Questions tagged [algebra-precalculus]
For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.
46,724
questions
465
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10
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Is this Batman equation for real? [closed]
HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-...
- 5,491
397
votes
33
answers
51k
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”:
$$ \frac{1}{a+b} \mathrel{\...
378
votes
20
answers
46k
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Find five positive integers whose reciprocals sum to $1$
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was ...
- 4,555
373
votes
23
answers
51k
views
Zero to the zero power – is $0^0=1$?
Could someone provide me with a good explanation of why $0^0=1$?
My train of thought:
$$x>0\\
0^x=0^{x-0}=\frac{0^x}{0^0}$$
so
$$0^0=\frac{0^x}{0^x}=\,?$$
Possible answers:
$0^0\cdot0^x=1\cdot0^0$,...
- 3,989
357
votes
31
answers
59k
views
Is it true that $0.999999999\ldots=1$?
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
295
votes
22
answers
48k
views
Why can ALL quadratic equations be solved by the quadratic formula?
In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
- 3,889
223
votes
10
answers
13k
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What does $2^x$ really mean when $x$ is not an integer?
We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
- 4,177
194
votes
14
answers
19k
views
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\\\\
\frac1{\sqrt{-1}} &= \frac1i \\\\
\frac{\sqrt1}{\sqrt{-1}} &...
- 2,143
187
votes
28
answers
19k
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Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction
I recently proved that
$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$
using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation ...
- 5,795
184
votes
20
answers
16k
views
How do people perform mental arithmetic for complicated expressions?
This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky.
The problem on the blackboard is:
$$
\dfrac{10^{2} + 11^{2} + 12^{...
- 6,660
171
votes
15
answers
353k
views
Is there a general formula for solving Quartic (Degree $4$) equations?
There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula:
$$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$
For cubic equations of the ...
- 3,451
162
votes
19
answers
21k
views
What actually is a polynomial?
I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial.
I wasn't in the advanced mathematics class in 8th grade, then in 9th ...
- 3,396
144
votes
40
answers
108k
views
Why is negative times negative = positive?
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like this:
...
- 2,043
141
votes
36
answers
304k
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 1,585
134
votes
17
answers
13k
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Division by zero
I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.
$\dfrac {x}{0}$ is Impossible ( If it's impossible it can't have neither infinite solutions or ...
- 1,577
134
votes
7
answers
96k
views
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=...
129
votes
14
answers
30k
views
Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$
$x,y,z >0$, prove
$$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$
Note:
Often Stack Exchange asked to show some work before answering the ...
- 4,351
118
votes
26
answers
36k
views
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
For example the square of $2$ is $2^2=2 \cdot 2=4 $ .
...
- 1,623
117
votes
15
answers
36k
views
Why rationalize the denominator?
In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there ...
- 5,169
112
votes
5
answers
118k
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 ...
- 1,339
107
votes
13
answers
12k
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 ...
- 1,079
103
votes
9
answers
9k
views
Division by $0$ and its restrictions
Consider the following expression:
$$\frac{1}{2} \div \frac{4}{x}$$
Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$.
But if we rearrange the ...
- 1,115
101
votes
14
answers
115k
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 beginning algebra students) in plain language. I just ...
- 3,345
101
votes
13
answers
12k
views
Can I think of Algebra like this?
This year in Algebra we first got introduced to the concept of equations with variables. Our teacher is doing a great job of teaching us how to do them, except for one thing:
He isn't telling us what ...
- 4,934
99
votes
15
answers
8k
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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:
$$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
- 963
98
votes
1
answer
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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.
$3883544142410_{10}=...
- 373k
97
votes
7
answers
174k
views
96
votes
12
answers
12k
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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:
$$ \frac{1}...
- 821
95
votes
17
answers
65k
views
What is the most elegant proof of the Pythagorean theorem? [closed]
The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield).
What's the most elegant proof?
My favorite ...
92
votes
7
answers
8k
views
$1=2$ | Continued fraction fallacy
It's easy to check that for any natural $n$
$$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$
Now,
$$1=\frac{1}{2-1}=\frac{1}{2-\cfrac{1}{2-1}}=\frac{1}{2-\cfrac{1}{2-\cfrac{1}{2-1}}}=\cfrac{1}{2-\...
- 5,011
88
votes
3
answers
5k
views
Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$
The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
- 85.9k
78
votes
8
answers
12k
views
Multiple-choice: sum of primes below $1000$
I sat an exam 2 months ago and the question paper contains the problem:
Given that there are $168$ primes below $1000$. Then the sum of all primes
below 1000 is
(a) $11555$ (b) $76127$ (c) $...
- 3,771
75
votes
7
answers
34k
views
Functions that are their own inverse.
What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
- 6,504
74
votes
14
answers
6k
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. ...
- 841
74
votes
13
answers
9k
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 ...
- 1,441
74
votes
5
answers
5k
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 ...
- 853
73
votes
22
answers
18k
views
Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
- 1,007
73
votes
15
answers
19k
views
Solving $DEF+FEF=GHH$, $KLM+KLM=NKL$, $ABC+ABC+ABC=BBB$
She visits third class and is $8$ years old (you can imagine how ashamed I felt when I said so to her). I helped her with lots of maths stuff today already but this is very unknowable for me. Sorry it'...
- 4,690
73
votes
6
answers
24k
views
$1 + 2 + 4 + 8 + 16 \ldots = -1$ 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 \...
- 841
73
votes
9
answers
31k
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 abstract ...
72
votes
17
answers
172k
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're factoring polynomials and had the same thought at the same time: when will we use this?
I feel a bit silly ...
- 861
71
votes
7
answers
7k
views
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers.
...
- 6,361
71
votes
17
answers
22k
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Why is a circle 1-dimensional?
In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position ...
- 5,471
71
votes
13
answers
9k
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 ...
- 1,521
69
votes
13
answers
14k
views
What is the definition of a set?
From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition.
My ...
- 3,133
69
votes
5
answers
17k
views
Nice expression for minimum of three variables?
As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive ...
- 16.2k
69
votes
15
answers
53k
views
Prove if $n^2$ is even, then $n$ is even.
I am just learning maths, and would like someone to verify my proof.
Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)...
- 575
68
votes
16
answers
52k
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
68
votes
6
answers
21k
views
Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$
I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$....
- 13.4k
67
votes
15
answers
8k
views
Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$?
I'm trying to solve this equation:
$$2-x=-\sqrt{x}$$
Multiply by $(-1)$:
$$\sqrt{x}=x-2$$
power of $2$:
$$x=\left(x-2\right)^2$$
then:
$$x^2-5x+4=0$$
and that means:
$$x=1, x=4$$
But $x=1$ is not a ...
- 1,016