Number Sense
Don’t go around saying the world owes you a living. The world owes you nothing, it was here first. – Mark Twain
Slipping between thousands, millions, billions and trillions without stopping to gasp in proper reverence for the sheer enormity of these relative sizes is like witnessing someone from the average family listening, simply nodding, as the president discusses a 7 trillion dollar federal debt and that average family’s income is less than $50,000 per year. And then it is too late. In fact, most people do just this they slip seamlessly between huge numbers without the sobering knowledge of the relative sizes they are discussing. Over 500 billion cigarettes are smoked yearly in this country. A ¼ of a million people die each day on this planet. While you can see over 2 thousand stars on a moonless night, there are some 100 billion stars in our galaxy alone, the nearest of which is about 26.5 trillion miles away from our sun, the brightest of which is about 1,000 trillion times brighter than our sun. The 45,000 American soldiers that died during the Vietnam War is approximately equal to the number of fatalities from motor vehicle accidents in this country each year. There are nearly 295 million United States citizens; the over 200 million privately owned firearms owned in the United States comes to nearly 2 guns for every 3 U.S. citizens; the nearly one-third of a trillion dollars of the Annual Defense Department budget comes to nearly 1 thousand dollars for every U.S. citizen. Over 50 trillion pounds of TNT equivalence of the world’s nuclear weapons co-exist on this planet with more than 6.4 billion people, over 8,000 pounds for every man, woman and child. There are roughly 2 million US millionaires, while 35.8 million US citizens live below the poverty level. World-wide, there are 7.3 million millionaires with a net worth of 27.2 trillion dollars, while over 1 billion people earn less than 1 dollar per day.
Let’s take any one of these statistics and mathematically think it through to its logical conclusion. Bear in mind, what we call mathematical, you may call intuition. It’s a technique we use every day. With a little mathematical approximation, we will see if it makes sense to accept the validity of this statistic that sadly 45,000 people die in traffics accidents each year. For starters, to make the estimation quick, let us begin by saying there are 50 states, thus 45000/50 gives us roughly 900 traffic fatalities per state yearly. Certainly, states with larger populations like California are prone to more accidents than the less populated states, like Wyoming. For the sake of expediency and ease of mental estimations, we can assume 900 traffic fatalities per year for each state. There are 365 days in a year, and if we average the 900 fatalities per year over the number of days in a year, we have 900/365, which is approximately 2 ½ . What does 2 ½ represent? For the average state, between two to three people are killed due to traffic accidents each day. Does this seem reasonable? Again, on any given day, for the larger states there may be more accidents and for the smaller states less. During holiday season or when there are severe accidents, more. The sad fact is, 45,000 traffic fatalities per year, unfortunately does seem accurate.
To be utterly honest, this is not a problem unique to mathematics. The vastness of time is to the geologist as the enormity of space is to the astronomer. For most of us, comprehending the sheer size of space or vastness of time seems mind boggling, and at first glance it may appear to be something beyond our comprehension. But let us take a moment to examine this wonderfully seemingly limitless world around us.
Geology: Vastness of Time Five thousand years of recorded human history seems like an awfully long time, but recall, humans first appeared nearly two million years ago. When we are talking about geologic processes, this amount of time is a mere blip on the geologic time scale. The earth itself is thought to be 3000 times older than when man’s first footsteps plodded forth.
Do you recognize the relative sizes of numbers like 5 billion, 700 million, 200 million, 2000, and 400? Sure, you can order them, but how much bigger is 5 billion than 700 million, or 700 million to 2000? In other words, do you have an ‘intuitive feel’ for the relative sizes of these numbers. Geologists have been battling this very crusade of trying to educate the public into realizing the comparative magnitude of geologic time. The most popular tool they employ is the comparison technique. They begin with comparing the earth’s 5 billion year age to one week’s worth of time.
Let’s compare. Let’s assume one week is the age of the earth. Starting with Monday at 12:00:01 am, the other celestial bodies are singing “Happy Birthday” to the earth. So, on this time line, present day (right now) is Sunday night, midnight. Well, from Monday through Saturday, very little of anything noteworthy occurred. To be honest, very little of the earth’s formational processes prior to evidence of life routinely makes the evening news or the front page, but a mere 700 million years ago, the first fossil animals appeared. Let’s ask ourselves, when did this “700 million years ago” occur in the course of our week? Fractions (which students and most members of society try to avoid) now become our ally. We are going to look at the ratio of what 700 million years is to 5 billion years and then we will know what proportion of time is equivalent to our one week time period. That period of time we will remove from the end of our week.
So, if 1 week is to 5 billion years, then what proportion or how much of that week is to 700 million years? We will estimate 700 million out of 5 billion. Working with the ratio 700,000,000/5,000,000,000 is preposterous to deal with. We will reduce this ratio to 700/5000 or 7/50. We can use 7/49 for a mental estimation, or 1/7 of a week, which translates to one day of the week. Early, very early Sunday morning, the last day of the week, the first fossil evidence of life appears on earth. See the power of estimation! Let’s continue.
Some 220 million years ago, the first dinosaurs appeared. Using our estimation technique again with respect to the age of the earth, we construct a ratio 220/5000. We can approximate this as 200/5000 = 2/50=1/25. OK, now what? We need to know how much time corresponds to 1/25 of a week and remove it from the end of the week. Well, 7 days divided into 25 parts is about 1 day for every 3 ½ parts or 24 hours for every 3 ½ parts or about 1 part per 7 hours. Thus, 7 hours before midnight, or around 5 pm Sunday evening, the dinosaurs first appeared.
Next, let’s tackle the number 2,000 from our introductory paragraph. The Birth of Christ was roughly 2,000 years ago. The question before us, in the development of our timeline, if the earth’s age compressed to a week, where is the Birth of Christ located on the timeline? We continue by reiterating the question, what is 2000/5,000,000,000 of a week? Or more specifically, what proportion of time is the ratio of 2000 years to 5 billion years with respect to one week? We will simplify the ratio by dividing 1,000 into the numerator and denominator, and re-ask the question, what amount of time is two-five millionths (2/5,000,000) of a week?
This is as an unfriendly a ratio as most of us can imagine, so we will enhance our comparison method by using a method chemists employ. Chemists use the ‘mole method’ or ‘stoichiometry’, mathematicians use a similar method to tackle unit conversions.
First, let’s firm up our language, specifically, what amount of time is 2/5,000,000 of a week ago? Including the units with the ratios, we have:
days ago.
Notice, the ‘weeks’ unit cancelled. In the process of providing a more familiar context for this small amount of time, we continue with the unit conversion. Now we have:
or about ¼ of a second ago. What does this ¼ of a second represent? If the earth’s age was compressed to a week, the Birth of Christ would have occurred within the last ¼ of a second. This means, the birth of Christ and any latter events, such as the wisdoms of Budha, works of Shakespeare, Beethoven or Picasso, arrival of the Pilgrims to Plymouth Rock, Revolutionary war, Babe Ruth’s 60th homer or when ex-vice president Al Gore invented the Internet, all would have occurred within time it took you to read this word, under a ¼ of a second ago.
All right, so with the use of setting ratios equal to each other, which in mathematics we recognize as proportions, we can get a feel for the relative magnitude of any numbers we encounter. If the numbers represent size, distance or some magnitude, the comparison and estimation method provides a practical tool that we can easily apply. Abstractly, we have just used the comparison method on a timeline. But, this comparison method is no different than placing the following visual in your mind: Picture the earth, a rock too big too comprehend. Imagine it in all it’s glory, picturesque, scenic, it’s soaring mountain ranges, it’s cascading valleys and vast oceans. Now, imagine shrinking it down to the size of a cue ball. This cue ball would be smoother than any cue ball you could ever construct. That is how large the earth is compared to the mountains and valleys you see.
Exercise Set
1
For problems 1 – 4, use the same breaks in time we used above and recall, the earth is 5 billion years old.
A. 700 million years ago, the first fossil animals appeared.
B. 220 million years ago, the first dinosaurs appeared.
C. 2000 years ago, The Birth of Christ
D. roughly 400 years ago, the writings of Shakespeare
Construct a timeline locating each item above on the time frame below, as we did in this text, for each of the following:
1. Compare the earth’s age to 1 month
(assume a 30 day month)
2. Compare the earth’s age to 1 year
3. Compare the earth’s age to 18 years
(the age of the average college freshman)
4. Compare the earth’s age to a century
For problems 5 to 9, imagine the celestial body below to be roughly the shape of a sphere, who’s circumference is given by the formula . If a ribbon was tied around the sphere in its middle, how long would the ribbon have to be? If a second ribbon was extended six feet from the object (sphere), how much longer would the new ribbon have to be?
5. Baseball, whose diameter is 2 ½ inches
6. Earth, whose diameter is 7,927 miles
7. Jupiter, whose diameter is 88,700 miles
8. The sun, whose diameter is 864,950 miles
9. Discuss the similarities of the answers
above. What have you learned?
1
Astronomy: Enormity of space
Let’s turn our attention to the enormity of numbers from the size of the earth and it’s fellow celestial bodies in contemplating the relative size of some astronomical record holders.
- The Sun’s diameter is 864,950 miles
- The Largest Planet is Jupiter, whose volume is over a thousand times that of the earth.
- The Smallest Planet is Pluto, which is 0.0017 as big as the Earth
- The Largest Star known is R136A in Large Magellanic Cloud, which is 400 to 1000 times the Sun’s mass
- The Most distant object is Quasar Q000-26, which is 18,000,000,000 (or 18 billion) light years from the earth.
In reality, these numbers are relatively pleasant magnitudes. Quick, try this. Which is smaller? 0.00000000007 or 0.00000000017? Quick, which is bigger? 1800000000000000000000000000 or 1080000000000000000000000000. Think these numbers are contrived? Are you thinking we might as well as asked how many angels could fit on the tip of a pin? The smallest organism of any kind is a viroid, a virus like plant pathogen. These little creatures are 0.00000000007 inches in diameter. The sun’s weight is 1,800,000,000,000,000,000,000,000,000 tons. The number of angels that could fit on the tip of a pin is answerable if you know the details as to the size of the angel and the diameter of the pin’s tip.
For most of us, when we first see numbers such as those representing the diameter of a viroid and the weight of the sun, we become mesmerized, paralyzed, and can not think. It is like being confronted by the weaving head of a cobra. Even mathematicians balk at using numbers with so many digits. Writing numbers with 28 decimal places becomes both tedious (too many zeroes) and impractical because we can not immediately tell the relative size of the numbers. And this is where scientific notation comes in. In mathematics and the related sciences, scientific notation is used to represent really big or really small numbers. The power of scientific notation is the immediate clarity of the size of the number, written succinctly.
Thus, to represent the sun’s weight of 1,800,000,000,000,000,000,000,000,000 tons, we will rewrite this number with just one nonzero digit to the left of the decimal point multiplied by an integer Using scientific notation this unmanageable 27 digit long number is succinctly rewritten as 1.8 x 1027. Similarly, negative exponents on the base of ten are used to indicate small numbers. That is, 0.00000000007 inches is rewritten in scientific notation is 7.0 x 10 - 11.
Why do we need such representations for large and small numbers? Answering this question is how basic numeracy helps us to understand the world around us. Why does the typical college graduate need to know large and small numbers? Where are we confronted by this in every day life? Ask yourself how much money per person the 7 trillion dollar federal debt comes to per person if there are about 295 million United States citizens. Ask yourself, if some one knew your user name for your hot mail account, how many passwords would the person need to go through if they were arbitrarily trying to enter your site before they were guaranteed access? We will actually see this huge number later, and we promise you, it is larger than you think.
Let’s try an exercise and restrict ourselves to our solar system.
Table 1. Our solar system, excluding the moons, satellites and other members.
Body / Body diameter in miles / Orbit diameter in milesSun / 864,950
Mercury / 3,000 / 72,000,000
Venus / 7,700 / 134,400,000
Earth / 7,927 / 186,000,000
Mars / 4,219 / 282,200,000
Jupiter / 88,700 / 986,600,000
Saturn / 75,100 / 1,772,000,000
Uranus / 29,300 / 3,560,000,000
Neptune / 31,200 / 5,580,000,000
Pluto / 3,700 / 7,340,000,000
Before we proceed with this exercise, let’s just take a moment to glance at the numbers in the table above and think about them. We’re not so busy that we cannot pause for a little reflection. Scan the above table, and think about the relative size of the planets and their orbits. Now, close your eyes and recall the last time you were in a planetarium. Does something strike you as odd? If not, numeracy has not yet set in. There is a great line in Rocky III, where Apollo Creed turns to Rocky at the end of the movie and says, “you fight great, but I am a great fighter.” Well, if what we have said up until now, has made perfect sense to you, but if the planetarium you envisioned doesn’t seem odd, then you think great, but let’s make you a great thinker.
We have a lofty goal, to read Table 1 and from it, put in context the precise size of the solar system. Now, what does it mean to contextualize the precise size of the solar system? Two visualizations must float through our minds. First, How large are the planets relative to each other? And second, how far apart are the planets from one another? For example, when we read Table 1, if we close our eyes, can we see a sphere whose diameter is 865,000 miles? If we could comprehend the enormity that is the size of the solar system, we would not need to use the comparison method to contextualize the solar system. The comparison method gives us the power to see numbers in the millions and billions while placing them in terms we can visualize. We can’t imagine the sun’s true size, however, we can imagine a scaled down version of the sun.
Guy Ottewell, in his THE THOUSAND-YARD MODEL, OR THE EARTH AS A PEPPERCORN illustrated this very notion. Succinctly and clearly, he provides us a visual in the form of a wonderful children’s activity that went something like this. First, let’s look at the table displaying the planets’ diameters and distances traveled in their orbits. Now we turn to the Comparison Method. Let’s begin with the sun whose diameter is approximately 865,000 miles. Can you visualize 864,950 miles? Neither can we. So, let’s reduce it to approximately 8 ½ inches, scaling down our solar system so the scale is 100,000 miles to an inch. This is a good scale to start with, because although we can not visualize 864,950 miles, we can visualize a soccer ball that has a diameter of about 8 or 9 inches.
Table 2. Clearly, if our scale was 100,000 miles for every inch, Mercury’s body diameter would be 3000/100000 inches or 0.03 inches or three hundredths of an inch. Below is a new table with the old and new measurements side by side for comparison purposes.
Body / Body diameterbefore and after the scale
Miles Inches
Sun / 864,950 8.65
Mercury / 3,000 0.03
Venus / 7,700 0.08
Earth / 7,927 0.08
Mars / 4,219 0.04
Jupiter / 88,700 0.89
Saturn / 75,100 0.75
Uranus / 29,300 0.29
Neptune / 31,200 0.30
Pluto / 3,700 0.03
Now, in relative terms, if the sun was a soccer ball, the other planets’ sizes would be: