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
269
votes
9answers
294k views
Is this Batman equation for real?
HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
73
votes
8answers
2k 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 ...
62
votes
14answers
4k 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 ...
62
votes
6answers
2k views
Find a real function, $f$, s.t. $f(f(x)) = -x$.
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:
Find a function $f: ...
56
votes
10answers
4k 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 ...
54
votes
11answers
3k 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 ...
47
votes
14answers
3k 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. ...
46
votes
17answers
2k views
Proving the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ 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 ...
40
votes
4answers
1k 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 ...
39
votes
4answers
2k 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,
...
38
votes
7answers
4k 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 ...
38
votes
12answers
2k 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 ...
36
votes
13answers
12k 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 ...
35
votes
4answers
3k 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 ...
31
votes
11answers
4k 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 ...
30
votes
1answer
548 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?
29
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 ...
29
votes
4answers
771 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 ...
28
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 ...
27
votes
6answers
571 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
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
26
votes
8answers
2k 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.
$x/0$ is Impossible ( ...
26
votes
2answers
883 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 -
$$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$
$$ 3 \cdot ...
25
votes
22answers
3k views
“Negative” versus “Minus”
As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and
not "minus $0.8$" to denote $-0.8$?
The so called "textbook answer" regarding this question reads:
...
24
votes
7answers
2k views
Do odd imaginary numbers exist?
Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
23
votes
7answers
14k views
Is there a general formula for solving 4th degree equations?
There is a general formula for solving quadratic equations, namely the Quadratic Formula.
For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each ...
23
votes
0answers
831 views
$4494410$ and friends
$4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
22
votes
9answers
1k views
Is this way of teaching how to solve equations dangerous somehow?
Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline.
There I learned a new way of solving equations, which is related to the principle that states that any ...
22
votes
4answers
448 views
Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$
Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical
$$
\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}
...
21
votes
9answers
2k views
Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
Prove that for all integers $n$, $n \geq 1$,
$$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$
How would I go about proving this?
21
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 ...
21
votes
7answers
823 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 ...
20
votes
8answers
1k 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 ...
20
votes
2answers
999 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 ...
19
votes
9answers
2k 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 ...
19
votes
3answers
485 views
Why everytime the final number comes the same?
I have come across an interesting puzzle.
Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with
$\text{Number}_{new} = xy/(x + y)$
OR
$\text{Number}_{new}= x + y + xy$
...
19
votes
3answers
639 views
A problem about floor function
Here's the problem
Suppose that
$$ a, b \in \mathbb {R^+},\qquad 0 < a + b < 1 $$
Prove or disprove that
$$ \exists n \in \mathbb{Z^+}: \left\{na\right\} + \left\{nb\right\} \ge 1$$
...
18
votes
6answers
1k views
Helping my daughter with her homework: solving an algebra word problem.
Three bags of apples and two bags of oranges weigh $32$ pounds.
Four bags of apples and three bags of oranges weigh $44$ pounds.
All bags of apples weigh the same. All bags of oranges weigh the ...
18
votes
3answers
674 views
I want to get good at math, any good book suggestions?
I have never been good at math. In my senior year I failed algebra 2 in the first semester because the teacher would refuse to show me how to solve problems saying that it's my fault for not paying ...
18
votes
9answers
1k views
$i^2$ why is it $-1$ when you can show it is $1$?
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 ...
18
votes
6answers
408 views
Solving $\sqrt{x+5} = x - 1$
I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it:
$$\sqrt{x+5} = x - 1$$
So I used this logic:
$$
\begin{align}
...
18
votes
1answer
532 views
Is this algebraic identity obvious?
If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then
$$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$
This identity follows from a probability calculation ...
17
votes
9answers
733 views
Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$
Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one ...
17
votes
1answer
145 views
To prove that $2^{3n}+2^n +1$ is not a perfect square.
Question: Prove that $2^{3n} + 2^n + 1$ cannot be a perfect square for any natural $n$.
I attempted this question and failed in two different ways.
1) I considered a polynomial $p(x) = x^3+ x + 1 - ...
17
votes
1answer
763 views
Averaging 2 roots of a cubic polynomial
consdier a cubic polynomial, p(x)=k(x-a)(x-b)(x-c) where k is some constant and a,b,c its 3 roots (not necessarily distinct, not necessarily real). It is very simple to show that if you average two ...
16
votes
10answers
988 views
Find the integer closest to $\ln(2013)$
I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck.
I tried to turn $\ln(2013)$ into ...
16
votes
16answers
2k views
Interesting calculus problems of medium difficulty?
I would like this to be a community wiki. But, I don't know how to do that here.
I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus ...
16
votes
4answers
639 views
Given $f(f(x))$ can we find $f(x)$?
Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
16
votes
3answers
545 views
How to answer a student objection to the use of “of” in pronouncing f(x)?
Once upon a time in elementary school, a student learned how to translate certain English words into math. For example, 'and' usually means 'plus' such as "If John has 3 oranges AND 5 apples, how ...
16
votes
5answers
563 views
How does one actually show from associativity that one can drop parentheses?
I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?
...
16
votes
6answers
428 views
$\log_9 71$ or $\log_8 61$
I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
