes
3answers
2k views

How to practically teach surds?

In teaching Middle School students (often around year 8 or 9), the topic of surds comes up here (I have to teach this topic) - and is often met with derision on commencement of the topic and during ...
14
votes
2answers
623 views

How the real numbers are taught?

I'm interested to know how the real numbers are introduced at beginner level in different countries. In my (old) experience of teacher in Italy there was some well defined steps: 1) Introduce the ...
17
votes
6answers
684 views

What is a number?

In a set theoretic point of view all mathematical objects are sets. We "call" some of them as numbers (e.g. sets in $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $Ord$, $Card$) but what is ...
22
votes
10answers
2k views

Redundant zeros

How to convince a middle school student that $0.50=0.5=0.500=\cdots$? I used the fact that $0.50=\frac{5}{10}+\frac{0}{100}=\frac{5}{10}=0.5$ but that far from intuitive. Then I tried to explain ...
12
votes
7answers
361 views

What different ways do people use to show students that $\mathbb{R}$ is uncountable?

In particular, if you use Cantor's diagonalization argument, do you ignore the repeating decimal annoyance? Or prove that it's not a problem? Is there another clean way that gives students intuition ...
25
votes
6answers
1k views

Good definition for introducing real numbers?

In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are. What are the advantages or disadvantages in the different "...