I think if you look at this animation and think about it long enough, you'll understand: Why circles and right-angle triangles and angles are all related. Why sine is "opposite over hypotenuse" and so on. Why cosine is simply sine but offset by $\frac{\pi}{2}$ radians.


My favorite: tell someone that $$\sum_{n=1}^{\infty}\frac{1}{2^n}=1$$ and they probably won't believe you. However, show them the below: and suddenly what had been obscure is now obvious.


This visualisation of the Fourier Transform was very enlightening for me: The author, LucasVB, has a whole gallery of similar visualisations at his Wikipedia gallery and his tumblr blog.


Here is a classic: the sum of the first $n$ positive odd numbers $= n^2$. We also see that the sum of the first $n$ positive even numbers $= n(n+1)$ (excluding $0$), by adding a column to the left.


A well-known visual to explain $(a+b)^2 = a^2+2ab+b^2$:


The sum of the exterior angles of any convex polygon will always add up to $360^\circ$. This can be viewed as a zooming out process, as illustrate by the animation below:


While attending an Abstract Algebra course I was given the assignment to write out the multiplication table modulo n. I forgot to do the homework until just before class, but it was so easy to write the program I was able to print the result between classes. The circular patterns in the tables fascinated me, and compelled me to replace the numbers with ...


This wasn't the first, but it's definitely awesome: This is a proof of the Pythagorean theorem, and it uses no words!


For me it was the Times Table of $9$. We are usually forced to memorize the multiplication tables in school. I remember looking at the table for $9$, and seeing that the digit in ten's place increased by one, while the digit in the one's place decreased by one. $$ \begin{array}{r|r} \times & 9 \\ \hline 1 & 9 \\ 2 & 18 \\ 3 & 27 \\ 4 & ...


“One of the ways to look at division is as how many of the smaller number you need to make up the bigger number, right? So 20/4 means: how many groups of 4 do you need to make 20? If you want 20 apples, how many bags of 4 apples do you need to buy? So for dividing by 0, how many bags of 0 apples would make up 20 apples in total? It’s impossible — however ...


Simple answer for "what is a radian": Logarithmic spiral and scale:


Here is a very insightful waterproof demonstration of the Pythagorean theorem. Also there is a video about this. It can be explained as follows. We seek a definition of distance from any point in $\mathbb{R}^2$ to $\mathbb{R}^2$, a function from $(\mathbb{R}^2)^2$ to $\mathbb{R}$ that satisfies the following properties. For any points $(x, y)$ and $(z, w)$,...


This is a neat little proof that the area of a circle is $\pi r^2$, which I was first taught aged about 12 and it has stuck with me ever since. The circle is subdivided into equal pieces, then rearranged. As the number of pieces gets larger, the resulting shape gets closer and closer to a rectangle. It is obvious that the short side of this rectangle has ...


Whether this is 'simple' enough is debatable... the method to generate the Mandelbrot set is likely to be far too complicated for the book in question, but the mathematical expression that's at its heart couldn't be much simpler. $z_{n+1} = {z_n}^2 + c$ After implementing the Mandelbrot set I learned about the Buddhabrot, which is basically a way of ...


A visual explanation of a Taylor series: $f(0)+\frac {f'(0)}{1!} x+ \frac{f''(0)}{2!} x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots$ or $f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots$ When you think about it, it's quite beautiful that as you add each term it wraps around the curve.


When I look up "area of a rhombus" on Google images, I find plenty of disappointing images like this one: which show the formula, but fail to show why the formula works. That's why I really appreciate this image instead: which, with a little bit of careful thought, illustrates why the product of the diagonals equals twice the area of the rhombus. EDIT: ...


As I was in school, a supply teacher brought a scale to lesson: He gave us several weights that were labeled and about 4 weights without labels (let's call them $A, B, C, D$). Then he told us we should find out the weight of the unlabeled weights. $A$ was very easy as there was a weight $E$ with weight($A$) = weight($E$). I think at least two of them had ...


I used to love naughty $37$. $37 \times 3 = 111;$ $37 \times 6 = 222;$ $37 \times 9 = 333;$ $37 \times 12 = 444;$ $37 \times 15 = 555;$ $37 \times 18 = 666;$ $37 \times 21 = 777;$ $37 \times 24 = 888;$ $37 \times 27 = 999;$


This is similar to Aky's answer, but includes a second drawing (and no math.) To me the second drawing is key to understanding why the $\mathrm c^2$ area is equal to the sum of $\mathrm a^2+\mathrm b^2$. Edit: comments requested an animation, so a simple gif is attached...


How about a line integral of a scalar field by http://1ucasvb.tumblr.com:


I found it completely amazing that the angles in a triangle always added up to 180 degrees. No matter how you drew a triangle, you could measure the angles with a protractor and they always add up to about 180 degrees, like magic. Even more amazing when I realized it wasn't some rule of thumb or approximation, but true in some deeper sense for the ideal, ...


Clearly the figure is a trapezoid because you can construct an infinite number of quadralaterals consistent with the given constraints so long as the vertical height $h$ obeys $0 < h \leq 9$ inches. Only one of those infinite number of figures is a square. I would email the above statement to the teacher... but that's up to you. As for the "politics" ...


I think this is a symptom of how students are taught basic algebra. Rather than being told explicit axioms like $a(x+y)= ax+ay$ and theorems like $(x+y)/a = x/a+y/a,$ students are bombarded with examples of how these axioms/theorems are used, without ever being explicitly told: hey, here's a new rule you're allowed to use from now on. So they just kind of ...


Draw a graph of the function on the blackboard, showing $a$ and $b$ and a crosshatched area representing the integral. Put an $x$ on the horizontal axis. Erase the $x$ and put a $z$ there. Does that change the area? Erase the $z$ and put a smiley face there. Does the area change? Why/why not?


My advice would be: $\bullet $ Do many calculations $\bullet \bullet$ Ask yourself concrete questions whose answer is a number. $\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!) $\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in ...


The first "math thing" that just blew my mind was the identity $$ e^{i\pi} = -1 $$ Namely the fact that the two independently discovered transcendent numbers and imaginary one so simply and elegantly bound. In the marginally rearranged form $$ e^{iπ}+1=0 $$ it uses absolutely nothing but nine essential concepts in mathematics: five of the most essential ...


This animation shows that a circle's perimeter equals to $2r*\pi$. As ShreevatsaR pointed out, this is obvious because $\pi$ is by definition the ratio of a circle's circumference to its diameter In this image we can see how the ratio is calculated. The wheel's diameter is 1. After the perimeter is rolled down we can see that its length equals to $\pi$ ...


Of course, you are right. Send an email to the teacher with a concrete example, given that (s)he seems to be geometrically challenged. For instance, you could attach the following pictures with the email, which are both drawn to scale. You should also let him/her know that you need $5$ parameters to fix a quadrilateral uniquely. With just $4$ pieces of ...

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