# Questions tagged [definitions]

For questions related to the issue of concepts of definitions.

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### Are there disadvantages to teaching complex numbers as purely geometrical objects?

Complex numbers are, or at least were to me, generally introduced like this: There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare ...
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### Online Definition/Theorem Statement Quizzes

This fall I'll be teaching a standard "Introduction to Proofs" course and administration is advising us to be ready at any time to switch from in-person to fully online (or anything in ...
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### Definitions of factors and terms

I have come across this question in a textbook How many factors are there in the term $5ab(x+y)$? State what they are It is being praised because it encourages thinking, which it does. However, I'm ...
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### Why is a translated exponential function considered an exponential function?

I am tutoring a student preparing to take Calculus 1 at a university. This student hasn't taken precalculus for a year, so I have been drilling him on definitions, rules, and theorems from a college ...
247 views

### How to define “axes with the same scale” in Secondary/High School?

It's easy to recognize visually when an orthogonal coordinate system has its axes in the same scale. See, for instance, the following image. But I'm trying to write down a precise definition of it. ...
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### Why is continuity defined as a local property?

The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the ...
389 views

### Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
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### What is the intuition behind the limit superior?

I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
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### Rigorously defining the concept of an angle for high school students

Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected. Google search provided me with many definition that are too technical or too vague IMO. ...
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### Can students tell the difference between the “definition if” and the “theorem if”?

The word "if" is used in two meanings in mathematics: Definition. A topological space is compact if every open cover has a finite subcover. Theorem. A topological space is compact if it is ...
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### What are the questions one should ask of oneself in trying to understand definitions and lemmas? [closed]

Definition (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then ...
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### how is volume different than capacity

I have in mind that volume is the amount of room or space a 3-d object takes up - its "outsideness" and that capacity with the amount of room or space a 3-d object can hold. Then I start thinking ...
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### What is a good way to explain the slightly different kinds of continuity?

What is a good way to explain the slightly different kinds of continuity to students? I have in mind these kinds of continuity: A function is continuous at a point. (This also has two sub-kinds: ...
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### Is multiplication by zero clear for and understood by K-3 students?

For K-3 students, perhaps it is not acceptable to introduce multiplication by zero as a property or definition. Instead, the child may think about multiplication as, e.g., repeated addition. Examples ...
I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define $e$ and $\ln$. For example, my ...
For some objects there are alternate definitions, which are "morphism-oriented". To give some examples, there are two definitions of a prime number: $p$ is prime if it is greater than $1$ and has no ...