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.

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465 votes
10 answers

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|-...
a_hardin's user avatar
  • 5,491
397 votes
33 answers

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

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 ...
Low Scores's user avatar
  • 4,555
373 votes
23 answers

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: $0^0\cdot0^x=1\cdot0^0$,...
Stas's user avatar
  • 3,989
357 votes
31 answers

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

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 ...
idonno's user avatar
  • 3,889
223 votes
10 answers

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 ...
David G's user avatar
  • 4,177
194 votes
14 answers

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}} &...
Wilhelm's user avatar
  • 2,143
187 votes
28 answers

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 ...
Fernando Martin's user avatar
184 votes
20 answers

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^{...
Vlad's user avatar
  • 6,660
171 votes
15 answers

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 ...
John Gietzen's user avatar
  • 3,451
162 votes
19 answers

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 ...
Travis's user avatar
  • 3,396
144 votes
40 answers

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: ...
Sev's user avatar
  • 2,043
141 votes
36 answers

Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
b1_'s user avatar
  • 1,585
134 votes
17 answers

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 ...
danielsyn's user avatar
  • 1,577
134 votes
7 answers

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

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 ...
HN_NH's user avatar
  • 4,351
118 votes
26 answers

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 $ . ...
bluebellae's user avatar
  • 1,623
117 votes
15 answers

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 ...
Reinstate Monica's user avatar
112 votes
5 answers

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 ...
barfoon's user avatar
  • 1,339
107 votes
13 answers

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 ...
user3818's user avatar
  • 1,079
103 votes
9 answers

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 ...
Devansh Sharma's user avatar
101 votes
14 answers

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 ...
Jeff's user avatar
  • 3,345
101 votes
13 answers

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 ...
Nico A's user avatar
  • 4,934
99 votes
15 answers

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: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
newcomer's user avatar
  • 963
98 votes
1 answer

$4494410$ 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}=...
Ross Millikan's user avatar
97 votes
7 answers

Is zero positive or negative?

Follow up to this question. Is $0$ a positive number?
user avatar
96 votes
12 answers

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}...
user86484's user avatar
  • 821
95 votes
17 answers

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

$1=2$ | 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-\...
Mher's user avatar
  • 5,011
88 votes
3 answers

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 ...
Paramanand Singh's user avatar
  • 85.9k
78 votes
8 answers

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) $...
Sufaid Saleel's user avatar
75 votes
7 answers

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 $ ...
Willemien's user avatar
  • 6,504
74 votes
14 answers

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. ...
Spacez Ly Wang's user avatar
74 votes
13 answers

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 ...
Vishnu Vivek's user avatar
  • 1,441
74 votes
5 answers

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 ...
Warren L.'s user avatar
  • 853
73 votes
22 answers

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
Ssegawa Victor's user avatar
73 votes
15 answers


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'...
cnmesr's user avatar
  • 4,690
73 votes
6 answers

$1 + 2 + 4 + 8 + 16 \ldots = -1$ 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 \...
Christian's user avatar
  • 841
73 votes
9 answers

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

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 ...
Dan's user avatar
  • 861
71 votes
7 answers

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. ...
Gaurang Tandon's user avatar
71 votes
17 answers

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 ...
mr eyeglasses's user avatar
71 votes
13 answers

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 ...
baum's user avatar
  • 1,521
69 votes
13 answers

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 ...
John Doe's user avatar
  • 3,133
69 votes
5 answers

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 ...
Oscar Cunningham's user avatar
69 votes
15 answers

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)...
user79612's user avatar
  • 575
68 votes
16 answers

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

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 $1-\sqrt{2}$....
JSCB's user avatar
  • 13.4k
67 votes
15 answers

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 ...
TheLogicGuy's user avatar
  • 1,016

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