Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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327
votes
10answers
352k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
225
votes
33answers
31k views

Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} ...
118
votes
16answers
11k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
110
votes
15answers
8k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
107
votes
9answers
4k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
83
votes
12answers
10k views

How to convince a math teacher of this simple and obvious fact?

I have in my presence a mathematics teacher, who asserts that $$ \frac{a}{b} = \frac{c}{d} $$ Implies: $$ a = c, \space b=d $$ She has been shown in multiple ways why this is not true: $$ ...
77
votes
1answer
3k views

$4494410$ and friends

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
68
votes
14answers
5k views

Why would I want to multiply two polynomials?

I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together? I flipped through some algebra books and ...
66
votes
17answers
4k views

Proving the identity $\sum\limits_{k=1}^n {k^3} = {\Large(}\sum\limits_{k=1}^n k{\Large)}^2$ without induction

I recently proved that $$ \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2 $$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
62
votes
13answers
5k views

What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.

Try to solve this puzzle: The first expedition to Mars found only the ruins of a civilization. From the artifacts and pictures, the explorers deduced that the creatures who produced this ...
59
votes
15answers
4k views

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
56
votes
21answers
18k views

Is $.999999999… = 1$?

I'm told by smart people that $0.999999999... = 1$ and I believe them, but is there a proof that explains why this is?
54
votes
14answers
4k views

What is $x^y$? How to understand it?

$x+y=z$ I have a pen. He has a pen. Total is two pen. This is plus. $x-y=z$ I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus. $x\cdot y=z$ I have two pens. ...
53
votes
4answers
8k views

Value of $\sum\limits_n x^n$

Why is $\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n$ equal $1/(1-0.7) = 10/3$ ? Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some ...
51
votes
4answers
3k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
45
votes
10answers
3k views

Division by $0$

I thought it was elementary to me, but I started to do some exercises and came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions. $\dfrac {x}{0}$ is ...
45
votes
7answers
12k views

Infinity = -1 paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...
45
votes
4answers
2k views

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
45
votes
12answers
4k views

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
43
votes
15answers
12k views

What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
42
votes
16answers
45k views

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We are factoring polynomials and we both had the same thought at the same time: when are we going to use this? I feel ...
42
votes
10answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
37
votes
4answers
1k views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
36
votes
8answers
50k views

Is there a general formula for solving 4th degree equations?

There is a general formula for solving quadratic equations, namely the Quadratic Formula. For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each ...
36
votes
6answers
1k views

Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
36
votes
1answer
595 views

What is that curve that appears when I use $\ln$ on Pascal's triangle?

I made a little program that generates Pascal triangles as images : I first tried it associating to each pixel a color whose intensity was proportional to the number in the Pascal triangle The ...
35
votes
25answers
7k 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: ...
34
votes
14answers
8k views

Why can't you square both sides of an equation?

Why can't you square both sides of an equation? I've been asked this many times and can never quite give a good, clear, concise answer (for algebra students) in plain language. I just searched the ...
33
votes
7answers
1k views

Is there any geometric way to characterize $e$?

Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing ...
33
votes
2answers
1k views

Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$

Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$ I have tried to fiddle with it as follows: $$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$ $$ 3 \cdot ...
32
votes
4answers
3k views

Does there exist rational $a,b,c$, such that $\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

Let $w = \sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}$. How to prove that there are no triples $(a,b,c)$, such that $a,b,c \in \mathbb{Q}$; $a \leqslant b \leqslant c$; $(a,b,c)\ne (1,2,4)$; $w = ...
32
votes
1answer
648 views

Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
31
votes
15answers
3k views

Which one is bigger: $\;35{,}043 × 25{,}430\,$ or $\,35{,}430 × 25{,}043\;$?

Which of the two quantities is greater? Quantity A: $\;\;35{,}043 × 25{,}430$ Quantity B: $\;\;35{,}430 × 25{,}043$ What is the best and quickest way to get the answer without using ...
30
votes
6answers
2k views

What is the larger of the two numbers?

What is the larger of the two numbers? $$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$ I solved this, and I think that is an interesting elementary problem. I want different points ...
30
votes
7answers
1k views

Pi Estimation using Integers

I ran across this problem in a high school math competition: "You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...
29
votes
2answers
1k views

A pencil approach to find $\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}$

What is the fastest, paper-pencil method of finding $$\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}?$$ This is actually a quantitative aptitude problem, and hence the ...
28
votes
9answers
3k views

Is this way of teaching how to solve equations dangerous somehow?

Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline. There I learned a new way of solving equations, which is related to the principle that states that any ...
28
votes
12answers
4k views

Algebra: What allows us to do the same thing to both sides of an equation?

I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. ...
28
votes
4answers
7k views

Are We Teaching Pre-Calc Wrong?

It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial. When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that ...
27
votes
6answers
1k views

Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $

How could we prove that this inequality holds $$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$ where $n \in \mathbb{N}$, I think we could use the AM-GM inequality ...
27
votes
11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
27
votes
6answers
2k views

$-1$ is not $ 1$, so where is the mistake?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
27
votes
7answers
1k views

Why is $\frac{1}{\frac{1}{0}}$ undefined?

Is the fraction $$\frac{1}{\frac{1}{0}}$$ undefined? I know that division by zero is usually prohibited, but since dividing a number by a fraction yields the same result as multiplying the number ...
27
votes
2answers
698 views

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
26
votes
7answers
3k views

Do odd imaginary numbers exist?

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
26
votes
2answers
1k views

Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?

I'm currently studying exponential functions, however I've come across this before and it was just a law that I was told to accept without much proof or explanation, could someone explain the ...
26
votes
2answers
2k views

Factorize $(x+1)(x+2)(x+3)(x+6)- 3x^2$

I'm preparing for an exam and was solving a few sample questions when I got this question - Factorize : $$(x+1)(x+2)(x+3)(x+6)- 3x^2$$ I don't really know where to start, but I expanded everything to ...
26
votes
10answers
3k views

$i^2$ why is it $-1$ when you can show it is $1$? [duplicate]

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with ...
26
votes
5answers
1k views
25
votes
12answers
2k views

Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$

Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)? $$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$ Using the calculator I can see that the first one ...