Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay
Large numbers are so common in science that a special notation, scientific notation,is used to deal with them.
Our familiar place notation is based on the powers of numbers. For example, 3421 =3x1000 + 4x100 + 2x10 + 1. Each digit represents a multiple of something ten times aslarge as the place to the right. We don't have to use 10 as a base. In binary notation,used in computing, powers of two are used. In this notation, 53 is written 110101; 1x32 +1x16 + 0x8 + 1x4 + 0x2 + 1x1.
For really large numbers, it gets cumbersome to write them out. A trillion dollarslooks like this: $1,000,000,000,000. In a typical room, there are10,000,000,000,000,000,000,000,000 air molecules. It's more convenient to write these as1012 and 1025, respectively. The 12 and 25 are called the exponentof the number. 104 means 10x10x10x10. Obviously 101 simply equalsten. Any number to the first power is itself.
The virtue of scientific notation, apart from compactness, is that it simplifies math.For example, 1000 x 100,000 = 100,000,000 = 103x105 = 108.In other words, to multiply numbers in scientific notation, just add their exponents.
Likewise, 10,000,000/10,000 = 1000 = 107/104 = 103. Todivive numbers in scientific notation, just subtract the lower exponent from theupper.
This rule applies even if the lower number is equal or bigger. 1000/1000 = 1 = 103/103= 100. This baffles a lot of people, but it is perfectly logical: anynumber to the zero power is one!. Negative exponents mean numbers less than one; 10-3= 1/1000 or .001.
Sometimes the exponents themselves are so large they are cumbersome. In that case wecan nest exponents. Sagan mentions a "googol" - 10100, and a"googolplex" - 10googol. Writing out a googolplex even in exponentialform is pretty daunting, and we could never write it entirely; it has more zeroes thanthere are atomic particles in the Universe. However, we can write a googolplex like this:
100 10 10
Who in the world ever needs numbers that big? There are only about 1080atomic particles in the Universe. But there's a branch of physics called statisticalphysics that uses such huge numbers. For example, the air molecules in a room have anaverage energy, but individual molecules have energies ranging from very low to very high.To account for the actual distribution of energies, we have to calculate, among otherthings, the total number of possible energy states the atoms can have. The exponentof that number is comparable to the total number of atoms; in a typical room that's about1025.
Infinity defies a lot of our common sense notions but there is a lot we can say aboutinfinity. In fact, there are different degrees of infinity.
You can't use ordinary mathematics to deal with infinity. One plus infinity is stillinfinity; ten times infinity is still infinity, and so on. One of the first modernthinkers to deal rigorously with infinity was Galileo. He pondered whether the collection[set, in mathematical parlance] of even numbers was smaller than the set of all integers.Obviously, every even number can be obtained by multiplying an integer by two. So forevery integer, there is an even number. In some strange way, there are as many evenintegers as integers in general, even though there are only half as many even integers!
Pairing up every even integer with an integer is called putting two sets intoone-to-one correspondence. We could do the same thing with all multiples of three, orten, or a million. Every number in those sets can be paired up with a correspondinginteger, and vice versa. Infinite sets are said to be equivalent if you can pair theirmembers up one-to-one.
Mathematician Georg Cantor went further. You can't pair integers up with the points ona line. Regardless how closely you space the integers, there are an infinity of points inbetween. The points on a line are "more" infinite than the set of integers.Cantor used the Hebrew letter Aleph to denote the different degrees of infinity (ortransfinite numbers, to use the correct term). The number of integers is calledAleph-zero, the number of points on a line is Aleph-one.
It is possible to draw an infinitely crinkly line that passes through every point in aplane, so the points in a plane can be put into one-to-one correspondence with the pointsin a line. Thus the points in the plane also have Aleph-one points. But the number ofpossible mathematical curves you can draw is greater yet. This set has Aleph-two members.And that's it so far; that's the highest degree of infinity we have been ever able toimagine.
How do we know there are atoms? For most of history, the idea was basically anintuitive one that we couldn't simply keep cutting things into tinier and tinier pieces.There had to be a limit.
The first solid clues came from chemistry. Once chemists began to get a handle on whatthe really fundamental materials of nature were, they found that they didn't mix inarbitrary proportions. You couldn't mix oxygen and hydrogen in weight proportions 5:1 or3.6: 1, only 8:1 would get water. Any other proportion would yield water plus somethingleft over. (For the chemists in the crowd, 16:1 will also work; that gives H2O2 orhydrogen peroxide.) The fact that materials always reacted in fixed proportions hintedthat matter came in discrete units.
Another clue came from certain minerals. When crushed, they always break into regularfragments. Salt, for example comes in tiny cubes because that's how salt breaks when it'scrushed. Chemists began to suspect that the cubic shape reflected some fundamental unitthat makes up salt, as indeed it does.
How big are atoms? One crude early guess noted that a cubic centimeter of oil couldcreate a film on water at least 10 meters in diameter. One cubic centimeter of volumecould spread out to cover an area of a million square centimeters. Thus the film of oilwas a millionth of a centimeter thick, and atoms had to be smaller than that. Actually themethod overestimates the size of an atom by roughly 100 times.
If we cut a pie in half repeatedly, Sagan points out it takes 90 cuts to get from applepie to atom. That is, an apple pie is 290 times bigger than an apple pie. Howbig is that? It so happens that 210 is 1024, or approximately 1000, a nicetie-in between binary and decimal math. When computer types talk about kilobytes of memory(thousands of bytes), they are actually talking about 1024 bytes. To do this problem, weneed another rule: to raise a number in scientific notation to a higher power, multiplythe two exponents. So 290 = 29x10 = (210)9 = 10009= 1027. That actually overestimates the number of atoms in a pie a bit. Notefor math-phobes: there will be no calculations on the exams.
Does it surprise you that there can be more atoms in a pie than in a roomful of air?Why do you suppose that might be so? Atoms can be thought of for many purposes as spheres,but you can't pack spheres together to fill space completely. In solids, about half thevolume of the solid is actually interatomic space. In liquids, about two-thirds of thevolume is interatomic space - the atoms are moving around a lot more freely. The same istrue, by the way, in the human body, so when your friends say you're not all there,they're absolutely right. In air, though, atoms account for only a couple of percent. Theatoms in air are widely spaced; a typical air molecule can travel about 500 times itsdiameter before hitting another one. Matter is indeed, as Sagan notes, "chiefly madeof nothing."
When we look at the interior of the atom, the emptiness becomes even more amazing. Onceit became clear that atoms could emit smaller particles, researchers began shooting atvarious targets. In experimenting with gold leaf, which is only a few hundred atoms thick,Ernest Rutherford found to his amazement that once in a great while particles were sharplydeflected, even occasionally flung backward. He likened this to seeing an artillery shellbounce off a sheet of tissue paper. Clearly there was some sort of hard object in thecenter of an atom, which Rutherford christened the nucleus. From the fraction ofparticles deflected, it was found that the nucleus is about 1/100,000 the size of theatom. In a scale model with the nucleus a centimeter across, the atom is a kilometeracross.
It was in the Cavendish Laboratory shown in the video that many of the discoveriesabout the interior of the atom were made. As Sagan notes, Physics and Chemistry reduce theworld to an astonishing simplicity of whole numbers. Chemical compounds have whole numbersof atoms, atoms have whole numbers of atomic particles. Pythagoras and his followers wouldhave been delighted.
Atoms consist of swarms of electrically negative particles, electrons,orbiting around a positively charged nucleus about 1/100,000 the diameter of atypical atom. It's the nucleus that determines what an element is. The nucleus consists ofmassive positively charged protons and electrically neutral neutrons. Tokeep everything electrically balanced, the number of electrons equals the number ofprotons. (To see what happens when the numbers get even slightly out of balance, shuffleacross a rug on a cold, dry day and then touch the doorknob.) Each chemical element isdetermined by the number of protons in the nucleus: hydrogen is 1, silicon is 14, iron is26, gold is 79, uranium is 92.
The electron cloud, though negligible in mass, is the part of the atom that interactswith other atoms. The electrons orbit in a shell structure and successive shells havesimilar electron arangements. Thus, if we arrange elements in a table, we can line upelements with similar properties in columns. This is the basis of the familiar PeriodicTable. The nucleus is pretty much inaccessible to the electrons of other atoms, andthe energy binding particles to the nucleus is thousands or millions of times greater thanthat binding electrons to the atom. That's why the alchemists never realized their dreamof changing base metals into gold. They were using only chemical processes to attack thenucleus, about like attacking a tank with a Nerf Bat. We can do it because we have thetechnology to accelerate nuclear particles to the necessary energies.
As Isaac Asimov once put it, had the alchemists succeeded, we would by now have a lotof gold around. But they failed, and in the process laid the basis for chemistry, so wehave dyes and plastics and synthetic fibers and new metals and antibiotics. Which wouldyou prefer?
There are four known fundamental forces in nature:
From time to time there are announcements or speculations about other forces in nature,but so far none have stood up to examination.
Most of us are familiar with the first two; they are long-range forces thatdrop off slowly with distance (one over distance squared to be exact). Many people wonderwhat keeps the protons in the nucleus from flying apart. The answer is that there are two short-rangeforces that are extremely strong within the nucleus but drop off so rapidly with distancethat they are undetectable even in a neighboring atom. We study these forces by slammingatomic particles into each other at high speed. The neutrons in an atom supply the strongnuclear force and help bind the nucleus together. But for nuclei with very large numbersof protons, even the strong nuclear force can't hold things together perfectly well.That's why heavy elements are unstable, and the heavier the atom is, the shorter itslifetime.
These four forces, by the way, help explain why most scientists are skeptical of claimsof the paranormal. To affect matter by mind power, you'd have to do one of two things:generate one of the known forces, or generate an unknown force. If your mind can generateone of the known forces (say transmit radio waves) it should be easily detectable. Ifthere exist unknown forces, then there are a lot of deep questions that nobody has everanswered satisfactorily. How are the forces created? How do they propagate across space?How do they interact with matter. Do they account for any of the effects we normallyascribe to known forces? If yes, which ones? How will we have to modify our view of theknown forces? If no, then what effects do they produce? These are serious issues becausethe four known forces seem to account for just about every physical process we observe.
Information on the Sun
Everything in the universe exists in a state of balance between gravity and some otherforce.
Note: you will have to visit some of the pages on stars and the Sun to answer these:
Return to Professor Dutch's Home Page
Created 23 March 1998, Last Update 9 April 1998