For questions related to the teaching and learning of mathematics. Note that Mathematics Educators StackExchange may be a better home for narrowly scoped questions on specific issues in mathematics education.

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24
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12answers
3k views

How do I make a student understand contradiction?

We were trying to prove that if $3p^2=q^2$ for nonnegative integers $p$ and $q$, then $3$ divides both $p$ and $q$. I finished writing the solution (using Euclid's lemma) when a student asked me ...
-15
votes
1answer
135 views

Solve $|z|+z=1+i$ [closed]

Solve the equation $|Z|+ Z=1+i$ please help me describe me how to solve it.
4
votes
2answers
90 views

I need a good reading list of the Masterworks of mathematics. [closed]

I'm interested in a solid list of the greatest textbooks in the major* fields as written by the "Masters". The problem I have is that after many years of a high-level, rigorous education in ...
0
votes
0answers
33 views

Are some Types of Proportional Relationships more useful than others?

Responses from a previous math stack exchange question (see link below) have lead me to this related question. Link to previous question Based on these responses, it seems that in just about any ...
-1
votes
1answer
45 views

a local government is planning to build new roads that will divide a forest into several fragments [closed]

a local government is planning to build new roads that will divide a forest into several fragments the roads can follow several different designs each of which design will minimize the edge effect on ...
2
votes
0answers
50 views

Is the precise definition of a limit worth covering in Calc 1?

I'm teaching calc 1 this semester and am trying to decide if I should cover the precise definition of a limit. My first impression was that I should. It helped me to gain a deeper understanding of ...
4
votes
4answers
129 views

Is the anti-derivative another way of saying the derivative?

This question is being posted so that I can understand a definition and concept. The definition - F(x) is an anti-derivative of f(x) if F'(x) = f(x) Therefore, $$F(x) = x^2$$ is an ...
0
votes
0answers
26 views

One Million Proficiency Efficiency Calculations

I have been playing the "Million" game on my apple device after discovering it through the trending applications in the search function. After beating the game I began to wonder to myself if there was ...
12
votes
9answers
803 views

Proof that doctors could relate to [closed]

I am supposed to present a mathematical proof to a lecture hall full of doctors in order to show them how mathematicians think. I'm having trouble picking a proof that will be easily followed by ...
1
vote
3answers
112 views

How to explain Clairaut-Schwartz's Theorem, $f_{xy}=f_{yx}$?

I am looking for a non-technical explanation of Clairaut's theorem which states that the mixed derivative of smooth functions are equal. A geometrical, graphical, or demo that explains the theorem and ...
2
votes
1answer
48 views

A different method for solving 2nd order ODEs

For constants $a,b$ there are many ways to find the solutions to $$y^{\prime\prime} + (a+b)y^\prime + aby = \phi(x). $$ Perhaps the most popular is to first solve the homogeneous case when $\phi(x) ...
0
votes
2answers
106 views

How complex can a Differential Equation be? [closed]

When I was first being introduced to differential equations, my teacher began by asking the other student and me to write down the most complex differential equation we could think of and gave us 10 ...
0
votes
0answers
14 views

Convention as to differentiability defined in terms of local extension?

Let $X \subset \mathbb{R}^{n}$. In differential topology or allied areas, people may define that a function $f: X \to \mathbb{R}^{m}$ is of class $C^{r}$ if and only if $f$ can be locally extended to ...
0
votes
1answer
105 views

calculate minimum of max value of set of numbers

Suppose to have a set of numbers: $$S:=\{ 1,2,3,9,10,56,58,60\}$$ How can i group this number like this: group 1:$\{1,2,3,9,10\}$ group 2:$\{56,58,60\}$ and then take "56" as minimum value of ...
-4
votes
1answer
131 views

Why is Distribution Prioritized Over Combining?

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + ...
0
votes
0answers
22 views

Identity and equality [duplicate]

What is difference between identity and equality in math? when to use identity and when equality? Most of math identity also defined as equality how to distinguish between identity and equality. in ...
0
votes
2answers
132 views

How do mathematicians find the underlying idea?

While reading through the lecture notes here (http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf , page 22, last paragraph), I came across the following " Thus there must be some ...
2
votes
2answers
61 views

Variable in Math

What is formal definition of "variable" ? I cannot able to understand variable because some times it is varying or some time used for approaching. Arbitrary constant also confusing me. In Multi ...
2
votes
3answers
54 views

Intuitive explanation of second derivative test for functions of two variables.

I will be teaching multivariable calculus again this semester, and I am not so happy with the explanation I have for the second derivatives test for functions of two variables. QUESTION: What is a ...
10
votes
9answers
552 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
0
votes
2answers
31 views

Define temperature by clustering with math operators

I can´t figure out how to cluster the temperature for the weather in 3 optimal cases: hot, mild, cold My data contains: air temperature(the average daily value), max air temperature(highest daily ...
2
votes
1answer
233 views

What are the (most) essential/interesting sections of The Princeton Companion to Mathematics a junior math student better have a read?

I am studying The Princeton Companion to Mathematics, but it is heavy, the sections have different difficulties, and time is limited for me. Assuming that I have only some knowledge of undergraduate ...
0
votes
0answers
38 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
0
votes
0answers
62 views

Good math websites for elementary geometry and algebra?

I would appreciate it if someone could suggest good websites where we can find English written geometry and algebra exercises for beginners, say junior and senior high school students. I have ...
0
votes
1answer
38 views

Program languages recommended for complexity theory

I am an undergraduate studying mathematics and one of my interests include complexity and computability theory. I have no experience in programming. The computability theory books I looked into didn't ...
4
votes
1answer
87 views

“A real polynomial of degree $n$ cannot have more than $n-1$ local extrema”: a proof without derivatives?

I am looking for a proof that does not use derivatives of the elementary theorem given in the title: Theorem: A polynomial $p:\mathbb{R}\to\mathbb{R}$ of degree $n$ cannot have more than $n-1$ local ...
2
votes
4answers
81 views

For what values of $p$ does this series converge?

This is a question we asked on a second semester calculus test. For what values of $p$ does this series converge? $$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$ I believe that it actually can be shown ...
6
votes
2answers
334 views

Why don't we start studying calculus via series instead of the calculus on finite expressions?

It seems that historically, there were two trends on the idea of integration: Newton's work which depended on infinite series. Leibniz work which depended on the dream of integration of elementary ...
2
votes
3answers
67 views

Are there any significant differences between studying functional analysis from a normed space perspective versus a metric space perspective?

Does it matter if functional analysis was introduced from a normed space versus a metric space formulation? Are all major theorems from functional analysis (such as Banach contraction mapping, Hahn ...
2
votes
1answer
69 views

How To Prove Irrational Square Roots and Inequalities In Courant's Calculus Book? [closed]

Here's the proofs questions in a screenshot The first questions ask about proving the irrationality of non perfect squares. Numbers 3,5, and 6 ask for inequality proofs. I find it daunting that the ...
1
vote
0answers
77 views

Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
4
votes
7answers
160 views

How do i convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

I would like to convince my student in high school level using a simple mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{z}$ which have only one integer solution that is $x=4$ . ...
0
votes
0answers
29 views

Hoeffding's inequality, number of samples required

I was deriving the number of samples required to qualify certain confidence bounds, at the end I am getting slightly different results from what is stated in my lecture notes. can anyone explain why ...
0
votes
1answer
46 views

Projecting functions onto planes

I understand the concept of projecting vectors onto the span of a vector but I'm having trouble projecting functions i.e How would I project the function cos(x) onto the vector that spans (1,1)?
2
votes
1answer
73 views

Alternative proof that base angles of an isosceles triangle are equal

The "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows: Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the mid ...
5
votes
8answers
1k views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...
1
vote
1answer
104 views

New idea to solve $\int x^n e^x dx$

I have this problem $$\int x^n e^x dx= x^ne^x -nx^{n-1}e^x +n(n-1)x^{n-2}e^x- \cdots+(-1)^nn!e^x $$ my try was to use integration by part . $$I_{n}=\int x^n e^x \, dx=e^x x^n -\int (nx^{n-1})e^x \, ...
4
votes
1answer
206 views

Books on complex analysis (Ahlfors, Conway and Lang)

To make my question slightly different from others, I would like to know how would you rate on the complex analysis books by Ahlfors, Conway and Lang? I had a basic course on complex analysis during ...
3
votes
7answers
301 views

Alternate ways to prove that $4$ divides $5^n-1$

I was working for various method to solve this: For all $n\in \mathbb N$: $4\;\mid\;(5^{n}-1)$. My try was: 1st: $$n=1 \to 4|5^1-1\\n \geq 2 \to 5^n=25,125,625,3125,...\\ n\geq 2 \to ...
2
votes
1answer
102 views

Difficult to read about different subjects simultaneously, should I leave one for now? [closed]

I learn math by reading books. Usually I read 3 books (about 3 different subjects) simultaneously and switch focus every couple of days. The books i'm studying right now are Rudin's functional ...
3
votes
1answer
67 views

Factoring bivariate quadratics with real coefficients (for high school students).

I was tutoring a Year 10 student last night (he's learning about quadratics). Unfortunately, we ran into a class of problems that I couldn't explain how to solve (beyond simply guessing and checking), ...
7
votes
5answers
435 views

New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...
0
votes
0answers
22 views

Proper name and access of the constituents of an equation.

In any math problems for example: ... B = (1 + B) A = (A + B) ... How can I definitely define a variable that points to the ...
3
votes
0answers
91 views

Opportunities to learn algebraic geometry outside of PhD education [closed]

After a lot of deliberation, I decided to go to professional school instead of pursue academic mathematics. I don't have research aspirations, but I do have an obsession with "getting" esoteric ...
-1
votes
1answer
31 views

Any idea how to approach this problem

A rectangular meadow will have a fence around it. The long side is $130$ m longer than the short side. The sides lengths can be written $x$ and $x+ 130$. Write a simplified expression for 1) ...
1
vote
1answer
43 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
15
votes
5answers
2k views

When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right ...
0
votes
0answers
84 views

Demonstrative geometry around the world and its significance.

This is not exactly a mathematical question. I am from Pakistan; and over here students are taught a subject 'demonstrative geometry' (as a part of mathematics) from secondary level education. ...
1
vote
1answer
46 views

How to calculate percentile?

I am reading a document issued by UK Gov about minimum salary for work visa, it is using word percentile, which I don't understand, Could anyone explain what would be the 50th or 75th percentile IF, ...
5
votes
3answers
816 views

Which background is more suitable to study “Cryptography” [closed]

I am a student of Pure Mathematics and also interested in programming .I have learnt C++,SAGE . Recently I have started learning "Cryptography" .But there are many definitions involved here like ...