Note Well: *Scientific Notation - The taming of the numbers*

If you are already totally familiar with Scientific Notation, the art of writing large and small numbers in a compact form, then feel free to move to the next, and last, introductory chapter.

If you are not, then you **need** to become familiar with it because it is virtually impossible to understand the scope of the Universe without it.

Luckily scientific notation is really easy.

I've already mentioned that you don't need to do any math to read, understand, and enjoy this book. You only need to understand scientific notation and the nature of large, and small, numbers.

But.

If you *want* to learn how to calculate with scientific notation I've included a small primer Scientific Notation math. And if you go that far then you will also want to learn Unit Conversion. When you have learned these two things, which are also not so difficult, you could then start making your own analogies and perspectives with the help of all the constants in the Reference appendix.

Ok. Onwards.

As we saw in Chapter 1, large and small numbers take on enormous proportions with breathtaking rapidity. We were only looking into, at most, 1 kilometer cubes and single snowstorms. You can imagine that as we move out into the Universe the numbers are going to become gargantuan (as well as uber-lilliputian). Using “word” numbers like “million”, “billion”, “trillion”, “trillionth”, “quadrillionth”, and so on will, literally, simply not do work. They are way too cumbersome.

Scientists have a shorthand method of writing very large, and very small, numbers which is compact and easy to use. It is called, unsurprisingly, Scientific Notation.

This chapter will present this notation * twice*. First, now, in a minimalist form, with some experiments. We will be using Scientific Notation throughout this book and so a passing acquaintance is

*required*.

Then, in the Focus section at the end of the chapter, this notation is examined in a more detailed, and slightly more mathematical fashion, for the interested reader. That discussion is purely optional, feel free to skip it if you want.

Scientific Notation is based on the powers of 10. Here “powers” means how many times 10 is multiplied by itself.

Lets take the number 1000. In Scientific Notation this is written as 10^{3}, note that 1000 has 3 zeros. In the case of 1,000,000 it is written 10^{6}. Note that 1,000,000 (1 million) has 6 zeros.

And so you see right away that all we are doing is replacing 0's in a number with 10 to the “power” of that many zeros.

One billion is 10^{9}, a trillion is 10^{12}, and a quadrillion is 10^{15}.

There is more information in these notations. Notice that each of these powers are 3 more than the previous number. And above we saw that 10^{3} is equivalent to 1000. So simply by looking at the powers we see a trillion (10^{12}) is 1000 times more than a billion (10^{9}), because 10^{9} × 10^{3} = 10^{12}, and a quadrillion (10^{15}) is 1000 times more than a trillion (10^{12}).

A quadrillion can be looked at another way as well: it is a million (10^{6}) times more than a billion (10^{9}).

So both 10^{6} × 10^{9} and 10^{3} × 10^{12} are 10^{15}. We'd known this from the discussion before, but now we can see it at glance.

So, generally, from now on we will write, for example, 10^{18} instead of 1,000,000,000,000,000,000, and you know at a glance that 10^{18} is a 1000 times bigger than a quadrillion (10^{15}).

You may asking, 'So we write 1,000,000 like 10^{6}, but how do we write 3,000,000?'

It is written: 3 × 10^{6}, which means 3 × 1,000,000 = 3,000,000. Some more examples:

30,000,000 = 3 × 10^{7}

300,000,000 = 3 × 10^{8}

3,000,000,000 = 3 × 10^{9}

1,015 = 1.015 × 10^{3}

1,100,000 = 1.1 × 10^{6}

10,400 = 1.04 × 10^{4}

1,200,900,000,000 = 1.2009 × 10^{12}

1,234,567,890 = 1.23456789 × 10^{9}

With this notation, so far, we literally cover all the large numbers we could possibly think of.

I say "so far" because we've not addressed small (meaning less than 1) numbers.

Notice that the powers we've used so far are positive numbers. In a wonderful symmetry we use negative numbers to indicate small quantities.

For instance the number 0.001 (one thousandth) is written 10^{-3}. One millionth (0.000001) is written 10^{-6}. The pattern is, as you can see, very similar.

The similarity to large number's notation does not stop there. For example, 3 millionths (0.000003) is written 3 × 10^{-6}. And some more examples:

0.0000003 = 3 × 10^{-7} (3 ten-millionths)

0.00000003 = 3 × 10^{-8} (3 hundred-millionths)

0.000000003 = 3 × 10^{-9} (3 billionths)

0.000000000003 = 3 × 10^{-12} (3 trillionths)

0.1 = 1 × 10^{-1} (1 tenth)

0.01 = 1 × 10^{-2} (1 hundreth)

0.001 = 1 × 10^{-3} (1 thousandth)

0.001015 = 1.015 × 10^{-3} (etc.)

In Chapter 1's Focus Section I mentioned that a snowstorm in Vermont was found to contain 1,200 quadrillion (1,200,000,000,000,000,000) snowflakes. We can now write this as 1.2 × 10^{18}, decidedly more compact.

Ok, but...

*...Caution*! When you're first learning scientific notation it is easy to fall into a couple of traps.

While we have tamed large numbers with a simple notation, do not be lulled by their deceptive simplicity.

10^{18} is a *HUGE* number (it is 1000 quadrillion, this is also equivalent to 1,000,000 trillion). Just remember each increase in the power (or decrease in the case of small numbers) makes increasingly accelerating differences in the quantities. For instance, knowing what you know now, numbers like 10^{24} or 10^{36} should inspire awe...they are simply massive! Similarly numbers like 10^{-24} are incredibly tiny.

In addition to the draw dots experiment from chapter 1, here are two more simple experiments you can do to prove to yourself how the "powers" of 10 increasingly increase:

1) While timing yourself, or having a friend do it, count to 100, as fast as you can. Now re-time yourself counting to 1000. And how long for 10,000? 100,000?

2) Grab a pile of uncooked rice and put it on a table. Now separate out 10 grains of rice, then 100, then 1000, then...and so on.

Each of the above steps is "only" 10 times more than the last but if you actually tried these you will have a real appreciation for what they actually mean.

This trend is also true of scientific notation. The difference between 10^{9} (1 billion) and 10^{10} (10 billion) is *big*, even though the numbers 10^{9} and 10^{10} look very similar.

Also, if one isn't paying attention one could fall into the "assumption trap" that 10^{4} is twice as big as 10^{2}. Remember it's not! It is 100 times as big (10^{4} divided by 10^{2} = 10^{2}).

Similarly 10^{12} (1 trillion) is not twice as big as 10^{6} (1 million), it is a *million* times bigger (the number that is twice as big as a million is 2 million).

And lastly, a word about "significance". When is a million insignificant? This is always relative of course and heavily dependent on how big the number you are comparing it with is.

For example, a million is insignificant, at least in my opinion, when compared to one quadrillion. A million is a billionth of a quadrillion (10^{15} divided by 10^{6} = 10^{9}) and most of the time a billionth just isn't worth taking into account.

This may seem like non-sense, but we see this all the time in science, when quantities that to us seem enormous are not even taken into account in the final number.

There are a couple reasons for this. One is often statistical, there are errors in measurement and technique (for example our "rulers" can only measure so far), this results in a "standard error" that is often published with the final data and that determines how "significant" we can be.

Another reason is that it is difficult to be exact when counting, for example, the snowflakes falling during a snowstorm. Being off by several trillion is not a problem, *and* not particularly significant!

Let's examine this. I said 1.2 × 10^{18} snowflakes fell during the Vermont snowstorm. What if, actually, I missed a trillion I didn't see? That means I need to add a trillion to my total. Let's do that:

1,200,000,000,000,000,000

__+ 1,000,000,000,000__

1,200,001,000,000,000,000

In Scientific Notation the answer is then written as 1.200001 × 10^{18}. You see the extra one is way off the main group of numbers. This is because a trillion is only one-millionth of 10^{18}. Since the trillion contributes so little, relatively, to the overall total we can (usually) just ignore it.

When you see numbers written out to such precision, and sometimes even more, in published scientific data you can be sure that scientists have taken into account all the un-surety and the number * is*, as far as possible, accurate that far out to the right of the decimal point.

As for me, I'll stand by the 1.2 × 10^{18} snowflakes and ignore the trillion or so, or even several quadrillion(!), that I may have missed. Remember a quadrillion in this number is about a thousandth of the total, so pretty safe to even ignore a quadrillion (10^{15}) error.

If all this is new to you, you may want to first re-read chapter 1, then re-read this section to solidify your understanding for these numbers.

The next section is the promised *detailed* discussion of Scientific Notation, feel free to skip it if I've already terrified you. But I encourage you to try it, when one understands what is behind the scenes it can often times help quite a bit.

In the next chapter we will have a very brief introduction to the units we will be using throughout the book and then it will be off to the stars...

*Focus:* Scientific Notation Exposed

As mentioned above Scientific Notation is based on the powers of ten. For example: 10^{3}.

Any number written like the “3” above (small typeface and raised up) is called a “power” or, more commonly, the “exponent”. This notation tells us to take whatever precedes it, here “10”, and multiply that thing times itself as many times as the exponent says.

Using our example above of 10^{3}, this is in plain English: multiply 10 times itself 3 times. This is 10 × 10 × 10 = 1000. All these ways of writing 1000 are exactly equivalent: 1000 = 10 × 10 × 10 = 10^{3}.

You can, and probably have, seen this method used in many, many ways. For example:

cm^{2} (centimeters × centimeters = square centimeters)

m^{3} (meters × meters × meters = cubic meters)

9^{2} (9 × 9 = 81)

2^{4} (2 × 2 × 2 × 2 = 16)

Scientific Notation is just the special term one uses when dealing with powers of 10.

This notation seems like a tiny thing, but its power (pun intended) to help us with big and small numbers is fantastic.

If 1000 is easily written as 10^{3} then how about one million? One million is 10^{6}. You may have already recognized the pattern this notation takes. Remember that a million is 1000 × 1000? And 1000 can also be written 10^{3}? And so: 10^{3} × 10^{3 }=^{ }10^{6} = one million.

The whole process is, really, counting zeros, simple as that.

Consider:

10^{1} = 10 » 1 zeros

10^{2} = 10 × 10 = 100 » 2 zeros

10^{3} = 10 × 10 × 10 = 1,000 » 3 zeros

10^{4} = 10 × 10 × 10 × 10 = 10,000 » 4 zeros

10^{5} = 10 × 10 × 10 × 10 × 10 = 100,000 » 5 zeros

10^{6} = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000 » 6 zeros...

__Special case:__ 10^{0} = 1 » NO zeros. *{any number, except 0 itself, when raised to the “0” power equals 1, it's a mathematical rule}*

One billion is 1000 million so it is written 10^{9}. One trillion is 1000 billion and so is written 10^{12}. One quadrillion is 1000 trillion written as 10^{15}. Each of these is 1000 times bigger than the one prior. 1000 is 10^{3}. Each of these has the exponent increase by 3.

Thousand, million, billion, trillion, quadrillion. 10^{3}, 10^{6}, 10^{9}, 10^{12}, 10^{15}. Scientific Notation is much more compact.

*Warning (again)!* Do not allow yourself to be lulled into thinking that this simple notation takes away any of the “power” of these numbers! The enormous increases continue with each increasing exponent. Remember our circles from Chapter 1? From the dot to the circle to the large circle. Exactly the same progression as above, each 1000 times bigger.

Another example is the difference between 1 second and 31.75 years: “only” 10^{9} steps. And:

between 1 second and 317.5 years - 10^{10}

between 1 second and 3,175 years - 10^{11}

between 1 second and 31,750 years - “only” 10^{12}.

If this still seems too theoretical for you, be sure to try the simple experiments described in the first part of this chapter and in chapter 1.

By the way, this kind of increasingly enormous progression is called a logarithmic scale. When something increases or decreases in size so fast it becomes impractical to use normal numbers to represent or diagram it. Sheets of paper just aren’t big enough, just like the circles we used.

So oftentimes the exponents are used as the scale rather than regular counting numbers. This is so in the Richter Scale used in measuring earthquakes for example. Each Richter Scale point indicates an Earthquake releases approximately 10 times as much energy as the preceding one. So the difference between a Richter 9 and a Richter 1 is a factor of 10^{8}: 100 million times stronger.

One seeming problem here is that so far we’ve written only notation for whole numbers that are multiples of 10. So while it is easy to notate 1000 as 10^{3}, how do we notate 3,200?

It is quite easy. We haven’t looked at the complete Scientific Notation yet. The notation also consists of a another part: a decimal or whole number portion multiplied times the power of 10. For example, 10^{3} written out *properly* is "1 × 10^{3}".

Since anything multiplied by 1 isn't changed scientists omit the "1 × " when it is a whole number like 1000. But, as we saw earlier, a number like 3,200 would be written: 3.2 × 10^{3}

This is telling us to multiply 3.2 by 1000 which is 3,200.

Extending this one would expect to write 32,000 as 32 × 10^{3} right? Well, no.

There's nothing wrong with that multiplication, it does equal 32,000. But, by convention, Scientific Notation (normally) requires only 1 digit be to the left of the decimal point so 32,000 is written as 3.2 × 10^{4}, which is 3.2 times 10,000.

Here are some other examples:

300 = 3 × 10^{2}

150 = 1.5 × 10^{2}

1,536 = 1.536 × 10^{3}

20.5 = 2.05 × 10^{1}

1,000,001 = 1.000001 × 10^{6}

12,000,000,000,000,000,000 = 1.2 × 10^{19}.

The rule is easy: move the decimal point to the left until only one digit remains while counting these moves. Then write this new decimalized number and multiply it times 10 raised to a power equal to the number of positions you moved the decimal point.

For example: taking 100,100 we move the decimal point to the left 5 times to get 1.00100. Trailing zeros add nothing (generally, but not always) to the number so they are usually discarded (unless they are significant) and our new decimalized portion is 1.001, now we multiply this times 10 raised to the number of positions we moved the decimal point: 5.

Voila, 100,100 becomes 1.001 × 10^{5}.

Let’s look at an example in nature of a number that is tedious to read and write.

A single water molecule is a very small thing and so we are able to pack a lot of them into a very small volume. In fact, in *one single drop* of water there are about 1,650,000,000,000,000,000,000 water molecules ( 1.65 billion trillion H_{2}O molecules).

This number is cumbersome, hard to understand, and confusing. However, it's Scientific Notation form takes away all doubt and makes it easy to grasp, and manipulate: 1.65 × 10^{21}.

For example, you know that 10^{15} is a quadrillion and you now have an appreciation for just how enormous it is. The difference between 10^{21} and 10^{15} is 6 factors of 10: 10^{6}. So you know, at a glance, that this quantity of water molecules is a *million* times bigger than a quadrillion!

Only one thing left to clarify before we move on the next chapter: how to express Scientific Notation for the incredibly tiny.

The same rules hold true for small numbers but with a decidedly different slant: the exponents are *negative*.

In chapter one we found that light could travel 0.3 millimeter in 1 trillionth of a second. Rewriting this using Scientific Notation: light can travel 3 × 10^{-1} millimeter in 1 × 10^{-12} seconds.

10^{-1} equals ^{1}/_{10}. Here the same principle applies as it did above except that instead of using the power of tens we will use the power of “tenths” and decimal places move to the *left*, instead of the right:

10^{0} = 1

10^{-1} = ^{1}/_{10} = 0.1 » 1 decimal place

10^{-2} = ^{1}/_{10} × ^{1}/_{10} = 0.01 » 2 decimal places

10^{-3} = ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} = 0.001 » 3 decimal places

10^{-4} = ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} = 0.0001 » 4 decimal places

10^{-5} = ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} = 0.00001 » 5 decimal places

10^{-6} = ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10} = 0.000001 » 6 decimal places

As before the "1" multiplication is understood: 10^{-6} is exactly the same as 1 × 10^{-6}. While 10^{6} means a million, 10^{-6} means a million*th*. And these numbers are just as "enormously" small as our other numbers were enormously big. 10^{-12} seconds, for example, is a trillionth of a second, vanishingly small, just as a trillion in our early discussions was vanishingly huge. And finally other numbers can be multiplied so that any small number can be written in Scientific Notation. For example:

0.33 = 3.3 × 10^{-1}

0.00012 = 1.2 × 10^{-4}

0.007 = 7 × 10^{-3}

0.00000106 = 1.06 × 10^{-6}

0.0000000000000034 = 3.4 × 10^{-15}

Notice that the method used here is a mirror image of the method used with positive exponents: move the decimal point to the *right* until we finally have only one non-zero digit to the left of the decimal point while counting these moves. Then write this new decimalized number and multiply it times 10 raised to a *negative* power equal to the number of positions you moved the decimal point.

For example: taking 0.00001001 we move the decimal point to the right 5 times to get our new decimalized portion: 1.001. Now we multiply this times 10 raised to the number of positions we moved the decimal point written negatively: -5.

Voila, 0.00001001 becomes 1.001 × 10^{-5}.

Now we have what we need to easily understand our Universe. We can soon begin to investigate the small...and the large...and the stuff in between.

The next chapter is the last introductory chapter. There we will be discussing the basic metric units we will use throughout the rest of this book.