Skip to main content
35 votes

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

To ask "Is $\frac{dy}{dx}$ a ratio or isn't it?" is like asking "Is $\sqrt 2$ a number or isn't it?" The answer depends on what you mean by "number". $\sqrt 2$ is not an Integer or a Rational number,...
Hawthorne's user avatar
  • 491
27 votes

Basic Geometric intuition, context is undergraduate mathematics

The viewpoint you're groping towards is not crazy, and at least historically you're in excellent company -- e.g., Leibniz had similar ideas when he viewed an integral as a sum of the heights of ...
hmakholm left over Monica's user avatar
26 votes

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

There are many answers here, but the simplest seems to be missing. So here it is: Yes, it is a ratio, for exactly the reason that you said in your question.
Toby Bartels's user avatar
  • 4,804
26 votes
Accepted

What are hyperreal numbers? (Clarifying an already answered question)

The usual construction of the hyperreal numbers is as sequences of real numbers with respect to an equivalence relation. For example, the real number 7 can be represented as a hyperreal number by the ...
Samuel's user avatar
  • 5,550
25 votes

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

I am going to join @Jesse Madnick here, and try to interpret $\frac{dy}{dx}$ as a ratio. The idea is: lets interpret $dx$ and $dy$ as functions on $T\mathbb R^2$, as if they were differential forms. ...
Dávid Kertész's user avatar
25 votes
Accepted

What is the use of hyperreal numbers?

Your point about successive extensions of a basic number system is very well taken. We use the successive extensions $$ \mathbb{N}\hookrightarrow\mathbb{Z}\hookrightarrow\mathbb{Q}\hookrightarrow\...
Mikhail Katz's user avatar
  • 43.3k
21 votes

What are hyperreal numbers? (Clarifying an already answered question)

Unfortunately, there is no "concrete" description of the hyperreals. For instance, there is no way to give a concrete description of any specific infinitesimal: the infinitesimals tend to be "...
Noah Schweber's user avatar
20 votes

Which branch of mathematics rigorously defines infinitesimals?

Your perception is wrong. Non-standard analysis is grounded on Logic and it's as solid as any other field of Mathematics. I suggest that you read Abraham Robinson's Non-standard Analysis.
José Carlos Santos's user avatar
18 votes

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

$dy/dx$ is quite possibly the most versatile piece of notation in mathematics. It can be interpreted as A shorthand for the limit of a quotient: $$ \frac{dy}{dx}=\lim_{\Delta x\to 0}\frac{\Delta ...
Joe's user avatar
  • 19.9k
16 votes
Accepted

Which branch of mathematics rigorously defines infinitesimals?

Most objections to non-standard analysis seem to be about the use of the axiom of choice in the construction of the field of hyperreals. Non-standard analysis is completely rigorous, but if you're a ...
Patrick Stevens's user avatar
16 votes
Accepted

Nonstandard infinite / hyperfinite sum in IST

Welcome to Math.SE! You did not indicate how much background you have in Internal Set Theory (IST) - but based on what you wrote, it seem like you have a few misconceptions about it. Let us start by ...
Z. A. K.'s user avatar
  • 11.5k
15 votes
Accepted

Basic Geometric intuition, context is undergraduate mathematics

This is less an answer and more of an extended comment. You seem to be struggling with the idea of a point as contrasted with an infinitesimally thickened point, and it sounds to me like you want to ...
ಠ_ಠ's user avatar
  • 10.8k
13 votes

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

The derivate $\frac{dy}{dx}$ is not a ratio, but rather a representation of a ratio within a limit. Similarly, $dx$ is a representation of $\Delta x$ inside a limit with interaction. This interaction ...
Gustav Streicher's user avatar
12 votes
Accepted

Dedekind completion of ordered fields

The existence of additive inverses fails in $\mathbb{S}^*.$ A member of $\mathbb{S}^*$ (a Dedekind cut in $\mathbb{S})$ is a subset A of $\mathbb{S}$ such that $A$ has an upper bound, $A$ has no ...
Mitchell Spector's user avatar
11 votes

How do we interpret and perform integrals using infinitesimals?

$\DeclareMathOperator{\st}{st}$I will explain one way to treat your definite integral in the Internal Set Theory (IST) approach to Nonstandard Analysis (NSA). Before beginning, please be aware this ...
GPhys's user avatar
  • 1,539
10 votes
Accepted

Can unlimited hypernaturals be represented by increasing sequences?

You certainly can't make $(a_n)$ be strictly monotonic, since that would mean that $a_n\geq a_0+n$ for all $n$. So, for instance, if $k_n=\lfloor\sqrt{n}\rfloor$ we cannot choose any such $a_n$, ...
Eric Wofsey's user avatar
10 votes
Accepted

Is $e$ transcendental when working with hyperreal numbers?

This is totally wrong. You cannot express $e$ as $(1 + \frac{1}{H})^H$; rather, $(1 + \frac{1}{H})^H$ will be infinitely close to (but NOT equal to) $e$. After all, $(1+\frac{1}{n})^n$ is never ...
Eric Wofsey's user avatar
10 votes

Cauchy sequence for $0$ in non-standard analysis

In the usual construction of the real numbers from Cauchy sequences, two Cauchy sequences are considered equivalent iff the term-wise difference converges to $0$. In the usual construction of the ...
Arthur's user avatar
  • 200k
9 votes

Basic Geometric intuition, context is undergraduate mathematics

I don't know if your post has much to do with "life advice", but the question of whether there should be an "infinitely small but non-zero width" is something that bears answering. The way math is ...
Ben Grossmann's user avatar
9 votes
Accepted

Hyperreals, other models and 1=0.999....

Let's make our notation more explicit. First, let's briefly recap the standard situation. Decimal representations are really just infinite sums, and in particular $$0.9999...:=\sum_{i\in\mathbb{N}}{9\...
Noah Schweber's user avatar
9 votes
Accepted

Does 3Blue1Brown's series on Calculus : Essence of Calculus approach it via limits or infinitesimals (or both)?

I'll flesh out @MichaelMorrow's comment, with a qualification: the standard modern approach is to describe everything in terms of limits. Historically, calculus grew out of a desire to understand what ...
J.G.'s user avatar
  • 116k
9 votes

Nonstandard infinite / hyperfinite sum in IST

You asked several questions that should really be separate posts, but let me start by answering one. You asked: "If anyone could provide a detailed proof that a sum indexed by an unlimited ...
Mikhail Katz's user avatar
  • 43.3k
8 votes
Accepted

Are there non-standard counterexamples to the Fermat Last Theorem?

I am going to respond to two questions quoted below, which come from this comment. Here "TP" means "transfer principle". I've switched the order of the questions. ... why it does not prove that ...
Carl Mummert's user avatar
  • 81.8k
8 votes

Why Cauchy's definition of infinitesimal is not widely used?

You ask "why Cauchy's definition of infinitesimal, along with his 'basic approach' was superseded?" The answer is that Cantor, Dedekind, Weierstrass and others developed a foundation for analysis to ...
Mikhail Katz's user avatar
  • 43.3k
8 votes
Accepted

Does absolute infinity invoke Cantor's Paradox?

First, it doesn't follow from Russell's Paradox that no set is a member of itself. It follows from the Axiom of Foundation, which is independent of the axioms necessary to prove Russell's Paradox. ...
Robert Shore's user avatar
  • 23.9k

Only top scored, non community-wiki answers of a minimum length are eligible