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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321
votes
10answers
346k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
215
votes
32answers
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} ...
104
votes
15answers
10k 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 ...
101
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 ...
100
votes
14answers
7k 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 ...
80
votes
11answers
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: $$ ...
74
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. ...
67
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 ...
65
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
4k 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 ...
56
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: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
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. ...
50
votes
4answers
7k 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 ...
49
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
12answers
3k 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 ...
44
votes
7answers
11k 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 ...
44
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 ...
42
votes
16answers
40k 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 ...
41
votes
15answers
10k 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 ...
41
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 ...
38
votes
4answers
941 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 ...
37
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 ...
36
votes
25answers
6k 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: ...
35
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 $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
35
votes
1answer
545 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 ...
34
votes
14answers
6k 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 ...
32
votes
19answers
17k 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?
32
votes
7answers
43k 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 ...
32
votes
4answers
2k 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
633 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
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 ...
30
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
4answers
6k 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
9answers
2k 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 ...
27
votes
12answers
3k 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. ...
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 ...
26
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 ...
26
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 ...
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 ...
25
votes
8answers
2k views

Is there a name for this strange solution to a quadratic equation involving a square root?

Here's an elementary question on solving the following quadratic equation (well, it's not a quadratic until the square root is eliminated): $$\sqrt{x+5} + 1 = x$$ Upon solving the above equation ...
25
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!
25
votes
2answers
5k views

Cannabis Equation

How can an equation for the following curve be derived? $$r=(1+0.9 \cos(8 \theta)) (1+0.1 \cos(24 \theta)) (0.9+0.1 \cos(200 \theta)) (1+\sin(\theta))$$ (From WolframAlpha)
25
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 ...
25
votes
2answers
1k 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 ...
24
votes
10answers
2k 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 ...