Nature’s most distant reaches have been probed by many new inventions within the last 25 years. Lasers, rockets, accelerators, fantastic photographic capabilities combined with space telescopes have tremendously expanded our vision of what surrounds us. In twenty years computer scientists have gone from speaking in terms of milliseconds (thousandths of a second), to nanoseconds (billionths of a seconds). Physicists can detect gravity waves by an amount of a million trillion smaller than an electron. That’s smaller than anything on the atomic scales. The launching of a space telescope was supposed to enable astronomers to see ten times farther than can be seen with present instruments!
Scientists are now looking at bigger, smaller, older, farther, and faster orders of magnitudes than have ever been known before in the natural world. Today, they are probing phenomena that are as tiny as 1/1,000,000,000,000 of a centimeter in an explorable universe whose edge lies at least 100,000,000,000,000,000,000,000 miles away. We are studying phenomena so short lived that they occur in 1/10,000,000,000,000,000,000,000 of a second. By contrast, astronomers tell us the universe is some 20,000,000,000 years old by using astrometry to mathematically measure stars and masses in space. For scientists, the frontiers of space hold the key to scientific questions about the existence of life on other planets, the real size of the universe, and Earth’s role in it.
The sheer scale of explorable nature has burst beyond our wildest assumptions with incredible proportions. Even as the U.S. surpasses a trillion dollar debt, this amount appears small when compared to our exploding microscopic and telescopic world. Increasingly, students and average people are bombarded with facts and figures of enormous proportions. Such numbers are very awkward to learn and are constantly getting more difficult to comprehend. As our need to deal with more and more zeros increases and decreases, our interest to handle these magnifications is being met by growing indifference. But since we must be prepared to use and manipulate large and small numbers, there must be a way to handle them.
The curriculum
Scaling the Natural World Using Dimensional Analysis
will address the sheer volume of incomprehensible numbers (speed, distance, age) in the natural world. The major goal of this interdisciplinary math and science unit is to enable students to understand the scale of the natural world using the concept of rates, proportions and dimensional analysis. These concepts greatly simplify the learning process for orders of magnitude problem solving in an innovative way. This curriculum should help clarify thoughts about the magnificent scale of the rapidly changing natural world and human beings place within it.
By the end of this curriculum unit, the student will be able to calculate problems such as the following: Measurements indicate that the continents of Europe and North America are separating (plate tetonics) at the rate of about 2 centimeters per year. If Columbus could repeat his famous voyage of 1492, about how many feet or yards must he travel further?
Curriculum Outline:
|
I.
|
Introduction—Magnitude
|
|
A.
|
Cosmological Time Scale
|
|
B.
|
The Universe: Galaxies, Stars, and Planets
|
|
C.
|
Powers of Ten
|
|
B.
|
Millions, Billions, and Trillions
|
|
E.
|
Exponential Notation
|
|
F.
|
Chart of Exponential Notation
|
|
II.
|
Why Use Dimensional Analysis?
|
|
A.
|
Teaching Dimensional Analysis
|
|
1.
|
Definitions
|
|
2.
|
Rules: Single Step Rate Problems
|
|
3.
|
Rules: Multi-Step Rate Problems
|
|
B.
|
Activity Sheets
|
-
1) Calculate the Circumference and Diameter of the Earth.
|
2.
|
How to Calculate a Light-Year
|
|
3.
|
Solar System Distances
|
A. Cosmological Time Scale
Event Years Ago
-
Big Bang 15,000,000,000
-
Solar System/Earth Formed 4,500,000,000
-
Oldest Rocks 3,900,000,000
-
Oldest Fossils 7,900,000,000
-
First Dinosaurs 200,000,000
-
First Flowers 135,000,000
-
End of Dinosaurs 65,000,000
-
Grand Canyon Began Forming 6,000,000
B. The Universe: Galaxies, Stars, and Planets
From the earth’s point of view, the most important aspect of the universe is its immensity. It is so large that the size is utterly meaningless to us. We can write it down, and even use the convenience of exponents (where 1,000,000 = 10
6
), but the figures are too large to have significance. We have all stood outdoors on a clear night and looked at the beauty of the stars and wondered with awe about the nothingness of the great space above us. But in terms of actual distance, all we can manage to understand is that the stars are far away. So is China, yet we know they are farther than China. What we cannot intuitively grasp, without great intellectual effort on our part, is how unbelievably far away they are.
Even the units used in astronomical distances seem beyond our immediate comprehension. A light-year is the distance traveled by light in one year (365 days, or 8,760 hours, or 525,600 minutes, or 31,536,000 seconds). Light travels at approximately 186,300 miles per second. To picture this speed, the circumference of the earth is about 24,000 miles; it would take light less than 1/7 second to go around the world (were it possible to do so). A light-year is approximately 5.9 x 10
12
miles, a distance which is already too great to imagine (See related activity about how to calculate a light-year).
If one light-year is too far to imagine, then what possible meaning can we derive from the fact that the farthest galaxy that can be observed with the most powerful telescope is estimated to be over two billion (2 x 10
9
) light years away? Clearly, the diameter of the entire universe must exceed this figure, but the distance is so large that making it any larger seems insignificant.
The universe contains stars. The stars are not uniformly distributed in space, but they form clusters called galaxies, which are about 100,000 light-years in diameter. In the universe there are thought to be some 10
15
galaxies, and each one of these contains on the average 10
8
stars. Not only are these distances beyond comprehension, but also these numbers.
The galaxies themselves are not evenly spaced throughout the universe; they tend to group together. Therefore, the distance between galaxies varies considerably, and the average distance is about one million light-year!
To continue our size description of the universe, the stars within a galaxy are separated by an average distance of 5 light-years. We are part of a galaxy which we recognize as the Milky Way. It has a diameter of roughly 100,000 light-years and contains about 2 x 10
11
stars. Galaxies have different forms, all presumed to be related to their movement, their rotation. They may be round, flattened elipses, or spirals of different configurations, and in some cases they form rather irregular shapes. Our galaxy is a flattened spiral in which one of the stars, the sun, is about two-thirds away from the center toward the edge. The fact that the Milky Way is a broad band across the sky is an index of the flatness.
The stars themselves vary tremendously in size as well as in their degree of brightness. The sun has a diameter of 864,000 miles. There are stars that are much smaller (1/10 the mass of the sun) and some which are much larger (10,000 times the mass of the sun).
Around each star there may be planets. The sun (diameter is 864,000 miles), for instance, has nine such planets, of which the earth is one. Planets are far smaller than stars, and they differ in their distance from their star, their rates of movement and rotation, their density, their chemical composition, their atmospheres, and in the number of satellites or moons that are in turn orbiting around them. In our solar system Jupiter is the largest planet, with a diameter of 86,000 miles, and Mercury the smallest, with a diameter of 3,100 miles. By comparison, the diameter of the earth is 7,918 miles and the moon 2,160 miles. Mercury is the planet closest to the sun, having a mean distance to the sun of 36 x 10
6
miles. The earth is the third planet from the sun, with a mean distance of 92.9 x 10
6
miles. Pluto is the farthest away, on the average 3,671 x 10
6
from the sun. Expressed in million of miles, the moon is on the average 0.24 x 10
6
miles from the earth.
We have in this picture of the universe a whole series of size levels, beginning with the entire universe, then the clusters of galaxies, the galaxies themselves, the stars, and finally the planets. At each level it is striking that the units are not evenly spaced or randomly distributed, but clearly clustered. This applies to groups of galaxies, stars, and planets.
Again the figures have little impact on our imagination. They do not give any appreciation of the immensity involved. A very vivid description is presented by Robert Jastrow in
Red
Giants and White Dwarfs
. He says:
“An analogy will help to clarify the meaning of these enormous distances. Let the sun be the size of an orange; on that scale of sizes the earth is a grain of sand circling in orbit around the sun at a distance of 30 feet; the giant planet Jupiter, 11 times larger than the earth, is a cherry pit revolving at a distance of 200 feet, or one city block; Saturn is another cherry pit two blocks from the sun; and pluto, the outermost planet, is still another sand grain at a distance of ten city blocks from the sun.
On the same scale the average distance between the stars is 2000 miles. The sun’s nearest neighbor, a star called Alpha Centauri, is 1300 miles away. In the space between the sun and its neighbors there is nothing but a thin distribution of hydrogen atoms, forming a vacuum far better than any ever achieved on earth. The galaxy, on this scale, is a cluster of oranges separated by an average distance of 2000 miles, the entire cluster being 20 million miles in diameter.
An orange, a few grains of sand some feet away, and then some cherry pits circling slowly around the orange at a distance of a city block. Two thousand miles away is another orange, perhaps with a few specks of planetary matter circling around it. That is the void of space.”
C. Powers of Ten
Measuring Tools Distance (Meters) Systems
Telescopes > 10
26
Universe
10
20
Galaxies
10
16
Light Year
10
15
Stars
10
12
Solar System
Eye 10
10
Sun
10
5
Moon
10
3
Kilometers
1
Humans
Microscopes 10
-8
Molecule
10
-10
Atom
Accelerators 10
-14
Nucleus
10
-15
Proton
10
-16
Nucleons
10
-17
Quarks/Leptons
D. Millions, Billions, and Trillions
It used to be that millions was the byword for a large number. The enormously rich were millionaires. The population of the earth at the time of Jesus was perhaps 250 million people. There were almost 4 million Americans at the time of the Constitutional Convention of 1787; by the beginning of World War II, there were 132 million. It is 93 million miles to the Sun. Approximately 40 million people were killed in World War I; 60 million in World War II. The global nuclear arsenals today contain an equivalent explosive power sufficient to destroy 1 million Hiroshimas. For many purposes and for a long time, “million” was the extremely big number.
But times have changed. Now the world has many billionaires—and not just because of inflation. The age of the Earth is well-established at 4.5 billion years. The human population of our planet is 5 billion people and, by the turn of the century, may be between 6 and 7 billion people. The Voyager spacecraft is roughly 2 billion miles from Earth. The U.S. defense budget is around $300 billion a year. The immediate fatalities in an all-out nuclear war are estimated to be around a billion people. There are billions of stars and galaxies. On the other hand, a few inches contains billions of atoms side by side.
While millions and billions have not faded, these numbers are becoming somewhat small-scale. The new number on the horizon and appearing in everyday language is the trillion.
World military expenditures are now over $1 trillion a year. The total indebtedness of all developing nations to Western banks is around $1 trillion. The annual budget of the U.S. government has reach $1 trillion. The national debt is more than $2 trillion (amount the U.S. government owes banks, businesses, and other countries). The distance from our solar system to the nearest star, Alpha Centauri, is 25 trillion miles.
Confusion among million, billions, and trillion goes on every day. An easy way to determine what large number is being discussed is simply to count up the zeros after the one. But if there are many zeros, this can get a little tedious. That’s why we put commas, or spaces, after each group of three zeros. So a trillion is 1,000,000,000,000 or 1 000 000 000 000. For numbers bigger than a trillion, you have to count up many triplets of 0’s there are. It would be much easier if, when we name a large number, we could just say how many zeros there are after the one.
E. Exponential Notation
Scientists and mathematicians, being practical people, have created a system to deal with extremely large numbers. It’s called scientific notation. You write down the number 10; then a little number, written above and to the right as a superscript, tells how many zeros there are after the one. Thus 10
6
= 1,000,000. 10
9
= 1,000,000,000. 10
12
= 1,000,000,000,000; and so on. These little superscripts are called exponents or powers; for example, 10
9
is described as “10 to the power 9” or equivalently, “10 to the ninth” (except for 10
2
and 10
3
which are called “10 squared” and “10 cubed,” respectively). This phrase, “to the power”—like “parameter” and a number of other scientific and mathematical terms—is creeping into everyday language, but with the meaning distorted.
In addition to clarify, exponential notation has a wonderful side benefit: You can multiply any two numbers just by adding the appropriate exponents. Thus 1000 x 1,000,000,000 is 10
3
x 10
9
= 10
12
. Or take some larger numbers: If there are 10
11
stars in a typical galaxy and 10
11
galaxies, there are 10
22
stars in all the galaxies.
But there is still resistance to exponential notation from people a little nervous about mathematics even though it simplifies, not complicates our understanding.
The first six big numbers that have their own name are in the chart on the next page. Each is 1 000 times bigger than the one before. Above a trillion, the names are almost never used. You could count one number every second, day and night, and it would take you more than a week to count from one to a million. A billion would take you half a lifetime. And you couldn’t count to a quintillion even if you had the age of the universe to do it in.
Once you’ve mastered exponential notation, you can deal effortlessly with immense numbers, such as the rough number of microbes in a teaspoon of soil (10
8
); of grains of sand on all the beaches of the Earth (10
20
); of living things on the earth (10
29
); of atoms in all the living things on Earth (10
41
); of atomic nuclei in the Sun (10
57
). This doesn’t mean you can picture a billion or a quintillion in your head—nobody can. But with exponential notation, you can think about such numbers when trying to understand the incredible scale of nature
F. Exponential Notation Chart
Number (U.S.)
One
Thousand Million
Billion
Trillion
Quadrillion
Quintillion
Number (written out)
1
1,000
1,000,000
1,000,000,000
1,000,000,000,000
1,000,000,000,000,000
1,000,000,000,000,000,000
Number (Scientific Notation)
10
0
10
3
10
6
10
9
10
12
10
15
10
18
How long it would take to count to this number from 0 (one count per second. night, and day)
1 second
17 minutes
12 days
32 years
32,000 years
32 million years
32 billion years
Larger numbers are called a sextillion (10
21
), septillion 10
24
), octillion 10
27
), Nonillion 10
30
), Decillion 10
33
), and 10
100
).
