A Million things to do: The nature of large numbers

The Universe is IMMENSE...

...vast distances filled with incredibly large objects consisting of extremely tiny things.

Of course, words like “vast”, “large”, and “tiny” are relative.

For instance, the distance to our Sun compared to the distance between your eyes is vast.

But the distance across our observable Universe compared  to the distance to our Sun is vastly greater.

Compared to your body a single cell is tiny. But a single atom compared to a single cell is vastly smaller.

Rating and comparing, then, requires a bit more precision than “vast” or “tiny”. To understand the difference in these quantities requires a series of relational steps and so this introductory chapter serves to give us those steps.

In order to do that we need to understand astronomically large, and microscopically small, numbers. We begin by comparing the relations between hundreds, thousands, millions, billions, and trillions.

First we note that words like "trillions" are not very convenient. Besides being open to differing interpretations, it is a word describing an abstract concept that can leave us wondering what it really means. Writing a trillion out like so: 1,000,000,000,000 is more exact but our eyes can find this difficult to read and it can be easily confused with a billion (9 zeros) or a quadrillion (15 zeros).

Even so we will use these "word" numbers like "trillion" in this first chapter for our first steps to understanding large numbers, carefully defining exactly what we mean so there is no confusion. In the next chapter we will simplify working with large and small numbers. This process is a necessary one if our goal is to see the Universe in perspective.

To make sure we all agree on what we will be talking about in this chapter I will here state things that may be obvious to some or all. But in this way we will all be at the same starting line.

We will be using the English system of denoting thousands and decimals. We will use the comma (,) to separate thousands in a number and the period (.) to denote the decimal portion. For example:

The "word" numbers, for example "1 billion", are open to interpretation depending on the country one lives in. One billion in some countries means a thousand million (1,000,000,000), in other countries it means a million million (1,000,000,000,000). We use the terminology of science which is as follows:

As I mentioned before these “word” numbers will, for the most part, be mainly used in this first chapter.

Most human beings have an innate innumeracy easily recognizing only the smallest numbers. We easily visualize 1, 2, or 3 things, but then it becomes increasingly difficult to distinguish more. Roman numerals are a good example. I was 1, II was 2 and III was 3. But they didn't use IIII for 4 or IIIII for 5 because they also recognized the increasing difficulty of distinguishing more than 3 things.

So all of us are intimately familiar with the number 1 and 2 and 3. We are also pretty good with 10 things, at least as a number and concept. We recognize the number 10 and know completely its meaning and size, we can visualize it, we can get a "handle" on 10 things. As we increase to 100 things, the meaning and concept is still quite understandable, though our visualization of 100 becomes a bit foggy. And as we move on to consider 1000 things -- concept: good, meaning: foggy, visualization: hard.

And this only gets worse, much worse, as we continue on to 10,000 - 100,000 - 1,000,000.

Similarly the notion of ¹/₃ and ¹/₄ are easy to visualize, and from that ¹/₁₀ and ¹/₁₀₀  make sense. But at ¹/₁₀₀₀ not to mention ¹/_1,000,000 our powers to concieve and visualize are poor, even though the notions are understood.

So let's try visualizing some of these numbers:

The most applicable, common everyday thing, for the thousandth scale is the millimeter (an uncooked rice grain is about 2 millimeters wide). A millimeter is ¹/₁₀₀₀ of a meter. Looking at a meter stick one can see, in one glance, 1000 things. That is: 1000 millimeters.

One million (1,000,000) is a 1000 thousands (1000 × 1000).  This means that if one wants to count to one million, one needs first to count to 1000 and then do that 999 more times. Makes sense? Yes. Helpful? No.

Can we visualize one million?

A qualified yes as we'll see. Easier is to visualize one milltionth (¹/_1,000,000). And for that we return to the millimeter.

Since there are 1000 millimeters in a meter, and there are 1000 meters in a kilometer, then a millimeter is one millionth of a kilometer.

And that is why I said it was a qualified yes, because one cannot, in a single glance, take in a kilometer and see the 1,000,000 millimeters. But, as you can see, it is easy to see something that is one millionth of a kilometer: the millimeter.

Another visualization of one million: the picture below is of a plexi-glass cylinder filled with 1 million small glass beads. 999,999 of the beads are clear glass, and only one bead is black. The cylinder is about a meter long, and can be rotated by people wishing to amuse themselves searching for the single black bead. I was fortunate to find it the evening I was there at the Museum. The picture, I believe, gives new meaning to the phrase “one in a million”.

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One million beads

One million beads (999,999 clear, 1 black) in a display at Mathematikum Math Museum, Gießen, Germany, www.mathematikum.de ( link opens in new tab/window). In the top picture is an overview of the million beads in the display. In the bottom picture you can see the detail of the single black bead near the red cover.

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Other examples of large numbers seen in daily life:

A single grain of sugar is about 0.5 millimeter on each side, which means it has a volume of 0.125 cubic millimeters. So 1 teaspoon of sugar (5 cubic centimeters) holds about 40,000 grains and so about 25 teaspoons worth will be a million grains of sugar.

Bed sheets are made by weaving threads in a criss cross fashion, at each junction of criss crossing threads a small square is formed. A typical “king” size bed sheet will have between 35 to 100 million squares in it depending on its thread count (how tightly woven it is).

So we are surrounded by large numbers in daily life. But, these numbers are actually quite small compared to the numbers we will find exploring the Universe where they run to the billions, trillions, and trillions of trillions of trillions...and bigger!

Large numbers just happen. And their consequent meanings and significance are mind boggling. Consider the simple, patient ticking of a clock. The table below lists times in seconds (a clock tick), each step increasing by a factor of 10. The steps are then translated to a more human readable form.

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Clock ticks

Increasing clock ticks and equivalent times. In the table we increase each step by 10 so we can see the power of multiplying by 10. But in practice levels are usually grouped by 1000's, a scientific convention. Here these thousand's divisions (one, a thousand, a million, and a billion) are shown in green.

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Clock Ticks

# of Seconds	Equivalent Time
1	one clock tick, one second
10	10 seconds
100	1 minute and 40 seconds
1,000	16 minutes and 40 seconds
10,000	2 hours and 47 minutes
100,000	1 day, 3 hours, 47 minutes
1,000,000	11 days, 14 hours
10,000,000	16 weeks, 4 days
100,000,000	3 years, 2 months
1,000,000,000	31 years, 9.5 months

 

The increase in time starts slowly and increasingly grows at a faster and faster rate. The difference, for example, between  10 and 100 seconds is only 90 seconds more. But the difference between 100 and 1000 seconds, just one factor of 10 more, is 900 seconds. Each successive step of 10 is increasingly out-racing the previous (an exponential increase) and is consequently harder to reach.

Here are the thousand's divisions summarized:

1 second vs. 17 minutes vs. 11.5 days vs. 32 years.

What this means is that it is 1000 times harder to reach 11.5 days than it was to reach 17 minutes, and it is 1000 times harder to reach 32 years than it was to reach 11.5 days. Consequently, it is a million times harder to reach 32 years than it was to reach 17 minutes.

Each successive step up these 10's "ladders" becomes successively harder to reach.

As an exercise to prove it to yourself take a piece of paper and a pencil and draw 10 dots. Not bad, no? It didn't take long. Now draw 100 dots. When you finish that try drawing 1000. Feeling frisky? Try 10,000. I think you get the idea.

The picture to the left shows this exponential increase up to a 1000. Think of the small circle having an area of 1. Then the second circle has 10 times the area, the third has 100 times, and the big one has 1000 times the area. Take a moment to imagine the circles growing up to 10,000 times, 100,000 times, and a million times.

 

Now lets see if your imagination did the job. The picture on the right of this page shows this relationship. The circle you can see entirely can hold 1000 of the small black dot. The big circle that can't fit on the page can hold 1000 of the full circles. This, of course, means the BIG circle can hold 1,000,000 black dots.

Full size versions of this are available to view at 2000, 4000, and 8000 pixels.

When opened they will probably be adjusted to fit in the window, just click the image to view it fullsize.

Keep in mind in the full size versions that the dot, the smallest circle, has 1 millionth the area of the BIG circle.

 

The relationships of these circles and the dot are the relationships we are building in our imaginations to understand the Universe in perspective.

This demonstration can easily(? :-o) be used at higher levels as well. Imagine shrinking the BIG circle down until it is the size of the dot. Our million dots now fit in that dot. This now means the full circle could hold 1 billion (1000 million) of the original small single dot and the BIG circle can hold 1 trillion (1000 billion) .

So now you have an idea of what a billion and a trillion means. These are large numbers.

We've seen above that a billion ticks of the clock will pass after a bit more than 31¾ years, about half a lifetime.

A trillion ticks of the clock takes more than 31,700 years, a much longer period of time than human written history (about four or five times longer).

As we've seen. a millimeter is a millionth of a kilometer. So, then, a millimeter is a billionth of 1,000 kilometers.

And it follows that a millimeter is a trillionth of 1 million kilometers. (about the distance from the Earth to the Moon and back to the Earth, and then back to the Moon).

Interestingly, it is with a trillion that we begin to work with interstellar astronomical distances. This is because it is used to describe how far away Alpha Centauri, the nearest star to us, is:  40 trillion kilometers. Think about that for a moment.

Real quickly, just long enough to blow your mind, the last number we will introduce in this first chapter is 1 quadrillion: this is 1000 trillion (or, equivalently, a million billion).

A trillion ticks of the clock takes 31,700 years. A quadrillion ticks takes 31 million 700 thousand years.

A millimeter is one quadrillionth of 1 billion kilometers.

The Sun is 150 million kilometers from the Earth, so this distance of 1 billion kilometers is about equal to going from the Earth to the Sun to the Earth to the Sun to the Earth to the Sun to the Earth to the Sun.

And here we stop. Further examples would only belabour the point which is that each increasing 1,000's level (the next is quintillion, a 1000 quadrillion) is an incredible increase. Simply keep these relations in mind, we will build on them later. Amazingly, as you can surely foretell, we will soon find that the quadrillion is tiny, almost "insignificant", compared to the numbers found in nature and the Universe.

Speaking of tiny...We've not yet examined the numbers in the other direction: to the very small. This one is difficult, at this point in the book, to make analogies for. So we will go a bit more abstract.

In the table below I list the diminishing distances light can travel in smaller and smaller amounts of time. Light, as I'm sure you know, is the fastest thing in the Universe, traveling 300,000 kilometers each second (equivalent to 300,000,000 meters / second).

It takes light only a bit more than a second to get from Earth to the Moon, and about 8 minutes to get from Sun to Earth.

By examining the table you can see the effect of the powers of 10 on something that is so speedy. Here, again, the 1000's levels are highlighted in green. The table leaves out going to a quadrillionth of a second because the small distance travelled will be meaningless at this point in the book.

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Increasingly small lengths of time (down to one trillionth second) and the resulting distance light can travel in that time. Each thousandth division is shown in green.

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Light travel at small times

# of Seconds	Distance light can travel
1	299,792.458 meters
Tenth	29,979,246 meters
Hundredth	2,997,925 meters
Thousandth	299,792 meters
10 Thousandth	29,979 meters
100 Thousandth	2,998 meters
Millionth	300 meters
10 Millionth	30 meters
100 Millionth	3 meters
Billionth	30 centimeters
10 Billionth	3 centimeters
100 Billionth	3 millimeters
Trillionth	0.3 millimeter

 

It is time to leave this world of “word” numbers behind. We will be looking at Scientific Notation in the next chapter, which is a millionbilliongazillion times easier to read, write and understand...

There is a mini-section after this, Focus: 3D. It deals with the world of 3 dimensions (length, area, and volume) and how large numbers can be generated there. If you are already thoroughly familiar with these topics feel free to move on to Chapter 2. But there may be something there you are not familiar with...

 

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Focus: 3D and more...

 

The three dimensions of space will be of importance and interest as we progress through the book. In this Focus section we will discuss how the three dimensions contribute to the large numbers around us.

As we investigate the scale of the Universe we will be talking in three general ways.

First we could be talking about a simple length, a line. For instance the distance to the Sun is a simple length in centimeters or kilometers (or some other unit of length), and so on.

Or we could be talking about an area as we did above for the circles, so we can see how many things can be inside a particular area. We will talk about areas of the World, and Suns. Areas are always squares. Even if it's a circle, or a sphere, or an undulating surface, areas can be calculated in units such as square centimeters (cm²), square kilometers (km²), and so on.

And finally we can talk about volumes. Volumes are always cubes. We will talk about the volumes of atoms and Suns. The units used for volumes are cubic: cubic centimeters (cc or cm³), cubic kilometers (km³), and so on.

These three dimensions of length, width, and height (or depth depending on your perspective), define our Universe, define our space. Sometimes calculating areas or volumes is complicated, but be assured there are always mathematical ways to do so, or, at least, approximate it to very accurate degree. The easiest shapes to calculate are straight line objects.

We will start here using 1 kilometer straight lines to investigate just how important dimensions become in scaling the Universe.

So consider a line 1 kilometer long. This line contains, as we have already seen, 1 million millimeters. Remember, a millimeter is about half the width of an uncooked rice grain. And so, in a simple length, we have found a large number that is easily understood by everyone.

As we now go into the other dimensions you will see just how powerfully dimensionality produces enormous numbers.

Areas, in the case of squares and rectangles, are calculated easily by multiplying the length times the width. Let's now imagine a square that is 1 kilometer on each side.

If we multiply the sides we find the area is 1 square kilometer by definition.

But, as we know, a kilometer is equivalent to 1 million millimeters. So looking at the area from the perspective of the millimeter we find it contains 1 trillion square millimeters (1,000,000 millimeters × 1,000,000 millimeters). And this, as we now know, is a very large number.

Simply by adding just one more dimension we find the numbers growing explosively. In an area of 1 square kilometer there a trillion square millimeters. If we had tiny 1 square millimeter tiles, we will need a trillion of them to cover the square kilometer. We will use techniques like this later to increase our understanding, and appreciation, of the Universe.

Volumes, like areas, are easy to calculate for straight line objects. Let's now imagine a cube that is 1 kilometer on each side. In the case of cubes the volume is simply the length times the width times the height, so in this case the volume is 1 cubic kilometer.

But what is it from the millimeter perspective? Each side is also equivalent to 1 million millimeters: the answer is an astounding 1,000 quadrillion cubic millimeters (1,000,000 millimeters × 1,000,000 millimeters × 1,000,000 millimeters). We already know that a quadrillion is a huge number, and this answer is a 1000 times bigger!

These numbers are large but the Universe delights in huge (and miniscule) numbers, as we’ll see in due time. But here is a another example where a quadrillion something’s was easily reached...and surpassed:

I visited the American state of Vermont, a place known for its snowstorms. We were having a moderately heavy storm and I was curious as to how many snowflakes fell per square centimeter each minute, so I counted them.

It turned out to be about 9 snowflakes per square centimeter each minute. A square kilometer contains exactly 10 billion square centimeters. So, over the course of an hour, using simple mathematics, we can calculate that, in one square kilometer, 5.4 trillion snowflakes fell.

Vermont often has storms that snow over the entire state, such as this particular storm did. Vermont has an area of about 23,830 square kilometers, which in an hour during this storm, then, received 128 quadrillion snowflakes.

It snowed at the same rate over the entire state for a total of 9 hours that day which makes a total of about 1,200 quadrillion snowflakes over the state of Vermont.

Distances and relationships and totals would very quickly grow so huge as to be cumbersome to use and understand if we are already having troubles with a simple 1 kilometer cube, or the number of snowflakes in a moderate snow storm. So, in the next Chapter, we will learn a method that brings these huge numbers under control and gives us back our ability to grasp and master them.

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