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
12
votes
19answers
7k views
Does .99999… = 1?
I'm told by smart people that 0.999... = 1 and I believe them but is there a proof that explains why?
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 ...
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 ...
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 ...
15
votes
19answers
7k views
Proof for formula for sum of sequence $1+2+3+\ldots+n$?
Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.
How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? ...
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 ( ...
11
votes
2answers
948 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}} ...
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!
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 ...
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 ...
10
votes
4answers
775 views
Highest power of a prime $p$ dividing $N!$
How does one find the highest power of a prime $p$ that divides $N!$ and other related products?
Related question: How many zeros are there at the end of $N!$?
This is being done to reduce ...
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?
10
votes
5answers
803 views
How to solve $x^3=-1$?
How to solve $x^3=-1$? I got following:
$x^3=-1$
$x=(-1)^{\frac{1}{3}}$
$x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
7
votes
4answers
678 views
What is the term for a factorial type operation, but with summation instead of products?
(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)
I'm aware of Sigma notation, but is there a function/name ...
4
votes
9answers
1k views
Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
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 ...
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 ...
11
votes
3answers
687 views
Simplification of expressions containing radicals
As an example, consider the polynomial $f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2)$ which clearly has a root $x = 1$.
But we can also find the roots using Cardano's method, which leads to
$$x = ...
13
votes
2answers
1k views
Significance of $\displaystyle\sqrt[n]{a^n} $?
There is a formula given in my module:
$$ \sqrt[n]{a^n} = a \text{ if $n$ is odd } $$
$$ \sqrt[n]{a^n} = |a| \text{ if $n$ is even } $$
I don't really understand the differences between them, ...
3
votes
3answers
270 views
Factoring $ac$ to factor $ax^2+bx+c$
I was watching a first-year high-school-algebra student struggle with factoring quadratics last night. Given a quadratic $ax^2+bx+c$ (I'll give you the exact example in a moment), her method — ...
9
votes
1answer
580 views
What is the standard interpretation of order of operations for the basic arithmetic operations?
What is the standard interpretation of the order of operations for an expression involving some combination of grouping symbols, exponentiation, radicals, multiplication, division, addition, and ...
23
votes
7answers
14k views
Is there a general formula for solving 4th degree equations?
There is a general formula for solving quadratic equations, namely the Quadratic Formula.
For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each ...
29
votes
7answers
1k views
Is there any geometric way to characterize $e$?
Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing ...
8
votes
4answers
277 views
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
39
votes
4answers
2k views
Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?
Problem:
Find $x$ in
$$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$
Trick:
$x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so,
$x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and,
...
16
votes
7answers
1k views
Explanation of method for showing that 0 / 0 is undefined
(This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes)
Ignoring limits, I would like to know ...
10
votes
3answers
679 views
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
5
votes
10answers
1k views
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am ...
4
votes
6answers
693 views
Do values attached to integers have implicit parentheses?
Given $5x/30x^2$ I was wondering which is the correct equivalent form.
According to BEDMAS this expression is equivalent to
$5*\cfrac{x}{30}*x^2$
but, intuitively, I believe that it could also look ...
2
votes
1answer
110 views
Summation of natural number set with power of $m$
Who knows about the summation of this series:
$$\sum\limits_{i=1}^{n}i^m $$ where $m$ is constant and $m\in \mathbb{N}$?
thanks
21
votes
6answers
1k views
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
How could we prove that this inequality holds
$$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$
where $n \in \mathbb{N}$, I think we could use the AM-GM inequality ...
7
votes
2answers
937 views
Integration by partial fractions; how and why does it work?
Could someone take me through the steps of decomposing
$$\frac{2x^2+11x}{x^2+11x+30}$$
into partial fractions?
More generally, how does one use partial fractions to compute integrals
...
20
votes
8answers
1k views
Is there a name for this strange solution to a quadratic equation involving a square root?
Here's an elementary question on solving the following quadratic equation (well, it's not a quadratic until the square root is eliminated):
$$\sqrt{x+5} + 1 = x$$
Upon solving the above equation ...
16
votes
5answers
563 views
How does one actually show from associativity that one can drop parentheses?
I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?
...
9
votes
7answers
995 views
Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
Dealing with a physics Problem I get the following equation to solve for $\alpha$
$1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
Putting this in Mathematica gives the result:
$a==2 ...
2
votes
1answer
510 views
Range scaling problem
I have a few ranges which I want to scale but I'm missing the formula (and common sense).
For example I have a scale range from 40 to 100, but I want my data to range from 0 - 100. What formula do I ...
8
votes
5answers
3k views
Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$
Is there any formula for this series?
$$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
6
votes
3answers
123 views
Apparently cannot be solved using logarithms
This equation clearly cannot be solved using logarithms.
$$3 + x = 2 (1.01^x)$$
Now it can be solved using a graphing calculator or a computer and the answer is $x = -1.0202$ and $x=568.2993$.
But ...
4
votes
2answers
307 views
How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?
This sum is difficult. How can I compute it, without using calculus?
$$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$
If someone can explain some technique to do it, I'd appreciate it.
Or advice using ...
19
votes
3answers
484 views
Why everytime the final number comes the same?
I have come across an interesting puzzle.
Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with
$\text{Number}_{new} = xy/(x + y)$
OR
$\text{Number}_{new}= x + y + xy$
...
4
votes
2answers
805 views
Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$
Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} ...
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 ...
11
votes
5answers
2k views
How can I write an equation that matches any sequence?
One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is:
x 0 1 2
y 1 5 9
If ...
2
votes
3answers
373 views
Create polynomial coefficients from its roots
Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients?
I know the Horner scheme and that we can just go backwards receiving those coefficients.
But I'm curious if ...
6
votes
4answers
322 views
How to detect when continued fractions period terminates
I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates?
Let me give you an example:
$\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
11
votes
3answers
475 views
An incorrect method to sum the first $n$ squares which nevertheless works
Start with the identity
$\sum_{i=1}^n i^3 = \left( \sum_{i = 1}^n i \right)^2 = \left(\frac{n(n+1)}{2}\right)^2$.
Differentiate the left-most term with respect to $i$ to get
$\frac{d}{di} ...
8
votes
2answers
556 views
Finding roots of polynomials with rational coefficients
I'm looking for a general approach (or approaches) for finding the roots of polynomials with rational coefficients of higher degrees than $4$. The problem is that I need to find the exact roots and ...
8
votes
2answers
430 views
Why is the even root of a number always positive?
Let $n \in \mathbb N$ be a natural number and $a \in \mathbb R$ be a real number. The $n$-th
root of the number $a$ is defined as follows:
Case I: $n$ is an odd number. In this case the ...
5
votes
4answers
276 views
How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying ...
4
votes
1answer
617 views
Why not write the solutions of a cubic this way?
For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as:
$$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + ...
