Mathematics Stack Exchange
Mathematics Stack Exchange

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.

This tag is for questions typically taught in precalculus, i.e. the prerequisite courses to the standard American first-year course in differential and integral calculus, including some parts of elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

43604 questions
465
votes
10 answers

Is this Batman equation for real?

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|-3}}+\left(\frac…
a_hardin
  • 5,499
  • 3
  • 13
  • 7
388
votes
32 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”: $$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$ $$ 2^{-3}…
Peter LeFanu Lumsdaine
  • 8,058
  • 5
  • 24
  • 49
364
votes
24 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: $x>0$ $0^x=0^{x-0}=0^x/0^0$, so $0^0=0^x/0^x=\,?$ Possible answers: $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined PS. I've read the…
Stas
  • 3,891
  • 3
  • 14
  • 7
339
votes
30 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?
Michael Haren
  • 308
  • 3
  • 7
  • 11
291
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 it. I have tried to figure it out by proving…
idonno
  • 3,815
  • 9
  • 24
  • 20
209
votes
9 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 please let me know what you think.
David G
  • 3,837
  • 8
  • 31
  • 38
181
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}} &= \frac1i \\\\ \sqrt{\frac1{-1}} &= \frac1i \\\\…
Wilhelm
  • 2,013
  • 2
  • 14
  • 8
181
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: $$ \dfrac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365} $$ The answer is easy…
Vlad
  • 6,497
  • 3
  • 31
  • 66
177
votes
27 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 of this property. I would also like to see any…
Fernando Martin
  • 5,445
  • 6
  • 29
  • 46
159
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 form $ax^3+bx^2+cx+d=0$, there is a set of three…
John Gietzen
  • 3,281
  • 4
  • 24
  • 25
155
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 grade I skipped the class and joined the more…
Travis
  • 3,328
  • 3
  • 20
  • 43
133
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 so: Assume $(-x) \cdot (-y) = -xy$ Then divide both sides by…
Sev
  • 1,909
  • 3
  • 17
  • 19
132
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 $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some…
AlexBrand
  • 355
  • 2
  • 5
  • 8
130
votes
18 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 even one. Nevertheless, both $1.$ and $2.$ are…
danielsyn
  • 1,527
  • 2
  • 11
  • 5
130
votes
35 answers

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

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".
b1_
  • 1,467
  • 2
  • 10
  • 6
1
2 3
…
99 100