Random math questions (modular arithmetic & notation)

Question

I found this amazing wall clock picture on the internet but I really don't know a few things. I don't know what's $B'_L$, 3, why $2^{-1}\equiv 4[7]$ and the one with the black and white circles.

For $B'_L$, I thought first that it could be Bernoulli numbers but figured it out that it was something completely different. For $2^{-1}\equiv 4[7]$, I've been thought modular arithmetic but never with fractions or inverses, so it doesn't make much sense to me. For the last one, I thought it was braille but found out that numbers was completely different.

So can anyone explain me what are these?

Answer 1

I actually own this physical clock. It came with a small slip of paper explaining each number as follows (hyperlinks added):

Geek Clock Cheat Sheet

1) Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function. Its value is now known to be exactly 1.

2) A joke in the math world: An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says, "I get it" and pours two beers.

3) A unicode [sic] character XML "numeric character reference."

4) Modular arithmetic, also known as clock arithmetic, is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value. The modular multiplicative inverse of 2 (mod 7) is the integer /a/ such that 2*/a/ is congruent to 1 modulo 7

5) The Golden Mean ... reworked a little.

6) Three factorial 3*2*1

7) A repeating decimal that is proven to be exactly equal to 7 with Cauchy's Convergence Test.

8) A graphical representation of a binary, or base-2 number

9) An example of a base-4 number, which uses the digits 0, 1, 2 and 3 to represent any real number.

10)A binomial coefficient, also known as the choose function. 5 choose 2 is equal to 5! divided by (2!*(5-2)!)

11) A hexadecimal, or base-16, number

12) A radical

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Answer 2

I don't know about the first two. But for the last two...

Actually, it is $2^{-1} \equiv 4 \pmod 7$. For $x \in \mathbb{R}\setminus\{0\}$, $x^{-1}$ is the unique real number such that $xx^{-1} = x^{-1}x = 1$. So $2^{-1}$ modulo $7$ is the unique number (modulo $7$) such that $2\times2^{-1} = 2^{-1}\times 2 \equiv 1 \pmod 7$. As $2\times 4 = 4\times 2 \equiv 1 \pmod 7$, $2^{-1} \equiv 4 \pmod 7$.

As for the dots, think about binary where a black dot is a one and a white dot is a zero.