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Geometric progression was studied over 3,600 years ago by the ancient Egyptians and Sumerians and much later by the Greeks, including Euclid (C. 300 B.C.) and Archimedes (287?–212 B.C.).

A Brief History of Discovery and Development

Especially after the Middle Ages, scientists became aware of many real-life objects that behave in a geometric

or exponential way, including the unrestrained breeding of animals; the cooling of hot objects; the shapes of natural spirals such as those found in pine cones, sunflowers, and ram's horns; the dimming of supernovae (exploding stars); the relationships between musical notes; and many others. Ancient records inscribed in clay show that in the Middle East, the Sumerians knew about the exponential properties of compound interest as early as 2000 B.C.

Our modern way of writing an exponent-placing a small number above and to the right of another number—was introduced in 1637 by the French philosopher and mathematician René Descartes (1596–1650). At abut that time the relationship between the logarithm and the exponent (namely, that they are inverses of each other) was finally clarified.

In the eighteenth century, Swiss mathematician Leonhard Euler (1701–1783) first devised the complex exponential function, where a base is raised to the power of an "imaginary" number containing the square root of −1. The square root of −1 was a radical new idea because it seemed impossible: what number, when multiplied by itself, could give −1? The square root of +1 is simply itself (because 1² = 1), but −1 cannot be its own square root (because −1² = −1 × −1 = +1). Nevertheless, mathematicians have found numbers containing the square root of −1, called "complex" numbers, to be very useful. Euler also explored the use of the number e = 2.7182818 as a "natural" (that is, highly convenient) base for exponents and logarithms.

Today, exponents are applied throughout mathematics. They are used in physics and engineering to describe phenomena that fade with time, such as radioactivity, or periodic (repeating) phenomena like waves. They are used in biology to model populations of bacteria, animals, and people; in medicine to model the breakdown of drugs in the body; and in business and economics to describe interest and inflation.

Real-life Applications

SCIENTIFIC NOTATION

With scientific notation you can write down a number greater than the number of atoms in the universe with just a few strokes of your pen. This is how:

Recall that for powers of ten, the exponent gives the number of zeroes: 100 = 10², 1000 = 10³, and so forth. Using this fact, an ugly number like 1,000,000,000,000, 000,000 becomes a user-friendly 10¹⁸. We can also write a number like 1,414,000,000,000,000,000 as 1.414 times 10¹⁸, namely, 1.414 × 10¹⁸. A number written in this form is said to be in "scientific notation." Scientific notation makes large numbers much easier to handle.

The same trick also works for small numbers, because numbers like .1, .01, .001, and so forth can be written as tens raised to negative exponents; for example,

We can therefore write 10⁻²⁰ instead of .0000000000 0000000001, and 1.675 × 10⁻²⁴ instead of .0000000000 00000000000001675 (which happens to be the mass in grams of a single hydrogen atom).

Another useful feature of scientific notation is that changing the exponent is shorthand for moving the decimal point. Thus we write 4.5 × 10⁻² for .045 and 4.5 × 10⁻⁴ for .00045.

To multiply two numbers written in scientific notation, all we have to do is multiply the numbers out front and add the exponents. So, 1.414 × 10¹⁸ times 1.675 × 10⁻²⁴ is 1.414 × 1.675 × 10¹⁸⁻²⁴ = 2.36845 × 10⁻⁶. This is so easy that when multiplying simple numbers like −1 × 10¹² and 5 × 10⁹, it's actually easier to do the math in your head than to punch buttons on a calculator provided you can add 12 and 9 in your head without blowing a fuse. (The answer is (1 × 10¹²) × (5 × 10⁹) = (1 × 5) × 10¹²⁺⁹ = 5 × 10²¹.) Division is equally easy, only you divide the numbers out front and subtract the exponents.

As for writing down a number larger than the number of atoms in the universe, this is it: Physicists estimate that there are fewer than 10¹⁰⁰ atoms in the Universe, perhaps only about 10⁸⁰. If every atom in the Universe were inflated into a universe full of atoms, there would still be only (10⁸⁰)² or 10¹⁶⁰ atoms in existence. So writing 10¹⁶⁰, or 10²⁰⁰, or 10³⁰⁰ gives a number much, much greater than the number of atoms in the Universe.

EXPONENTIAL GROWTH

A useful fact in science (and banking) is this: Any quantity that grows by a fixed percentage during each interval of time grows exponentially.

Consider a pair of rabbits. Say that this pair has two offspring by the end of one year. There are now four rabbits: the population has doubled in one year. Assume that both pairs will breed the following year, each producing two more offspring, and that their offspring will also breed, and so on, so that the total population keeps on doubling every year. This is the same as saying that the population increases by a fixed percentage every year, namely 100%: 2 rabbits plus 100% of 2 rabbits equals 4 rabbits (first year's population growth), 4 rabbits plus 100% of 4 rabbits equals 8 rabbits (second year's growth), and so forth. The growth of this rabbit population is described by an exponential equation. If we label the first year Year 0, then the number of rabbits at the beginning of each year, R(t). is given by the following series of numbers: R(0) = 2, R(1) = 4, R(2) = 8, R(3) = 16, and so on. In general, 2R(t) = 2 × 2^t, where t is years.

Figure 3 shows the number of rabbits as an exponential function of time, starting with two rabbits and assuming a doubling time of 1 year. This curve is similar to the part of Figure 1 to the right of x = 0.

Do rabbits really do this? Sure—when they can; that is, if they have food to eat, room to live in, and air to breathe, and no enemies or diseases nasty enough to keep the population from growing. In reality, rabbits cannot have all these things, certainly not in infinite amounts. There are predators and diseases; there is only so much food, water, and room. So in one sense the exponential equation is not realistic—but in another sense, it is terribly realistic. It forces conflict between growing populations and their environments. If a population seeks to increase by any fixed percentage per year (that is, if it seeks to grow exponentially), then at some point

deaths—whether from hunger or other causes—will inevitably outpace births. To see why this cannot be avoided, let's use the master rabbit equation, R(t) = 2 × 2^t, to look at what would happen if our imaginary rabbit population was somehow, magically, able to keep on growing.

Let's calculate how long it takes to get a given number of rabbits, N. To do so, we find the "solution" to the equation 2 × 2^t = N, that is, that unique value of t for which the equation is true. Let's pick a nice, big value for N—say, enough rabbits to completely fill the Solar System. Pluto, usually regarded as the outermost planet, has an average distance from the Sun of 5.914 × 10¹² km. Because the volume of a sphere of radius r is

(exponents again!), the volume of the Solar System is

where 1 m³ is a cubic meter (the amount of space in a cube 1 meter across). If we can pack 50 rabbits into each cubic meter of space, then the number of rabbits that can fit into the Solar System is

Because the number of rabbits at the start of year t is N = 2 × 2^t, to find out the number of years till there are

rabbits we need to find t such that

To find the t that satisfies this equation, we must perform the mathematical operation known as "taking the logarithm" of both sides. Taking the logarithm undoes

exponentiation (applying exponents) in much the same way that subtraction undoes addition or division undoes multiplication.

Solving using logarithms, we find that t equals approximately 129 years. This means that a rabbit population that doubled every year would fill the whole Solar System with long-eared rodents in only 129 years. It would take the first 128 of those years to fill half the Solar System, but just one more year to fill the other half!

A Solar System full of rabbits is, of course, physically impossible. The moral is that exponential population growth always runs up against physical limits, most often getting eaten or starving to death.

The equation R(t) = 2 × 2^t is an example of the general exponential equation R(t) = R₀b^t, where b is some positive number and R₀ is the value of R(t) at t = 0 (because R(0) = R₀ × b⁰ = R₀ × 1 = R₀). We'll use this form of the exponential equation in the following application also.

ROTTING LEFTOVERS

Any quantity—say a population of rabbits, or of people—that grows by a fixed percentage each year, no matter how small, will double over some period of time. It will then double again after an equal period of time, and so on forever. Every exponentially growing quantity grows in this way, so every exponentially growing quantity is said to have a "doubling time."

Suppose you leave a dish of food out at midnight. The dish happens to have 10 bacteria sitting on it. Suppose also that this population of bacteria increases by 4% every minute. How long will it be before the number of bacteria in the dish doubles? And how many bacteria will you be consuming when you finish off the leftovers at noon the next day?

If the population is growing by 4% every minute, then it is growing exponentially, and can be described by an exponential equation of the form R(t) = R₀b^t, just like the rabbit population in the previous example. We already know R₀, the number of bacteria at time t = 0; it's 10. But what is b?

Besides the fact that R₀ = 10, we also know that the bacterial population at the end of 1 minute, R(1), is 4% greater than at t = 0, because we're told that the population is growing by 4% every minute. This fact can be written down as R(1) = 10 + (.04 × 10) = 10.4.

We also know that R(1) must be given by the exponential equation R(t) = R₀b^t with 10 plugged in for R₀ and 1 plugged in for t, namely, R(1) = 10b¹ = 10b. We can now set this expression for R(1) equal to the number found in the previous paragraph: 10b = 10.04.

Dividing both sides by 10 to solve for b, we find that b = 1.004.

We now have both R₀ and b, and so can write down the exact exponential equation that describes this bacterial population: R(t) = R₀b^t = 10 × (10.004^t).

We can now return to our first question: What is the doubling time? Let us call that unknown number of minutes T_D. Because we start out with 10 bacteria, the number of bacteria after the first doubling time will be 20 (double). So the population at time T_D is given by R(T_D) = 20 = 10 × 1.004_TD. To solve for T_D, we must "take the logarithm" of both sides of this equation. Taking the logarithm undoes the exponential operation much the way that subtraction undoes addition or division undoes multiplication (see chapter on Logarithms). We find that T_D = 173.63 minutes (about 2 hrs 54 minutes).

The equation R(t) = 10 × (10.004^t) also tells us how many bacteria you'll be eating at noon the next day. All we need to know is the number of minutes between midnight and noon, which is 12 hours times 60 minutes per hour, or 720 minutes. Thus, R(720) = 10 × 1.004⁷²⁰, or about 177 bacteria.

Not bad, really. As you read this, your intestines contain trillions of bacteria. But remember the power of exponential growth. After 24 hours there will be 3,137 bacteria; after 48 hours, 984,205 bacteria; and after 3 days, 3.088 × 10⁸ bacteria, about as many people as there are in the United States. Look out for a stomach upset—or worse.

EXPONENTS AND EVOLUTION

Predators and sickness often keep populations from growing exponentially, but if they don't there is one thing that certainly will: hunger. There can never be an infinite food supply—even if you could somehow turn the whole Earth into a ball of food, it would be limited. Therefore, sooner or later, any exponentially growing population must run out of food and either stop breeding or start starving.

This principle was first clearly explained English economist and minister Thomas Robert Malthus (1766–1834). In his 1798 book, An Essay on the Principle of Population, Malthus wrote: "Population, when unchecked, increases in a geometrical ratio [exponentially]. Subsistence [that is, food supply] increases only in an arithmetical ratio [like a straight line]. A slight acquaintance with numbers will show the immensity of the first power in comparison of the second….This implies a strong and constantly operating check on population from the difficulty of subsistence."

By "a slight acquaintance with numbers" Malthus meant a knowledge of exponents. Population increases exponentially ("in a geometrical ratio") whenever it can, rising in a curve that gets ever steeper; but food supply increases (if it increases at all, say by the clearing and planting if more cropland) approximately as a straight line, that is, according to a "linear" function or "arithmetical ratio." And any exponential function will eventually outrun any linear function. Accordingly, any freely-breeding population must eventually outrun its food supply.

Malthus was talking about the human population, but his logic applies to any biological population. Two English biologists, Charles Darwin (1809–1882) and Alfred Russel Wallace (1823–1913)—realized this when they read Malthus's book in the mid-1800s. Both Darwin and Wallace were trying to think of a mechanism to explain biological evolution, the process whereby new species of animals and plants arise from older ones. People had been suggesting theories of evolution for years, but none of them could explain why evolution happened. However, Malthus's reasoning about population growth triggered a fresh insight for both Wallace and Darwin. Working separately, they realized that the potential of every species for exponential population growth guaranteed struggle between organisms. In biology, "struggle" usually means not fighting, but competition to leave more offspring than one's rivals. In an article published jointly with Wallace in 1858, Darwin said, echoing Malthus: "[T]he amount of food for each species must, on an average, be constant, whereas the increase of all organisms tends to be geometrical, and in a vast majority of cases at an enormous ratio ….Now, can it be doubted, from the struggle each individual has to obtain subsistence, that any minute variation in structure, habits, or instincts, adapting that individual better to the new conditions, would tell upon its vigour and health? In the struggle it would have a better chance of surviving; and those of its offspring which inherited the variation, be it ever so slight, would also have a better chance….Let this work of selection on the one hand, and death on the other, go on for a thousand generations, who will pretend to affirm that it would produce no effect … "

Wallace and Darwin's insight was that competition (made inevitable by exponential population growth) is more than just a check or limit on population: it forces nature to filter the chance changes that show up in every generation of creatures and so acts as a creative force, helping sculpt such marvels as the gull's wing, the eagle's eye, and the human brain.

RADIOACTIVE DECAY

In Nevada, about 90 miles (145 km) northwest of Las Vegas, stands an unremarkable-looking ridge of dry, brown rock, owned by the Federal government and known as Yucca Mountain. This is where the United States government hopes to bury 77,000 tons (69,853 tonnes) of highly radioactive nuclear waste from around the U.S. (about 60% of the total amount that had built up as of 2004).

This waste is what is left over when a nuclear power plant has finished using uranium fuel to produce electricity. Plans call for it to be mixed with molten glass and cooled to a solid ("vitrified"), sealed inside rust-resistant metal containers, and parked along 73 miles (117 km) of branching tunnels located 1,000 feet (305 m) below the surface of Yucca Mountain. When the tunnels are full, they will be sealed off and hopefully not entered again—especially by water, which might carry the waste back to the surface—for at least 10,000 years.

Why does the waste need to be put in such a special place at all? And why for as long as 10,000 years—or for only 10,000 years? Why not forever?

The Reason is Radioactive Decay

A "radioactive" substance is one whose individual atoms break apart (fission) at random (chance) times, releasing energy. This energy takes the form of fast-moving particles or invisible rays that can cause cancer or other sickness. Some radioactive atoms are mixed naturally with the environment, whereas some are human-made. Regardless of where they come from, the fewer radioactive atoms we come in contact with, the better for our health. (Some medical tests and treatments do use radioactive substances, however, where the gain is thought to be larger than the risk.)

Radioactive substances disappear over time as their atoms change into other types of atoms. This change in atoms is called "radioactive decay." Like the curve in Figure 2, radioactive decay can be described by an exponential equation with a base between 0 and 1. Someday, therefore, all the nuclear waste generated today will be harmless, but that date is tens or hundreds of thousands of years in the future.

Just as every exponentially increasing process has a doubling time, so every exponentially decaying process has a halving time. In the case of a radioactive substance, this halving time is called the substance's "half-life." Half the atoms in a lump of any radioactive substance will have changed into other substances after one half-life of that substance. Different radioactive substances have different half-lives. Half-lives vary from a tiny fraction of a second up to billions of years.

Consider the substance plutonium 238. Plutonium 238 (also written²³⁸Pu) is both poisonous and radioactive. It can be used as fuel for nuclear reactors or to make nuclear bombs. It is one of the ingredients in radioactive waste of the type that may someday be stored beneath Yucca Mountain (perhaps starting in 2010). ²³⁸Pu has a half-life of about 25,000 years. If the amount of ²³⁸Pu that we start out with at time t = 0 is Q₀ tons, then the amount at some later time t will be described by the exponential equation Q(t) = Q₀k^t. Because we know that after the first 25,000 years there will be half as much ²³⁸Pu as at time t = 0, we can write the following:

Because Q₀ appears on both sides of the equal sign, it cancels out. We can then solve for k using logarithms. We find that k = .999972. The radioactive decay of 1 ton of ²³⁸Pu is shown in Figure 4.

In Figure 4, the radioactive decay of 1 ton of plutonium 238 (²³⁸Pu). Time is shown in units of half-lives. Notice that at t = 1 (one half-life) the amount of ²³⁸Pu is down to

1/2 ton. To read the time axis in units of years, multiply by 25,000. Note that this curve is exactly the same (except for vertical scale) as the part of Figure 2 to the right of x = 0.

As a rule of thumb, experts often say that we should wait at least 10 half-lives of a radioactive substance before considering that a chunk of it is harmless. In the case of plutonium, 10 half-lives are about 250,000 years. How much of an original quantity of any radioactive substance, say 1 ton, is left after 10 half lives? Since waiting one half-life cuts the amount in half, and waiting two half-lives cuts that amount in half, and so on, we can use exponents. A little thought shows that the fraction that is left after ten half-lives is

That is, after 10 half-lives, only .0009766 tons (about one one-thousandth of the original ton) will be left.

The wastes intended for Yucca Mountain contain many radioactive substances besides plutonium. Many of these have much shorter half-lives than plutonium, so in about 300 years 99% of the radioactivity of the nuclear waste will have disappeared. After 10,000 years, the amount of time that the Yucca Mountain storage tunnels are supposed to be guaranteed for, these shorter-lived substances will be essentially gone. On the other hand, after 10,000 years only about one-fourth of the plutonium will be gone.

RADIOACTIVE DATING

"Radioactive dating" is an essential technique in geology and archaeology. By looking at the amounts of radioactive substances embedded in rocks or other objects and at the amounts of breakdown products (substances left over from radioactive decay) that are mixed with them, scientists can tell how much radioactive material must have been originally present in the object—and thus, by the exponential equation, how old the object is. For example, if a rock contains 1 gram of radioactive uranium 238 (²³⁸U) and 1 gram of lead, which is a breakdown product, it is probable that the rock originally contained 2 grams of ²³⁸U. Since ²³⁸U has a half-life of about 4.5 billion years, it takes about 4.5 billion years for 2 grams to decay to 1 gram, so we deduce that this particular rock is about 4.5 billion years old.

In practice, breakdown products and radioactive dating are more complex than this. Scientists must look at many different samples of rock (or wood, or whatever material they are dating) and at a number of different radioactive substances and breakdown products, rather than just one. But by combining methods and measuring many different objects, error can be minimized. Through radioactive dating, scientists have verified that the Earth, the Moon, and most meteorites are about 4.5 billion years old. That is, about half the ²³⁸U that was present when the Solar System formed has turned into other elements. Some of it, in fact, has ended up in your car battery.

INTEREST AND INFLATION

Let's say that you earn $100 at a weekend job. Your parents insist that you put it in a nice, safe bank until you're 18. They try to comfort you with the idea that your money will earn interest, so you will end up with more money if you wait. How much more? And what exactly does it mean to "earn interest," anyway?

"Interest" is money that is paid to you by a bank in which you have deposited money. The bank invests the money in enterprises that it thinks will be worth more in the future. Banks make profit by taking in more money on their investments than they pay in interest to their depositors (that's you), but regardless of how well a bank's investments are doing, it is obliged to pay you the agreed-upon interest.

To pay interest, the bank looks at the money in your account at regular intervals, say every three months. It then calculates a fixed percentage of that amount (your interest) and adds this money to your account. (At the words "regular intervals" and "fixed percentage" your ears should prick up: "Regular intervals? Fixed percentage? My money will grow exponentially?" Yes, but wait.) The percentage used to calculate your interest is called the "interest rate."

After another three months (or whatever the interval happens to be), the bank calculates the fixed percentage again and adds it to your account. The interest from the previous time interval—also called a "conversion period"—earns interest during the next time interval, assuming that you haven't taken any money out. This arrangement, where interest earns interest, is called "compound" interest.

Let's go back to your $100. Assume the conversion period is three months (which is one quarter of a year, so it's also called a "quarter"). You get a quarterly interest rate of 1.5%, so the end of the first quarter, the bank adds 1.5% of $100 to your account, namely, __BODY__.50. Your account now contains $101.50. At the end of the second quarter, the bank calculates 1.5% of $101.50, which is __BODY__.52 (rounding down), and it adds that to your account. Your account now contains $103.02. Notice that the amount of interest you receive at the end of the second quarter is larger than the interest you receive at the end of the first quarter. The reason is that you've begun to earn interest on your interest.

Not surprisingly, this is an exponential process. Its equation is S(n) = P (1 + r)ⁿ. Here S(n) is the amount of money in your account after n quarters, P is your principal (the money you start off with, in this case $100), and r is the quarterly interest rate (1.5%, in this case). Since time, n, is in the exponent, this is an exponential function. Putting in our numbers for P, r, and n, we find that S(n) = 100 (1 + .015)ⁿ = 100 × 1.015ⁿ.

For the end of the second quarter, n = 2, this gives the result already calculated: S(2) = $103.02.

This equation for S(n) should look familiar. It has the same form as the equation for a growing population, R(t) = R₀b^t, with R₀ set equal to $100 and b set equal to 1.015.

If $100 is put in the bank when you're 14, then by the time you're 18, four years or 16 quarters later, it will have grown exponentially to $100 × 1.015¹⁶ = $126.90 (rounded up). If you had invested __BODY__,000, it will have grown to __BODY__,268.99. That's lovely, but meanwhile there's inflation, which is exponentially making money worth less over time.

Inflation occurs when the value of money goes down, so that a dollar buys less. As long as we all get paid more dollars for our labor (higher wages), we can afford the higher prices, so inflation is not necessarily harmful. Inflation is approximately exponential. For the decade from 1992 to 2003, for example, inflation was usually around 2.5% per year. This is lower than the 6% per year interest rate we've assumed for your invested money, so your $100 of principal will actually gain buying power against 2.5% annual inflation, but not as quickly as the raw dollar figures seem to show: after four years, you'll have 26% more dollars than you started with ($126.90 versus $100), but prices will be 10.4% higher (i.e., something that cost $100 when you were 14 will cost about $110 when you are 18).

Furthermore, 6% is a rather high rate for a savings account: during the last decade or so, interest rates for savings accounts have actually tended to be lower than inflation, so that people who keep their money in interest-bearing savings accounts have actually been losing money! This is one reason why many people invest their money in the stock market, where it can keep ahead of inflation. The dark side of this solution is that the stock market is a form of gambling: money invested in stocks can shrink even faster than money in a savings account, or disappear completely. And sometimes it does.

CREDIT CARD MELTDOWN

When you deposit money in a bank, the bank is essentially borrowing your money, and pays you interest for the privilege of doing so. When you borrow money from a bank, you pay the bank interest, so if you don't pay off your debt, it can grow exponentially. Exponential interest growth is why credit-card debt is dangerous. A credit-card interest rate, the percentage rate at which the amount you owe increases per unit time, is much higher than anything a bank will pay to you. (Fifteen percent would be typical, and if you make a late payment you can be slapped with a "penalty rate" as high as 29%.) So if you only make the minimum monthly payments, your debt climbs at an exponential rate that is faster than that of any investment you can make. This is why you can't make a living by borrowing money on a credit card and investing it in stocks. If you could, the economy would soon collapse, because everyone would start doing it, and an economy cannot run on money games; it needs real goods and services.

Those high credit-card interest rates are also the reason credit-card companies are so eager to give credit cards to young people. They count on younger borrowers to get carried away using their cards and end up owing lots of fat interest payments. And it seems like a good bet. In 2004, the average college undergraduate had over __BODY__,800 in credit-card debt.

The good news is that to avoid high-interest credit-card debt, you need only pay off your credit card in full every month.

THE AMAZING EXPANDING UNIVERSE

The entire Universe is shaped by processes that are described by exponents.

All the stars and galaxies that now speckle our night sky, and all other mass and energy that exists today, were once compressed into a space much smaller than an atom. This super-tiny, super-dense, super-hot object began to expand rapidly, an event that scientists call the Big Bang. The Universe is still growing today, but at different times in its history it has expanded at different speeds. Many physicists believe that for a very short time right after the Big Bang, the size of the Universe grew exponentially, that is, following an equation approximately of the form R(t) = Ka^t, where R(t) is the radius of the Universe as a function of time and t and K and a are constants (fixed numbers). This is called the "inflationary Big Bang" theory because the Universe inflated so rapidly during this exponential period. If the inflationary theory is correct, the Universe expanded by a factor of at least 10³⁵ in only 10⁻³² seconds, going from much smaller than an electron to about the size of a grapefruit.

This period of exponential growth lasted only a brief time. For most of its 14-billion year history, the Universe's rate of expansion has been more or less proportional to time raised to the 2/3 power, that is, R(t) = Kt^2/3. Here R(t) is the radius of the universe as a function of time, and K is a fixed number.

Most scientists argue that the Universe will go on expanding forever—and that it's expansion may even be accelerating slowly.

WHY ELEPHANTS DON'T HAVE SKINNY LEGS

The two most common exponents in the real world are 2 and 3. We even have special words to signify their use: raising a number to the power of 2 is called "squaring" it, while raising it to the power of 3 is called "cubing" it. These names reflect the reasons why these numbers are so important. The area of a square that is L meters on a side is given by A = L², that is, by "squaring" L, while the volume of a cube that is L meters on a side is given by V = L³, that is, by "cubing" L.

These exponents—2 and 3—appear not only in the equations for the areas and volumes of squares and cubes, but for any flat shapes and any solid shapes. For example, the area of a circle with radius L is given by A = πL² and the volume of a sphere with radius L is given by 4/3 πL³. The equations for even more complex shapes (say, for the area of the letter "M" or the volume of a Great Dane) would be even more complicated, but would always include these exponents somewhere—2 for area, 3 for volume. We say, therefore, that the area of an object is "proportional to" the square of its size, and that its volume is proportional to the cube of its size.

These facts influence almost everything in the physical world, from the shining of the stars to radio broadcasting to the shapes of animals' legs. The weight of an animal is determined by its volume, since all flesh has about the same density (similar to that of water). If there are two dogs shaped exactly alike, except that one is twice the size of the other, the larger dog is not two times as heavy as the smaller one but 2³ (eight) times as heavy, because its volume is proportional to the cube of its size. Yet its bones will not be eight times as strong. The strength of a bone depends on its cross-sectional area, that is, the area exposed by a cut right through the bone. The bigger dog's bones will be twice as wide as the small dog's (because the whole dog is twice as big), and area is proportional to the square of size, so the big dog's bones will only be 2² (four) times as large in cross section, therefore only four times as strong. To be eight times as strong as the small dog's bones, the big dog's bones would have to be the square root of 8, or about 2.83 times wider.

You can probably see where this is leading. An elephant is much bigger than even a large dog (about ten times taller). Because volume goes by the cube of size, an elephant weights about 10³ = 10 × 10 × 10 = 1000 times as much as a dog. To have legs that are as strong relative to its weight as a dog's legs are, an elephant has to have leg bones that are the square root of 1,000, or about 31.62 times wider than the dog's. So even though the elephant is only 10 times taller, it needs legs that are almost 32 times thicker. If an elephant's legs were shaped like a dog's, they would snap.