Tag: algebra-precalculus

algebra precalculus – Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

The Question : 108 people think this question is useful $x,y,z >0$, 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 question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the

algebra precalculus – 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?

The Question : 116 people think this question is useful 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 $2^2=2 \cdot 2=4$ . But square root of $2$ is not $\frac{2}{2}=1$

sequences and series – Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

The Question : 115 people think this question is useful 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? PS: This problem is known as “The sum of the first $n$ positive integers”. The Question Comments : This an important example of a finite integral,

algebra precalculus – Why is negative times negative = positive?

The Question : 121 people think this question is useful 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 so: Assume $(-x) \cdot (-y) = -xy$ Then divide both sides by

algebra precalculus – Division by $0$

The Question : 126 people think this question is useful 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 even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only

sequences and series – Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$

The Question : 126 people think this question is useful 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 $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some values of $x$ for which the above formula is invalid? What about if we take only a

algebra precalculus – What actually is a polynomial?

The Question : 145 people think this question is useful 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 grade I skipped the class and joined the more advanced class.

algebra precalculus – Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

The Question : 165 people think this question is useful 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}} &= \frac1i \\ \\ \sqrt{\frac1{-1}} &= \frac1i \\ \\ \sqrt{\frac{-1}1} &= \frac1i \\ \\ \sqrt{-1} &=

sequences and series – Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

The Question : 164 people think this question is useful 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 of this property. I would also like to see any other proofs. The Question Comments : math.stackexchange.com/questions/61798/… Look at this

algebra precalculus – Why can ALL quadratic equations be solved by the quadratic formula?

The Question : 279 people think this question is useful 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 it. I have tried to

real analysis – What does $2^x$ really mean when $x$ is not an integer?

The Question : 199 people think this question is useful 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 please let me know what you think. The Question

algebra precalculus – How do people perform mental arithmetic for complicated expressions?

The Question : 178 people think this question is useful 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: $$\dfrac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365}$$ The answer is easy using paper and pencil: