Quantum mechanics is one of two cornerstones of modern physics (the other is general relativity), and has been precisely confirmed in thousands of very exacting experiments.
For these reasons, scientists have considerable confidence in these dates when they are measured properly in accordance with procedures that have been developed and refined over several decades.
However, usually it is not possible to apply this formula directly, because, for instance, in many cases we do not know the original amount of the radioactive isotope when the rock was solidified.
Also, such a calculation does not provide us with any statistical error margin to double-check the result.
By some simple algebraic manipulation of the basic radioactivity formula above, one can show that the following formula must hold at any time t: (Sr87/Sr86) is the ratio of these two isotopes at time t.
Note that this equation is in the simple form y = b m x, namely the formula for graph of a straight line with slope m and with y-intercept b: here y = (Sr87/Sr86).
Of course, in real scientific research, scientists do not rely on manually drawing points on graph paper to determine a best-fit straight line or to determine the line's slope or y-intercept.
Instead, they use a statistical technique known as linear regression, which computes the least-squares best fit of a straight line through a sequence of points.
This cannot be used for radiometric dating, but it does suggest advanced technology such as this is rapidly advancing and soon will be available to consumers.
In mathematical terms, radioactive decay is governed by a simple exponential formula, taught in many high school math classes: P is the amount after time t, and L is the decay constant for the radioactive isotope.
This decay constant L can be expressed in terms of the half life T (the time it takes for one-half of the material to decay) as L = log(2) / T, where log(2) = 0.693147... In other words, if we know P, or even merely their ratio, we can solve the above equation for the time t.
Fortunately, scientists have developed several methods that not only circumvent the difficulty of not knowing the original amounts, but also provide a very reliable means of statistical validity checking.
For example, the rubidium-strontium isochron method, one of the most widely used schemes, is based on the radioactive decay of rubidium-87 into strontium-87 by the emission of a high-energy electron.