The two statements are equivalent, assuming care with quantifiers. In the original form $\frac{n}{m}=\sqrt{7}$ we have an equation over the real numbers. We trade this for the second form $7m^2=n^2$ ...

As a department chair, I think this is the most difficult and expensive course to staff, unlike college algebra, elementary statistics, calculus, etc. I want the best faculty who can bring their own ...

I really appreciate your question. It gets to something I think about often when teaching calculus. You should read about Ed Dubinsky's notion of reflective abstraction, rooted in Piaget. At one ...

Mathematically a horizontal line is asymptotic to itself. So for example the horizontal asymptote of the constant function $f(x)=3$ is the line $y=3$. This issue of whether the definition of a ...

The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to ...

You can also use Wolfram Alpha to look up definitions of words. So by reductive reasoning, there is no need for infants to learn vocabulary because when they are old enough to use a smartphone, people ...

I think the best "real" application of the mean value theorem for integrals is to make a rigorous proof of the fundamental theorem of calculus.

Keep in mind that the Law of Sines and Law of Cosines are not identities in the same sense that your items 1-21 are identities. Rather these two laws are geometrical facts that are particular to ...

In many respects this is what is done in European universities, where the degree is often a three year program. First year students reading maths take analysis. Of course they probably learned ...

The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing ...

In many respects this problem reminds me of the sort of exercises that came out of the calculus reform movement of the 1990s. The problem is "conceptual" in the sense that it cleverly does not rely on ...

The "high level" answer is that a polynomial algebra is a free commutative algebra generated by the indeterminates. But this would probably not be a very satisfying answer to 8th graders. A possible ...

Indeed we could get by without referring to Taylor series, and instead just always refer to a "power series." But then maybe we should also stop referring to Fourier series and simply refer to "...

Although I have serious issues with "outcomes based" education, its philosophy does combat the pernicious notion that grades should have a normal distribution centered on "average." It is not ...

The reason why learning mathematical induction is difficult is developmental. Your audience of under-prepared college students probably accept "proof by verifying the first few cases" as more ...

Perhaps the convention is rooted in how the big dipper rotates around the north star--in a counterclockwise direction. Also think about how the earth rotates around its axis--in a counterclockwise ...

The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is ...

Going beyond the good answers here that have been given, and which work for teaching in a classroom, one could instead focus on pliable strategies that work well on GRE exams. When I tutor students ...

One approach is to start with the fact that a 90-degree rotation results from a reflection across the $x$-axis followed by a reflection across the line $y=x$. (In general, reflecting across two ...

Coxeter, in his book titled Projective Geometry, describes a dictionary game called Visch (short for "viscious circle"): Point=that which has position but not magnitude Position =place occupied by ...

The choice of i versus -i is correlated with the geometric orientation of the the plane. One choice, i, correponds to counterclockwise orientation which matches the righthand rule for orientation. ...

There are two formulations for definite integrals: $$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$ and the one you state: \int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))...

You are close. Floor($\log_{10}(N))+1$ is the number of digits.

Gauss said "You have no idea how much poetry there is in a table of logarithms." The first paragraph of this paper might get you pointed in the right direction ON THE DISTRIBUTION OF PRIMES—GAUSS’ ...

It is a poorly written question because it presumes there is some sort of universal definition of what is meant by an "exponential function." You can closely replicate the graph with \$f(x)=3+e^{-1/...

I often use the module "How to Tune a Radio--Trigonometric integrals explain tuning a radio." It is in Volume 3 (Applications of Calculus) of the MAA Resources for Calculus Collection (MAA Notes ...

This might be a good opportunity to inject some history into your course. Aristotle thought that cannonball trajectories were line segments, up at an angle then straight down. Look in Chapter 5 of "...

I have a colleague whose hobby is to always figure out how to a evaluate any given limit without L'Hospital's rule. Every week in the lunch room, there is yet another example. I think a very good ...