This is the third post in the math/science part of my "Gen Eds in a Nutshell" series. It's a series of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm half way through the world history subject on Wednesdays. I'm combining the last two into one series on Fridays.
Thus far in the math/science subjects:
1. Way back in the 400s BC, a Greek named Democritus suggested that everything we see around us, all matter, might consist of certain fundamental, uncuttable units, a certain number of core units from which everything is made. He called them "atoms," meaning uncuttable. He even suggested there might be "soul atoms," which made things live.
The idea wasn't fully picked up again until the 1800s with John Dalton (1766-1844). Newton had suggested that light was made up of particles, and the century after him generally had begun to think of the world consisting of particles. But it was Dalton who saw that these particles combined in a way that showed all of matter consisted of a certain fixed number of fundamental units. We call these fundamental units, "elements."
Everything around us that is matter--gases, liquids, solids--all are made up of less than a hundred distinct elements. If we took the whole world and somehow boiled it all down and put its fundamental components into baskets, we would find that there is a finite number of building blocks of everything around us.
2. So all of matter--stuff--breaks down into 90 basic elements that occur in nature. These are 90 basic types of "atom" that combine in various ways to form "molecules" and compounds. Then those molecules get mixed in various ways to form everything we see around us that is a gas, a liquid, or a solid.
As an example, two of the most basic elements of the universe are hydrogen (H) and oxygen (O). Both of them can bind to form "molecules" like hydrogen gas (H2), oxygen gas (O2), water (H2O) and hydrogen peroxide (H2O2). If you know these compounds, you know they each behave quite differently. Hydrogen gas can be used to fly the Hindenberg, but don't let any sparks come near it. Oxygen is necessary for us to breathe, but it is also what keeps fires burning. Water is a liquid we need to survive. Add a second oxygen per molecule and you have something that helps clean a wound.
3. The initial idea of an atom was that it was the smallest building block of matter, an indestructible unit. However, in the late 1800s, experimenters began to wonder if there was a level of reality even smaller than the atom. Did the atom itself have parts--or "particles"--inside them, if you would?
For example, Ernest Rutherford discovered in 1909 that when you bombard gold foil with something called alpha rays , most of them went through but occasionally one would bounce backward. From this observation he concluded that the center of an atom was more dense than the rest of it. He called this center of an atom the "nucleus," and inferred that most of the rest of the atom was empty space.
Twelve years earlier, in 1897, J. J. Thompson had also figured out that "cathode rays" were smaller components inside an atom and that they had an electrical charge. They would come to be called electrons. With the discovery of the nucleus by Rutherford, Niels Bohr would suggest in 1913 that the much smaller electrons orbit the nucleus like planets around the sun.
4. Over the next few decades, a more precise sense of the atom would unfold. There are three fundamental particles in the atom: electrons, protons, and neutrons. Protons and neutrons are in the nucleus. Electrons pertain to the space beyond the nucleus. The electron has a negative charge, the proton a positive charge. The neutron is neutral and has no charge.
The electrical charge in general holds the negatively charged electrons to the positively charged nucleus. Opposite charges attract, like charges repel.
Then why don't the positive charges in the nucleus cause the nucleus to fly apart? It would be the 1970s before the current theory came together. Protons hold together because of a force, the "strong nuclear force," one of the four basic forces of the universe, the strongest of them all. But it only works on a very, very small scale. 
As the "standard model" of particles continued to develop, physicists would conclude that protons and neutrons themselves were made up of smaller particles called "quarks" (three a piece). The strong nuclear force that holds protons and neutrons together also holds these quarks together to form protons and neutrons.
5. At the same time that particle physics was developing--even faster in fact--a sense of the nature of electrons was also developing. Late in the year 1900, Max Planck would give birth to modern physics with his proposal that energy existed in discrete passages or "quanta" of energy. It was a revolutionary suggestion and Planck himself probably did not realize the full implications.
So electrons could only occupy specific energy states around the nucleus. To complicate matters, Louis de Broglie was able to solve certain problems by suggesting that electrons were both waves and particles. Werner Heisenberg discovered an uncertainty principle. You couldn't know both a particle's location and its momentum. In a sense, it didn't even have a specific location until you tried to measure it.
So an electron is really more like a certain probability field around the nucleus. That is to say, an electron is not really in just one place at a time. In a sense, it is more or less in a zone. If you try to locate it, you make it take on a location, and there is a greater likelihood that it will be in some places more than others. It probably will not be in a different galaxy, although there is a very very small possibility.
The first two electrons of an atom are most likely found in a spherical zone close to the nucleus (the 1s orbital). In keeping with the next energy states that are possible, it is as if there is assigned seating area for each additional electron an atom might have. After the first spherical zone is a second one for two more electrons (the 2s orbital). Then there are three perpendicular zones for the next six electrons (the 2p orbital).
The zones for additional electrons are now well known, that is, the fields they occupy. 
6. Quantum physics implied a dramatic shift from the physics that Isaac Newton developed in the 1600s. Up till the twentieth century, scientists assumed that everything in the universe was determined. That is to say, many believed that if we had all the data and knew all the laws of nature, we could determine everything that would happen for the rest of history.
Quantum physics suggested that you simply couldn't predict what nature would do, at least not on its smallest scale. Richard Feynman often used the so called double slit experiment to explain the crazy quantum world. If you fire a series of photons randomly at a screen through two slits, they will eventually form a pattern on the screen known as an interference pattern. This is a pattern typical of a wave.
But if you put a detector by each slit, to see which one each particle is going through, the interference pattern goes away. Now you have two collections of dots from the photons, as if the photons are particles. So you cannot predict where any one photon will go, although you can predict the interference pattern if you don't connect the detector. And if you connect the detector, the act of measurement eliminates the wave like property of the photon.
Now the atomic world is not determined. It is unpredictable on its most fundamental level.
7. This seems like an appropriate place to mention the atomic bomb, dropped on Japan in 1945. One of the discoveries of the early twentieth century was that the same element can have more than one form. The same element can have different numbers of neutrons in its nucleus. We call these different "isotopes" of an element.
To back up a little, it is the number of protons that identifies an atom as a certain element. If an atom has one proton in its nucleus, it is hydrogen. If it has two, it is helium. If an atom has 8 protons in its nucleus, then the element is oxygen.
But the same atom can vary in the number of neutrons--or even electrons--it has. So with regard to electrons, sodium can lose an electron and take on a positive charge. Or chlorine can gain an electron and take on a negative charge.
It is differing numbers of neutrons that make different "isotopes" of an element. For example, most hydrogen does not have any neutrons in its nucleus. But there is a rare isotope of hydrogen with one neutron (deuterium) and there is an even rarer isotope with two (tritium).
"Radioactive" isotopes are forms of an atom that tend to deteriorate into other elements. For example, uranium has an "atomic number" of 92. That means it is an element with 92 protons in its nucleus. However, it has more than one isotope. Uranium 238 is relatively stable. It has 238 particles in its nucleus (its atomic weight). So a little subtraction suggests that this isotope has 92 protons and 146 neutrons.
There is, however, another isotope called Uranium 235. It has 143 neutrons and is unstable. If you shoot a neutron at it and hit the nucleus just right, it begins a reaction that ends with uranium breaking apart into Krypton and Barium. The atomic bomb took less than a kilogram of Ur 235 and created a nuclear chain reaction, releasing the immense amount of energy that destroyed Hiroshima and Nagasaki.
8. A standard model for particle physics reached its mature form in the 70s and 80s. There are six types of quark. There are six smaller particles called leptons that include the electron and a small particle called a neutrino. There are four bosons that are the mediators of the fundamental forces: gluons mediate the strong force, photons the electromagnetic force, the W and Z mediate the weak force. In 2013, the Higgs boson was also tentatively confirmed, which is thought to give matter its mass.
9. It is not agreed what may underlie these smallest of particles, if anything. String theory suggests that all these particles and forces can be explained by way of fundamental strings that vibrate in certain ways. However, not all agree and there is no experimental confirmation of string theory. A competing theory is loop quantum gravity, which functions on the assumption that space itself has a smallest size, the Planck unit.
Next Week: The Periodic Table
 Now we know that alpha rays are two protons and two neutrons, basically the nucleus of a helium atom.
 The electromagnetic force is a second. Then the "weak" nuclear force works in something called beta decay. Finally, there is gravitation.
 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d10, 4p6, 5s2, 4d10, 5p6, 6s2, 4f14, 5d10, 6p6...