0

When playing with my calculator I found that $$\tan 3 + \pi \approx 3$$ Is there a mathematical reason for this?

Mahesh s
  • 53
  • 3
    The numbers are not that close. $3-\tan(3)-\pi\approx 9.539\cdot 10^{-4}$. Really, I would think it stranger if there was not a number that could get it that close that is easily represented. "Is there a mathematical reason for this?" Beyond coincidence and the fact that that so happens to be what the value of $\tan(3)$ and $\pi$ happen to be? – JMoravitz Feb 21 '16 at 04:55
  • To be honest the statement $x \approx y$ is extremely subjective. Nonetheless I would encourage you to keep "playing" with the calculator. If you do you may discover that there are many examples of "strange coincidences." – cpiegore Feb 21 '16 at 05:05
  • 1
    Apologizes in advance... $22/7$ is also approximately equal to $\pi$. Is there any mathematical reason for this? Yes, it's close to $\pi$ – Yeah.. Feb 21 '16 at 05:37
  • @Yeah.. : there are an integral and a series that evaluate to $\frac{22}{7}-\pi$ $$\begin{align}\frac{22}{7}-\pi &= \int_0^1\frac{x^4(1-x)^4}{1+x^2}dx \ &= \sum_{k=1}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)} \end{align}$$

    http://math.stackexchange.com/questions/1654104/series-and-integrals-for-inequalities-and-approximations-to-pi

    The integral shows that this difference is small because the numerator is a product of relatively high power of quantities between $0$ and $1$. The series shows that this difference is small because it is a sum of small numbers.

     – Jaume Oliver Lafont Feb 21 '16 at 06:34
  • 2
    @Yeah because $\frac{22}{7}$ is one of the convergents of the continued fraction for $\pi$, e.g. These always form good approximations. – Henno Brandsma Feb 21 '16 at 08:40

1 Answers1

6

Expand $\tan (x)$ near $\pi$ I find $$\tan(x)=(x-\pi)+\frac{1}{3}(x-\pi)^3+O((x-\pi)^5)$$, and $abs(\frac{1}{3}(x-\pi)^3)\left.\right|_{x=3}<0.001$. I hope this might help

Alexis
  • 941