These definitions were developed over many years of teaching music to children and adults. They are meant to capture the essence of their terms in a simple way.
The standard tuning frequency for musical instruments is A440, which is the pitch you hear when a sound vibrates exactly 440 times each second. In fact, if you could clap 440 times per second, then you could tune an entire orchestra!
It is common practice to designate the particular octave with a number. For example, if the first C on a standard, 88-key piano is C1, then middle C is C4. This is the octave system used on aarongrooves.com.