Radioactive Dating: How do we know the initial amount of radioactive atoms present in the object? I'm currently reading a book about Earth's geological history and the authors mentions radioactive dating as on of the methods used to estimate the age of given fossils. It obviously does makes sense to me and I fully accept it as a valid scientific method, but : 


*

*How do scientists measure the initial amount of the radioactive
material present in a given object, for instance let it be C-14? Do
they compare them to other objects of identical chemical structure in
which the decay process is yet to start, and if so, how do they know
if the process itself hasn't started already?

*What method has been used to calculate the half-life for each isotope, 
because obviously it did not happen through scientific examination,
since in many cases the radioactive decay takes thousands, if not
millions of years.

 A: The C-14 dating method was calibrated by comparing its results
with the results from another independent dating method
(the counting tree-rings - dendrochronology).
Quoted from Radiocarbon dating - Calibration:

To produce a curve that can be used to relate calendar years to radiocarbon years,
   a sequence of securely dated samples is needed which can be tested to determine
   their radiocarbon age. The study of tree rings led to the first such sequence:
   individual pieces of wood show characteristic sequences of rings that vary
   in thickness because of environmental factors such as the amount of rainfall
   in a given year.
   These factors affect all trees in an area, so examining tree-ring sequences
   from old wood allows the identification of overlapping sequences.
   In this way, an uninterrupted sequence of tree rings can be extended
   far into the past. The first such published sequence, based on bristlecone pine tree rings,
   was created by Wesley Ferguson.
   Hans Suess used this data to publish the first calibration curve for radiocarbon dating in 1967.


The K-Ar dating method is based on the half-life of $^{40}K$ which is $1.248\cdot 10^9$ years.
This half-life could be determined in the laboratory by measuring two things:


*

*The isotope mixing ratios of natural potassium
can be determined with a mass spectrometer.
It contains $0.0117 \text{%}$ of $^{40}K$.
The other isotopes are not radioactive.

*The radioactive decay rate of a certain amount of natural potassium
can be determined with a radioactivity counter (e.g. a Geiger-Müller counter).
The decay rate is around 44 per second and per gram of potassium.


From these two measured numbers and Avogadro's constant
the half-life of $^{40}K$ could be calculated in a straightforward way.
