Algebra classes always teach the Binomial Theorem – writing out the expanded form of a binomial raised to a power. What they don’t do is teach Newton’s brilliant generalization of the Binomial Theorem, usually called the Binomial Series, in which the exponent can be anything, not just an integer. The Binomial Series is much more important and interesting, and in this post I will describe it, including how Newton figured it out.

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Radioactivity was discovered in 1896 – so long ago that it now seems rather ordinary. Actually, it’s a remarkable thing; the nucleus of the atoms of some elements spontaneously explodes. This is a random event for the atom, and has nothing to do with how long the atom has existed, or with anything external to the atom. It just happens, although the disintegration rate is different for each type of radioactive atom. In fact, the rate is different for each isotope of an element; the number of neutrons in the nucleus makes a difference in how stable it is.
The shrapnel that emerges includes an alpha particle (2 protons and 2 neutrons, which is a helium nucleus) traveling at about 1/10 the speed of light. Alpha particles have enough energy that they can be observed individually as little flashes of light, emitted when the particle hits a suitable target.
After the disintegration, the nucleus has a smaller atomic number than the original, so that the atom has changed to another element (which may itself be radioactive).
In this post, I will just review some of the basic math involved with radioactivity. Radioactivity is a perfect example of exponential decay, which is an important thing for a mathematically literate person to know about.

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