Linked Questions

961
votes
20answers
95k views

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
53
votes
24answers
16k views

“Negative” versus “Minus”

As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and not "minus $0.8$" to denote $-0.8$? The so called "textbook answer" regarding this question reads: ...
46
votes
9answers
5k views

Is $|1-i|$ larger than $1$?

I am confused about complex numbers. Does $1-i$ lie outside the unit circle? How do I show that this is larger than $1$ in absolute value?
15
votes
16answers
12k views

Abusing mathematical notation, are these examples of abuse?

I have often seen notation like this: Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by $$f(x,y)=x^2+83xy+y^7$$ How does this make any sense? If the domain is $\mathbb{R}^2$ then $f$ should be ...
14
votes
8answers
2k views

Strangest Notation? [closed]

While this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?
21
votes
3answers
4k views

$-1 = 0$ by integration by parts of $\tan(x)$

I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation. A friend of mine forgot that such a primitive could easily be ...
8
votes
3answers
438 views

Interpretation of “not equal” notation

This will be a short question. Let $x$, $y$, $z$ be three elements from any set. Is the following: $$x \ne y \ne z \tag{1}$$ Equivalent to: $$x \ne y, ~ ~ y \ne z, ~ ~ z \ne x \tag{2}$$ Or simply: ...
11
votes
2answers
1k views

Notation regarding different derivatives

I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). For example, if $f:\mathbb R\to\mathbb R,x\mapsto f(...
3
votes
4answers
331 views

Notation for the derivative of a function: $f'$ or $f'(x)\;$?

The derivative of a function is often defined as $f'$ and $f'(x)$. So which one is it? $f'(x)$ is the output of the function $f'$, so why do I see people using $f'$ and $f'(x)$ interchangeably to ...
2
votes
4answers
244 views

$A=\{A,\emptyset\}$ and axiom of regularity

The axiom of regularity says: (R) $\forall x[x\not=\emptyset\to\exists y(y\in x\land x\cap y=\emptyset)]$. From (R) it follows that there is no infinite membership chain (imc). Consider this set: $...
3
votes
2answers
1k views

Which rule is applied to define the operator precedence for factorial

Please apologize the question, I struggled with finding a good formulation in the first place: Looking at $\binom{2n}{k}$ it is very clear that for n,k integer and n>k we can solve it by calculating: ...
1
vote
2answers
152 views

Proof verification: prove $A\subseteq B$ if and only if $A\cap B=A$.

Can someone verify whether my proof is logically correct? :) Proof: Assume $A\subseteq B$. Then for every element that belongs in A, such element also belongs in B. Then $A\cap B \subset A$. If $x \...
3
votes
1answer
180 views

Does $f^{-1}(Y)$ make sense if $Y$ is “bigger” than $X$

My textbook asks me to decide whether or not this expression is true: Given the function $f: X \to Y$ with $B_1 \subseteq Y $. $ f^{-1}(Y $ \ $ B_1) = X $ \ $f^{-1}(B_1) $ I was confused because ...
1
vote
3answers
150 views

How does cardinality map a finite set to $(\{0\} \cup \mathbb{N})$?

In reading Kevin Houston's "How to Think Like a Mathematician", there's a line stating the following: Let $X$ be the set of finite sets. Then the cardinality of a set is a function on $X$, that is $|....
2
votes
2answers
67 views

About the additive property of little-o(h)

So I encountered this definition in Salas Hille Etgen's One and Several Variables Calculus: (This definition is for single variable case) Definition: Let $g:\Bbb R \to \Bbb R$ be a function defined ...

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