338
votes
Accepted
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?
taking square root means reversing the effect of squaring. Dividing a number by itself does not do that (but rather always ...
- 76.2k
272
votes
Accepted
If there are $74$ heads and $196$ legs, how many horses and humans are there?
A hypercentaur is a creature with $2$ heads and $6$ legs; an anticentaur is a creature with no head and $-2$ legs.
Since $74$ heads make for $37$ hypercentaurs, with $74\cdot 6/2=222$ legs, you have $(...
- 237k
190
votes
Accepted
Why can a quadratic equation have only 2 roots?
Suppose there are three distinct roots $x,y,z$. One has
$$\begin{cases}ax^2+bx+c=0\\ay^2+by+c=0\\az^2+bz+c=0\end{cases}\Rightarrow\begin{cases}a(x^2-y^2)+b(x-y)=0\\a(x^2-z^2)+b(x-z)=0\end{cases}\...
- 28.7k
168
votes
Accepted
Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$?
This is because the equation $\;\sqrt x=x-2$ is not equivalent to $x=(x-2)^2$, but to
$$x=(x-2)^2\quad\textbf{and}\quad x\ge 2.$$
Remember $\sqrt x$, when it is defined, denotes the non-negative ...
- 175k
167
votes
Why is negative times negative = positive?
This is pretty soft, but I saw an analogy online to explain this once.
If you film a man running forwards ($+$) and then play the film forward ($+$) he is still running forward ($+$). If you play ...
- 3,762
165
votes
Accepted
Why does $\frac{1}{x} < 4$ have two answers?
You have to be careful when multiplying by $x$ since $x$ might be negative and hence flip the inequality. Suppose $x>0$. Then
$$\frac{1}{x}<4\iff4x>1\iff x>1/4.$$
If $x>0$ and $x>1/4$...
151
votes
Accepted
Multiple-choice: sum of primes below $1000$
The sum of the first 168 positive integers is $\frac{168^2+168}{2}=14196$, which is greater than answer (a). The sum of the first 168 primes must be even greater than that.
- 3,973
141
votes
In a village, $90\%$ of people drink Tea, $80\%$ Coffee, $70\%$ Whiskey, $60\%$ Gin. Nobody drinks all four. What percentage of people drinks alcohol?
If you add up the percentages, they come out to $300\%$. This means that the average number of beverages per person is $3$. No one drinks more than that, so no one can drink less than that, either. ...
- 11.9k
133
votes
Why does the discriminant in the Quadratic Formula reveal the number of real solutions?
Think about it geometrically $-$ then compute.
Everyone knows $x^2$ describes a parabola with its apex at $(0,0)$. By adding a parameter $\alpha$, we can move the parabola up and down: $x^2+\alpha$ ...
- 29.6k
122
votes
Why does $\frac{1}{x} < 4$ have two answers?
Here is the solution $$\frac { 1 }{ x } <4$$$$ \frac { 1-4x }{ x } <0$$$$ \frac { x\left( 1-4x \right) }{ { x }^{ 2 } } <0$$$$ x\left( 1-4x \right) <0$$$$ x\left( 4x-1 \right) >0 $$
...
- 21.5k
116
votes
What actually is a polynomial?
A polynomial (in one variable) is an expression of the form $$ p(x) = a_0+a_1x+a_2x^2+\ldots+a_nx^n$$ where the coefficients $a_i$ are some kind of number (or more generally they're elements of a Ring)...
- 56.8k
115
votes
Find the height of a bar, given the lengths of shadows cast by it and another bar
Here is a different visualization.
- 57.7k
103
votes
Accepted
Solution to the equation of a polynomial raised to the power of a polynomial.
Denote $a=x^2-7x+11.$ The equation becomes $a^{a-5}=1,$ or equivalently* $$a^a=a^5,$$ which has in $\mathbb{R}$ the solutions $a\in \{ {5,1,-1}\}.$ Solving the corresponding quadratic equations we get ...
- 8,236
100
votes
What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?
The number is clearly a multiple of $5$ and $2$. We look for the smallest, so we assume that it has no more prime factors.
So let $n=2^a5^b$. Since $n/2$ is a square, then $a-1$ and $b$ are even. ...
- 64.8k
96
votes
Accepted
Can I think of Algebra like this?
My question is: Is there any disadvantages to thinking about Algebra
like this? Is there anything later in my math education that will
require me to know that I am subtracting or adding 2x to get ...
- 8,189
94
votes
Why does cancelling change this equation?
When you cancel out $t$ on both sides you are assuming $t\neq0$.
To be rigorous you need to divide into two cases,
$t\neq0$, then you can cancel out $t$.
$t=0$, then you cannot cancel out $t$ as ...
- 12.5k
93
votes
Accepted
Systems of linear equations: Why does no one plug back in?
You wrote this step as an implication:
$$\begin{cases} -2x-y=0 \\ 3x+y=4 \end{cases} \implies \begin{cases} -2x-y=0\\ x=4 \end{cases}$$
But it is in fact an equivalence:
$$\begin{cases} -2x-y=0 \...
- 76.2k
93
votes
$r=\pm1$ are the only rationals with $\,r+1/r\in \Bbb Z$ (sum with its reciprocal is an integer)
It seems like you are asking for a rational number $n$ with the property that
$$n+\frac{1}{n}$$
is an integer. Let $z$ be an integer. Then we have
$$n+\frac{1}{n}=z$$
and
$$n^2+1=zn$$
$$n^2-zn+1=0$$
...
- 39.4k
91
votes
Is there a clever solution to Arnold's "merchant problem"?
At the end the tea cup is as full as at the start. This implies that the added wine is exactly outweighed by the tea that has disappeared.
- 225k
90
votes
Accepted
What is the definition of a set?
Formally speaking, sets are atomic in mathematics.1 They have no definition. They are just "basic objects". You can try and define a set as an object in the universe of a theory designated ...
- 390k
88
votes
Accepted
Are polynomials with the same roots identical?
No, they are not.
For instance, $2x^2-2$ and $x^2-1$ have the same roots, yet they are not identical.
And, depending on what you mean by "the same roots", we have that $x^2-2x+1$ and $x-1$ have the ...
- 197k
87
votes
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
Expanding on Christian Blatter's answer.
There are a few key points.
The arithmetic mean of two rational numbers is always rational.
The harmonic mean of two non-zero rational numbers is always ...
- 1,256
86
votes
Accepted
Is $\sqrt{64}$ considered $8$? or is it $8,-8$?
Your new teacher is wrong. $\sqrt{\cdot}$ is the principal square root operator. That means it returns only the principal root -- the positive one. $\sqrt{64}=8$. It does NOT equal $-8$.
On the ...
- 2,700
85
votes
Accepted
Mathematical symbol for 'slightly greater than'?
More often it is used as $b=a+\epsilon$ where $\epsilon$ normally stands for a small positive quantity. That provides b slightly greater than a. Similarly $-\epsilon$ for slightly below.
- 5,044
83
votes
Intuition for why the difference between $\frac{2x^2-x}{x^2-x+1}$ and $\frac{x-2}{x^2-x+1}$ is a constant?
Would you be surprised that the difference of $\dfrac{2x^2+x+1}{x^2}$ and $\dfrac{x+1}{x^2}$ is $2$?
- 1,087
81
votes
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
The simple reason this,
$$\sqrt{\frac{-1}{1}}
=\frac{\sqrt{1}}{\sqrt{-1}}$$
is not valid is because of a branch cut that must be taken as a result of the square root function.
Using exponentials:
$$
\...
- 1,654
81
votes
Accepted
What is an intuitive approach to solving $\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)$?
Intuition should say:
the denominator grows with $n^2$, the numerator grows with $n$. However, the number of fractions also grows by $n$, so the total growth of the numerator is about $n^2$.
And ...
- 122k
80
votes
What actually is a polynomial?
There are lots of good answers here and they are all essentially correct, even though they are different! I will try to contribute another, which is somewhat more abstract than the others. I normally ...
- 91.8k
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