sin(x) almost equal to x, ok but to start with? does this not depend on which units are used? — universeness

Sin(x) is a real number, not a degree; this is in radian measure. And it's true only near zero.

The three symbols you cite are used in a looser sense. I use them to mean approximately. The symbol in s(n) ~ t(n) means "behaves like", so its a bit loose also. Here's an example:


$\frac{n!}{{{n}^{n}}}\sim \sqrt{2\pi n}\cdot {{e}^{-n}}$

This is a form of Stirling's formula.

Yes, but it not stays in that state, and two states are necessary for quantum computing. There was a computing done in which about 70 (I don't remember the exact number) qubits were involved, facilitating 2exp70 possibilities in parallel. — Cornwell1

Yes, and I see now that you're referring to qubit coherence. I found an interesting summary of the historical and projected improvements in coherence times for quantum computing systems:

In the graph below are coherence times for notable studies of the last 20 years, the most recent being a time of 22 milliseconds. A future trend line is also projected to 2040. Based on this rate of progress, it appears feasible that a quantum computer will achieve coherence of one second or more before the end of this decade. Assuming that trend continues, we could see 10, 100, or even 1,000 seconds during the 2030s. A major milestone will be the cracking of RSA-2048 encryption keys (among the world's most secure algorithms), which a quantum computer with 4,100 stable qubits could achieve in 10 seconds. That same task would take a classical computer around 300 trillion years. — Quantum coherence times, 2000-2040

As the valves became tiny transistors, a decimal system would require 30 transistors to represent the range 0..999 but can such a problem not be overcome using something like floating-point representation? — universeness

A representation that was later used was binary-coded decimal. That required four transistors per digit, for a total of twelve transistors for a 3-digit decimal.

So sounds like bcd reduces inefficiency significantly from 30 vs 10 to 12 vs 10 valves/transistors.
So again, I think the strongest reason for using binary and not decimal in computing is the complexity of decimal arithmetic compared to binary arithmetic and the error handling which would be required if you used 10 states instead of 2.