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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How can I explain a Zero Knowledge Proof with minimal mathematics

I asked this earlier on how to explain a Zero Knowledge Proof to a layman. but I'm looking for a mathematical analogy that might "enhance" the superpower explanation. In that linked superpower, that ...
4
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0answers
134 views

I have a free summer before university. What should I learn? [closed]

Note: This is a soft question. It may be a bit early to be thinking about this, but I figured I'd ask now and see what responses I get. I'm currently a high school senior, and I quite like pure ...
1
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0answers
89 views

Calculus as a structure in the sense of Model theory

I am not a specialist in Logic (my field is Functional Analysis), so excuse me my ignorance. I suppose there must be texts where Calculus is presented as a structure in the sense of Model theory. I ...
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0answers
52 views

What to do when finding “complex” result by accident?

This happened a couple of times today. While working on some problems, you find unexpected relationship between objects, which are not obvious to find directly. For example today I found out that ...
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7answers
1k views

Can someone provide the formal definition of the tangent line to a curve?

I was recently explaining differentiation from first principles to a colleague and how differentiation can be used to obtain the tangent line to a curve at any point. While doing this, my colleague ...
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3answers
138 views

Is it possible for extremely ingenious but elementary proofs for famous problems to exist?

As Erdős put it, "Mathematics is not ready for such problems." when faced with the great conjecture of Collatz. So is it impossible altogether for simple but ingenious proofs for famous problems ...
31
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3answers
1k views

Non-Euclidean Geometry for Children

I should've asked this question two years ago when my son (at that time, 9 years old) came to me and said: "Dad, today in school our teacher drew a line on a paper and said this is a straight line, it ...
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1answer
55 views

Subtlety in integration by parts

We all know the rule of integration by parts: $$\int a(x)b'(x)dx=a(x)b(x)-\int a'(x)b(x)dx$$ But most calculus textbooks lay it down without proper discussion, since what happens if the product ...
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1answer
88 views

Johann Carl Friedrich Gauss [closed]

I've been asked to give a brief over view of the mathematician Carl Gauss' life, so i should include his birth, family, education, occupation, death, who influenced him, his major contributions and ...
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5answers
350 views

Show $\lim\limits_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log

Taking as our definition of exponentiation repeated multiplication (extended to real exponents by continuity), can we show that the limit $$\lim_{h\to 0}\dfrac{a^h-1}{h}$$ exists, without ...
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1answer
24 views

A “Simple Chain of Regions” and Compactness in the Continuum

Let me just start by saying that I'm basically trying to prove this: How to prove every closed interval in R is compact? Except that I need to do it in a very strange way... I'm teaching an ...
6
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1answer
82 views

Visual questions for 6th graders

I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed ...
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1answer
20 views

Annuities-calculating interest

Janet receives a $ 10,000 life insurance benefit. If she uses the proceeds to buy an n-year annuity immediate, the annual payout will be 1613.36. If a 2n-year annuity due is purchased, the annual ...
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1answer
20 views

Prove some divisibility with deductive way

I have some proves But I want to prove them with deductive not inductive. Here are my proves: 1) $2^{3n} - 1 $ is divisible by 7. 2) $2^n + (-1)^{n+1}$ is divisible by 3. 3) $n^2 + 2$ is not ...
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3answers
28 views

derive binomial distribution for a classroom

I am supposed to give a 20-min presentation to a group of kids aged around 14-15, so they have very limited knowledge about the topic in advance -- at the same time I want the talk to be something ...
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0answers
16 views

In-depth resources

How to learn mathematical topics in-depth? For instance, if you want to learn about Symmetry, you would use Google, you'll get results. Unfortunately, what you'll find is tutorials for beginners. But ...
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1answer
25 views

Linear problem using the simplex algorithm

Maximise $8x_1+4x_2+5x_3$ subject to $x_1+2x_2+x_3\le10\\2x_1-3x_2\le8\\x_1,x_2\ge0;x_3\text{free}$
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0answers
23 views

Find basic solutions using matrices? (and determine which ones are bfs)

I'm trying to find all basic solutions using matrices. Then I'll try to determine which ones are bfs. Minimize $3x_1-x_2$ subject to ...
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2answers
34 views

How to convert linear programming model to standard form?

This might be a stupid question. For that, I'm truly sorry. But I appreciate any help!
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1answer
52 views

find the derivative of (cosx)^(sinx)^x

Solve to find the derivative of the following function: (cosx)(sinx)x Do not simplify the answer.
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2answers
35 views

How to convert mathematical program into an LP?

This might be a stupid question. For that, I'm truly sorry. But I appreciate any help! minimize (max {2x-4, |x|}) subject to -3 <= x <= 6
2
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2answers
98 views

Showing that $\cos (\sqrt x)=e^x-2$ has solution in $(0,1)$

I rephrased this problem as finding a solution of $f(x)=e^x-2-\cos(\sqrt x)$ in $(0,1)$. I apply the intermediate value theorem: $f(0)=e^0-2-\cos0=-2$. The goal is to show $f(1)>0$ (which is true) ...
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0answers
30 views

Function satisfing : $h(x)=f(2x-1)$ with $f'(-1)=0 $ and $f'(2)=-2$ then what is $h'(x) $?

I find in some book this function defined as follow $h(x)=f(2x-1)$ . with $f'(2)=-2 $ and $f'(-1)=0$ , we would like to know $h'(3/2)$ ? In the book take $h'(x)=f'(2x-1)=(f'(2))x+f'(-1)=-2x $ but ...
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0answers
29 views

Powerful pictures or plain proceedings?

After tutoring someone about polynomial manipulation, my mind went back to this equivalence: $(a+b)\times (c+d)=ac+ad+bc+bd$ I realized that, while it can be memorized as it is, it has a very simple ...
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2answers
206 views

What is the right way to define a function?

Most authors define functions this way: Given the sets $A$ and $B$. A relation is a subset of $A\times B$. Then given a relation $R$, we define $Dom_R=\{x|(x,y)\in R\}$ and $Img_R=\{x|(y,x)\in R\}$. ...
3
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3answers
127 views

mathematical rigore for an engineer! [closed]

I recently bought a used copy of "Mathematical Analysis" by Apostol for \$1.0 and "Probability and Measure Theory" by Robert Ash for \$3.0 (well another \$3.99 for shipping)! When I read the first few ...
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1answer
19 views

Slope in algebra I

What is a good project for teaching y=mx + b and having students discover slopes of lines in classroom or on the classroom buildings?
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2answers
196 views

False notion of Limit in Stewart's “Calculus”

In calculus, when we take limits of functions, say $\lim_{x\to a}f(x),$ do we require that $x$ tends to $a$ from within the domain? For example, I would say $\lim_{x\to 0} \sqrt{x}=0$ since I am ...
4
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1answer
95 views

Publish pedagogical results as an undergraduate

As an undergraduate, I became fond of real analysis and complete metric spaces. Regarding completeness (mostly in R), I proved the same theorem in perhaps 10 different ways, using different approaches ...
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0answers
52 views

Undergrad looking for study material/advice for applied mathematics.

I am an undergraduate math student (junior) who is looking to get a masters degree in Applied Math. I like pure math, but I want to use my education to get a great-paying job. Here are a few questions ...
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1answer
49 views

Can we find $ a + b $? [closed]

How can we find the value of $ a + b $ in the following question? a & b are integers. Question: If $ a^{2} \times b^{3} = 216 $, find $ a + b $.
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2answers
109 views

What (previously and currently unsolved) problems motivate the study/development of analysis?

As I had ever know there are at least two (previously unsolved) problems motivate the study/development of abstract algebra: (1) the ancient Greeks' three problems in compass-and-straightedge ...
2
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0answers
38 views

Flat Tax vs. National Income Tax Average

If the national income tax average under a progressive system is 30%, will tax revenue change if the progressive system is changed to a flat tax also at 30%? In other words, if the mean average ...
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0answers
27 views

in nested brackets which would be outermost bracket

i wanna know that in an algebric equation of 2 level nested brackets which format of nested bracket is correct {4[3+5(3+2)]} [4{3+5(3+2)}] is curly would be outermost or square
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0answers
61 views

What to do with java-based math demos? [closed]

A large volume of mathematical demos were written in Java. Then came security warnings of all sorts. Currently it does not seem viable to direct students to such demos. Do you think an organization ...
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2answers
36 views

problem proof of induction $2\cdot3^0+2\cdot3^1+2\cdot3^2+…+2\cdot3^{n-1}=3^n-1$

I need to prove that $2\cdot3^0+2\cdot3^1+2\cdot3^2+...+2\cdot3^{n-1}=3^n-1,\, n=1,2,3,4...$ I know when n = 1 the left side $=2\cdot3^{1-1}=2$. The right side is $2^n - 1=2^1 - 1 = 1$. Thus the ...
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1answer
53 views

Could someone make the proof into a hinted exercise?

Let $(c_n)$ be a sequence of positive numbers. Could someone make the proof of the inequality $\displaystyle\limsup_{n\to\infty}\sqrt[n]{c_n}\leq\limsup_{n\to\infty}\frac{c_{n+1}}{c_n}$ into a ...
1
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1answer
55 views

For an introductory analysis class, why is it that many people avoid teaching it using sequences?

I am currently taking an honors introduction to analysis course, and it seems to me that to me defining all of the concepts in terms of sequences allows for much cleaner proofs, and the concepts are ...
1
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1answer
63 views

Variation on the “Number of non-bald people in NYC” problem

I have two questions about the following problem, taken from Challenging Problems in Algebra by Posamentier and Salkind: (1) Why is the answer not 1 person? (2) The answer given, without solution, is ...
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1answer
43 views

Subtraction and borrowing for dummies

Regarding the subtraction and borrowing a digit from upper digits, I know how that works for more than one digit numbers. However, I can not figure out for one digit numbers! It is an obvious thing ...
3
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0answers
57 views

On teaching elementary co-ordinate geometry. [closed]

I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's ...
0
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1answer
64 views

Looking for interesting problems to solve numerically

I am giving tomorrow an introduction to Python to undergraduates and have to present how it can be used to solve some mathematical problems. I have been looking for some nice or challenging problems ...
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1answer
14 views

building blocks

My little sister used her alphabet blocks to build a wall 8 blocks long, 8 blocks high, and 2 blocks wide. This took her 8 minutes. At this rate how long would it take her to build a wall 4 blocks ...
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1answer
65 views

Find a polynomial $p$ in $\Bbb P_2$ that spans the kernel of T, and describe the range of T.

Define $ T: \Bbb P_2 \to \Bbb R^2 $ by $ T(p) = \begin{bmatrix} p(0) \\ p(1) \end{bmatrix} $. For instance, if $ p(t) = 3 + 5t + 7t^2 $, then $ T(p) = \begin{bmatrix} 3 \\ 15 \end{bmatrix} $. Find a ...
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3answers
26 views

Cannot understand this property about column space

Here is a property given to me in my textbook. $ ColA=\Bbb R^m \text{ if and only if the equation } Ax=b \text{ has a solution for every } b \text{ in } \Bbb R^m $. What does it mean by every $b$?
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1answer
70 views

How does a $2$ horizontal row ABACUS work? Till what number can u Add & Subtract…can it be used for Multiplication & Division? SEE DETAILS

For design class (grade $10$), we have to make a Learning Product/Tool. I decided to make an abacus with 2 horizontal rows (it s easier) like the one in this picture: The abacus isn't for very ...
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1answer
126 views

how to find the distance between two points in the polar coordinate system?

Help me, please! how to find the distance between two points $ A( x_1,y_1 )$ $ B( x_2,y_2 )$ in the polar coordinate system?
4
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2answers
136 views

Good example showing why limits must exist in limit product rule

I'm looking for a way to show my calc 1 students not to use the limit laws without knowing that the individual limits exists. I could use $$\lim_{x\to 0} x^{2} \sin(1/x),$$ but by doing it wrong, one ...
3
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1answer
58 views

Being ready to tackle the math courses in my CS program

Here's my (long) story cut short. I was awful at math in high school. I did 4 years in the service and now I'm going to start college in just a few weeks. I am really nervous because I will have 5 ...
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52 views

Circular Logic and Continuity

So, I was doing a Calculus problem a few minutes ago and just recalled something that my real analysis professor said during a lecture years ago... To provide context, take the function $f$ defined ...