Why are there 44 sine curves on $\sin(n)$ with $0 < n < 10,000$ and $n$ integer (Gilbert Strang's "thousand points of light") On Gilbert Strang's Calculus book (available on the following link: http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf), at page 34 (with subtitle "A thousand points of light"), according to the book's enumeration, (page 40 according to pdf reader enumeration) he starts reasoning about the graph of $\sin n$ with $0 < n < 10,000$ ($\sin n$ is $\sin x$ with $x$ an integer, wich means its graph will not be continuous but rather a "cloud of points").

I do not understand why the reasoning he makes leads to the final conclusion.
He reasons as follows (in topics):


*

*Even though the graph of sin(n) is a cloud of points, when
"looking from far away" (that is, with the graph in small
proportions) it looks like there is more than one curve on it;

*I want to know how many curves there are on it;

*The points at $n = 22$ and at $n = 44$ are close to 0 (because they're close to multiples of $\pi$, whose sine is 0);

*The point 44 starts the middle sine curve;

*There are 44 sin curves.


I do not understand the reasoning that leads to 4 and 5 and would appreciate if somebody could help.
Thanks in advance.
 A: This section was confusing at first for me, too, perhaps because of the author's informal style of writing. After one day of experimenting with a graphing calculator, it has got a little bit clearer, though I still do not think I have fully understood it. (I never think I fully understand anything.)
First, $y = sin(x)$, for $x \in [0, 10000]$, when plotted, is not like graph 1 in that section. As the author mentioned, the "1591 oscillations would be so crowded that you couldn't see anything". Here is what it looks like for $x \in R$ with viewing window adjusted for $x$ from 0 to 1000, with 159 oscillations in between:

In fact, graph 1 is $y = sin(n)$ for $n \in \{n: n \le 10000, n \in N\}$. The points are so crowded that they look like continuous curves. Graph 2 is $y = sin(n)$, for $n \in \{n: n \le 1000, n \in N\}$. Here is what it looks like for $n \in N$ in a graphing calculator, with viewing window adjusted for $n$ from 1 to 1000:

Since $44 \approx 14\pi + 0.0177$, $sin(n)$ changes a little bit for every 44 increase in n so that the points $(44, sin(44)), (88, sin(88)), \dots$ seem to form a sine curve that starts from the origin. It is called the middle sine curve. I have marked the points red in my graph.
Now, we see the middle curve, which begins with $(44, sin(44))$. Then, there are other curves that begin with $(1, sin(1)), (2, sin(2)), \dots, (43, sin(43))$. Note that $(45, sin(45))$ is a point on the $sin(1)$ curve (the curve that begins with $(1, sin(1))$), so we have 44 curves.
I hope I have answered your question. A final note about a point that may confuse you (and confused me at first): When the author said "The first point to come close is sin 22." does not mean that the first point to come close to $(0, 0)$ is $(22, sin(22))$. Compared with $(22, sin(22))$, for which the distance is $\sqrt{22^2+sin^2(22)} \approx 22.0031$, $(1, sin(1))$, for which the distance is $\sqrt{1^2+sin^2(1)} \approx 1.0001$, is actually closer. What the author means is the first value of $sin(n)$ to come close to $sin(0)$, as he mentioned "A point near (0, 0) really means that sin n is close to zero."
