2.1.2 - A Primer on Scientific Notation
2.1.2
A Primer on Scientific Notation
PREV>> 2.1.1 - Introduction
INTRODUCTION
You may have heard of "scientific notation" from math class. Scientific notation is simply a compact notation for expressing very large and very small numbers, specifically exponential class numbers. It's an alternative to decimal notation which is used quite extensively in science and engineering. However, you don't need to be a scientist or engineer to understand it. This article will go over the notation in detail to help you better understand it.
ORIGINS
The first known occurrence of the term "scientific notation" dates back to 1961 in the third edition of the New International Dictionary of the English language according to Mark Huber , Assistant Professor of Mathematics and Statistics [1] . The term most likely floated around before this date until it's use was wide spread enough to grant it an entry in an international dictionary. Scientific notation began as a special number format for computers for expressing very large numbers, and then the term was later applied to the notation that scientists use to conveniently describe very large and very small numbers.
WHAT IS IT?
But what is Scientific notation ? Not everyone is well versed in scientific notation. It is a topic usually visited briefly in remedial math classes. If you need some brushing up, or if your not that familiar with the notation, then this article is meant to help give you a better foundation with the notation (If your already familiar with scientific notation you can skip most of this article. You might want to check the heading "E-Notation" however, as this is the format I'll be using most frequently throughout Section II)
Scientific notation is basically a format for expressing certain very large and very small numbers that would be cumbersome to express in decimal notation. A typical example would be something like ...
2.34 x 10^5
Note that this is simply a mathematical expression composed of numbers and operations. The key to understanding scientific notation is the way it uses exponents. So before we go into the format, let's first understand exponents.
EXPONENTIAL NOTATION
Most people are familiar with addition and multiplication, but exponentiation , or "powers" as they are better known, are a little more vague. The best way to understand exponents is by analogy. Recall that multiplication can be understood as a "group of groups"...
for example 3x5 is "5 groups of 3"
This implies that 3x5 = 3+3+3+3+3
This is what we mean when we say multiplication is repeated addition . The following table should get the idea across ...
3 x 1 = 3
3 x 2 = 3 + 3
3 x 3 = 3 + 3 + 3
3 x 4 = 3 + 3 + 3 + 3
3 x 5 = 3 + 3 + 3 + 3 + 3
etc.
Note that you can use this expanded form to solve multiplication problems even if you haven't committed them to route memory. For example, to solve 3 x 5 we can expand it as repeated addition, and then we can add up the terms to find the sum ...
3 x 5
=
3 + 3 + 3 + 3 + 3
=
3 + 3 + 3 + 6
=
3 + 3 + 9
=
3 + 12
=
15
Simple so far, right ? Now we just apply the same concept to exponents. Firstly , you may recall that in school, when you want to raise something to a power, you write the power as a smaller number raised on the right side of the base number...
3 2
The above expression would be read as "3 raised to the 2nd" or as "3 squared". Sometimes we want to write expressions in a linear format however. This is where the "^" symbol comes in, known as a "caret" (we'll be making extensive use of this symbol later when we study the hyper-operators in Section III). The expression ...
3^2
Means the same thing as the expression above. Keep in mind that it's the number on the right that represents the power.
You may recall that "squaring" a number is equivalent to multiplying it by itself. So ...
3^2 = 3 x 3 = 9
The key here is to think of exponentiation (powers) as repeated multiplication. The number of repetitions is equal to the exponent. Thus when we "cube" a number (raise to the 3rd) we can solve it by expanding it into a repeated multiplication ...
3^3 = 3 x 3 x 3
Again we solve by performing operations until we are left with a single result ...
3 x 3 x 3 = 3 x 9 = 27
Now look at the following table and let the idea sink in ...
3 ^ 1 = 3 = 3
3 ^ 2 = 3 x 3 = 9
3 ^ 3 = 3 x 3 x 3 = 27
3 ^ 4 = 3 x 3 x 3 x 3 = 81
3 ^ 5 = 3 x 3 x 3 x 3 x 3 = 243
As you can see, exponents are powerful. The results on the right are increasing 3-fold every time we raise the value of the exponent by 1.
Note that we can choose any number for the base. If we choose 10 as the base we get a convenient short hand for powers of 10 ...
10 ^ 1 = 10 = 10
10 ^ 2 = 10 x 10 = 100
10 ^ 3 = 10 x 10 x 10 = 1,000
10 ^ 4 = 10 x 10 x 10 x 10 = 10,000
10 ^ 5 = 10 x 10 x 10 x 10 x 10 = 100,000
10 ^ 6 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
Now observe that the expression 10^6 is shorter than 1,000,000 yet they are equivalent. In essence we can use a mathematical expression to represent it's number value. This idea will be very important to our further study of large numbers in chapters to come.
The first thing to know about scientific notation is that it will always involve an exponential expression whose base is 10, as in the example ...
2.34 x 10^5
So is that all there is to scientific notation ? Not quite.
Scientific notation assumes an understanding of the concept of the decimal point, as well as negative exponents, for example ...
6.54 x 10^-3
How do you compute a negative exponent ? and what does 6.54 mean ? To understand this we need to go beyond a discussion of the counting numbers. Up until this point I have basically ignored other "kinds" of numbers. Now we are going to need to understand a "larger" set of numbers[2] .
INTEGERS
Up until this point we have mainly been working with the set of Counting numbers. In mathematics these are also known as the Natural Numbers, or typically as the positive integers. These are the familiar numbers starting with 1,2,3, and continuing indefinitely.
Mathematically we can symbolize this as ...
N = { 1 , 2 , 3 , ... }
Where N represents the set of Naturals.
The Integers include all of these as well as zero, and what are called the "negative numbers". Basically these numbers allow us to count indefinitely backwards as well as forwards.
Counting backwards with 3 , 2 , 1 , 0 , we can continue with " - 1 " , called negative one, then " -2 ", called negative two, and so on, with every counting number having a negative counterpart.
To symbolize this mathematically we write ...
Z = { ... , -3 , -2 , -1 , 0 , 1 , 2 , 3 , ... }
Where Z represents the set of Integers[3] . The triple dots are at both ends, meaning that the list can be expanded indefinitely in both directions.
THE DECIMAL POINT
Most people should have some familiarity with the concept of the decimal point.
In daily life we know that a dime is 10 cents, but it is also 0.10 US Dollars. Here the decimal point is seperating dollars from cents. Also note that a dime is one tenth of a dollar. A decimal point basically allows us to go beyond single units and go into even smaller sub-divisions.
Recall that each position in decimal notation has a base value 10 times larger than the previous position.
Thus we have ...
0th position ( ones ) 1.
1st position ( tens ) 10.
2nd position ( hundreds ) 100.
3rd position ( thousands ) 1000.
etc.
Notice that I have put a decimal point at the end of each of the numbers. Not only can we have digits to the right of the decimal point, but to the left as well.
To understand this, try to think of the above process in reverse. Each position in decimal notation has a base value 10 times smaller than the next position.
So we have ...
2nd position ( hundreds ) 100. is 10 x smaller than 3rd position ...
1st position ( tens ) 10. is 10 x smaller than 2nd position ...
oth position ( ones ) 1. is 10 x smaller than 1st position ...
So the next logical step would be ...
-1st position ( tenths ) .1 is 10 x smaller than oth position ...
-2nd position ( hundredths ) .01 is 10 x smaller than -1st position ...
-3rd position ( thousandths ) .001 is 10 x smaller than -2nd position ...
and so on ...
We can now combine whole units with these further subdivisions. For example, the decimal expression ...
234.567
means " 2 hundreds , 3 tens , 4 ones , 5 tenths , 6 hundredths , and 7 thousandths ".
What people have the most difficulty with regards to the decimal point however is how to move it.
Basically it moves 1 digit to the right if you multiply by 10, and 1 digit to the left if you divide by 10. Take the number in the above example. Let's say we multiply it by 10 ...
234.567
x 10.
=
2345.67
Basically what's happening is that each unit is getting multiplied by 10 fold, so that ones become tens, tens become hundreds, and so on, for every place value. Thus all the digits end up moving up one position. This is what they mean by "the decimal point moves 1 to the right". It may be better to understand that it is the digits which are moving, not the decimal point.
When we divide by 10 the same thing occurs in reverse. Say we divide our example number by 10 ...
234.567
/ 10
=
23.4567
Now every unit is being divided 10 fold so that hundreds become tens, tens become ones, and so on. Thus every digit is now moving back 1 position. Again this is equivalent to saying that the decimal point moves one to the left.
NEGATIVE EXPONENTS
Now we can talk about negative exponents. Recall that when we first went over exponents we looked at a table showing the powers of ten. Consider that pattern in reverse ...
10 ^ 3 = 10 x 10 x 10 = 1000.
/ 10
10 ^ 2 = 10 x 10 = 100.
/10
10 ^ 1 = 10 = 10.
Notice that each row has a value which is 1/10 that of the one above. Thus we find 10^0 by dividing by 10 once again ...
/ 10
10 ^ 0 = 10 / 10 = 1.
/10
10^-1 = 1 / 10 = .1
/10
10^-2 = .1 / 10 = .01
/10
10 ^ -3 = .01 / 10 = .001
etc.
In this way negative exponents can be used to express very small numbers.
You now have some knowledge of the basic concepts involved in scientific notation. Let's now apply this knowledge to it.
THE FORMAT
Before we start translating scientific notation, we first need to understand the format. Only certain expressions are allowed in scientific notation.
Any expression in scientific notation will always be of the form ...
m x 10^n
Where a is a decimal number, and b is an integer. a is called the mantissa, and b is called the exponent. The mantissa is further restricted in that it can only be a value equal to or greater than 1 and less than 10...
1 =< m < 10
The reason for this will become clear later when we compute these expressions.
Note that the mantissa can also be a whole number, for example ...
4. x 10^6
as long as it's between 1 and 10. Also if it is equal to one, it is permissible to "drop it" and leave only the exponential expression ...
1. x 10^8
=
10^8
Now all that remains is understanding how to interpret these expressions.
CONVERTING BETWEEN NOTATIONS
Let's begin with a simple example. Let's use everything we've learned to convert this expression to decimal ...
3. x 10^2
The first step will always be to resolve the exponent. Never carry out the multiplication first, because this will lead to an incorrect answer. Order of operations basically says that the more powerful operation ( exponents in this instance ) always takes precedence.
We know that ...
10^ 2 = 10 x 10 = 100
So this means ...
3 x 10^2 = 3 x 100
Then simply carry out the multiplication ...
3 x 100 = 300
So " 3. x 10 ^ 2 " simply means 300. At this point you may ask, why go through all the trouble. The practical application of scientific notation only becomes apparent when we deal with some very large numbers.
For example let's take a very large number and convert it into scientific notation. Let's say we have a number like ...
600,000,000,000,000,000,000
That's 600 quintillion. First factor out the 6 so that you are left only with a power of 10 ...
6oo,ooo,ooo,oo0,000,000,000
=
6 x 100,000,000,000,000,000,000
Now convert the power of 10 into exponential form. The easy way to do this is simply to count up the zeroes. The number of zeroes corresponds to the exponent. In this case we have 20 zeroes, thus ...
6 x 100,000,000,000,000,000,000
=
6 x 10^20
Notice how much simpler " 6 x 10^20 " is compared to 600,000,000,000,000,000,000. In decimal form we are certainly aware that it is a big number, but it's difficult to judge how big without actually counting up zeroes. Furthermore it is tedious to write out 20 zeroes every time you want to express the number. In constrast, the scientific notation for the same number is relatively compact and means exactly the same thing. It's is also easier to judge the size of the number. All you need to know is the size of the exponent (20) and you instantly know what kind of number this is.
Let's use a negative exponent this time. Convert this expression to decimal ...
5. x 10^-7
Firstly we convert 10^-7 ...
5 x 10^-4 = 5 x .0000001
Then multiply ...
5 x .0000001 = .0000005
This is a pretty small number, in the same way that a million is pretty big. Because this site is about large numbers we won't be using negative exponents much, but they will come up occasionally in calculations.
Now let's try an example where the mantissa is not a whole number ...
2.3 x 10^6
First convert the exponent ...
2.3 x 10 ^ 6 = 2.3 x 10 x 10 x 10 x 10 x 10 x 10 = 2.3 x 1,000,000
Now multiply ...
2.3 x 1,000,000 = 2,300,000
In otherwords 2.3 x 10^6 literally means 2.3 million ( or 2 million 3 hundred thousand ).
The most important thing here is that you understand the size of these numbers. It's all based on the exponent. The higher the exponent the greater the number. Scientific notation can be deceptive, because it makes the numbers seem much more managable then they really are. For example consider the numbers 10^30 and 10^80. By comparison there may not seem to be much of a difference. One might reason that 30 and 80 are roughly in the same "ball park" so to speak, and so 10^80 isn't much larger than 10^30. This reasoning is entirely faulty however. 10^80 is vastly larger than 10^30 by 50 orders of magnitude. Try to understand that every order of magnitude represents a ten fold increase. Thus even something like 10^33 is already a 1000 times greater than 10^30. Now look at their decimal expansions ...
10^30 = 1,000,000,000,000,000,000,000,000,000,000
10^80 = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000
Much larger as you can see, but even this doesn't totally bring the scale difference home completely. It's kind of like comparing the mass of the earth to that of the universe ! And that is not an exageration. When it comes to large numbers it's almost impossible to exagerate.
E NOTATION
There is also a notation which I will be using quite regularly from here on out. This is known as "E Notation". Basically let:
mEn = m x 10^n
This is just an alternative form of scientific notation. When m=1, we can use the truncated form:
En = 1 x 10^n
In this form we can let n be a real number, instead of just an integer, to allow us to express any positive real number. I'll be using this form extensively throughout the rest of Section II. It also forms the basis of a notation I develop in Section III and Section IV.
Typically I will include the sign of "n" throughout 2.1, when "m" is specified. The reason is that expressions like:
2.341334E13
don't seem to sufficiently make the exponent stand out. Instead I would write:
2.341334E+13
The plus allows one to better see where the exponent begins. Only when "m" is omitted, will I refrain from using the plus:
E13
Real-Valued E-Notation
There is one more variant way that I will be using E-Notation. Sometimes I will use real exponents, instead of restricting them to integers. This allows one a greater range of expressible values without having to specify an "m". This variant is sometimes appealing for me, as it more closely matches my use of E-Notation in my Hyper-E Notation (We'll learn about this extended version of scientific notation in Section III).
To understand how this works however, I'll have to explain a little bit about the real numbers, real exponents, roots, and logarithms. First off the real numbers, form a continuum of values between all of the integers. Any real number can be expressed as an endless series of digits, where the first non-zero digit has a finite place value. Example:
0.142857142857142857...
It's fine if there are in fact an infinite number of non-zero digits, as long as there is a greatest non-zero digit. Consequently, it's not required for a real number to have a least non-zero digit. Unspecified digits have the default value of zero. It is possible to express any integer as an endless series of digits. ie:
...0000001.000000...
The above expresses the number 1. When there are only a finite number of non-zero digits, the number is known as a terminating decimal. We can also have non-terminating decimals:
0.333333333333333333333333333333333333333...
When there is a non-terminating decimal, two things can happen. Either the sequence eventually falls into a infinitely repeating pattern, like above. This is known as a repeating decimal. Every repeating decimal expresses a particular rational number and any rational number can be expressed as a repeating decimal. There are also non-repeating decimals. For example:
1.41421356...
The above is the first few digits of the square root of 2. The square root of 2 never falls into an infinite repeating pattern! Such numbers are known as irrationals. Every non-repeating decimal expresses some irrational number, and every irrational number can be expressed as a non-repeating decimal. The real numbers are the collection of all the rational and irrational numbers.
Although this is an informal definition of the reals it is sufficient for our purposes here to think of reals as decimal expansions.
In real-valued E-Notation, the argument (exponent) can take on the value of any real number. It's convention to use "x" to represent a real variable. So we can say that:
Ex = 10^x
We already know how to compute 10^x when x is an integer. How would we compute something like:
E0.5 = 10^0.5
For this we need to go back to the exponential laws. We let one of the laws be generalized to real values. Let:
10^x * 10^y = 10^(x+y)
For all real x,y. This gives us a means to compute 10^0.5. Simply observe that:
10^0.5 * 10^0.5 = 10^1 = 10
so let 10^0.5 be some real quantity, x and say:
x * x = 10
x^2 = 10
So x is some real quantity such that the square of x is 10. We say that x is the "square root" of 10. By the Continuity Axiom we can assert that such an x exists. The continuity axiom is basically that if a continuum of values has no breaks, and part of it's continuum passes below and above a given value, then that value must be included in the continuum of values. Since 3^2 = 9 and 4^2 = 16, by the Continuity Axiom there exists an x^2=10 which is greater than 3 and less than 4. We can even estimate it's value by sandwiching it between better and better lower-bounds and upper-bounds. ie:
3.1^2 = 9.61 , 3.2^2 = 10.24
3.16^2 = 9.9856 , 3.17^2 = 10.0489
3.162^2 = 9.998244 , 3.163^2 = 10.004569
etc.
In this way we can compute E0.5 to arbitrary accuracy. However we'll never actually get the precise value because E0.5 is irrational. It should not be too difficult to see what the upshot of this is. We can, in principle, compute Ex to arbitrary accuracy for any terminating decimal we might want to use. To compute E0.1 we simply perform the calculation:
E0.1 = 10^0.1
(10^0.1)^10 = 10^1 = 10
By finding the tenth root of 10 we solve for E0.1. We can then compute E0.2, E0.3, E0.4, etc. simply by raising E0.1 to the appropriate power. Example:
E0.1 ~ 1.26
E0.2 ~ 1.26^2 = 1.5876
E0.3 ~ 1.26^3 ~ 2.0004
E0.4 ~ 1.26^4 ~ 2.5205
E0.5 ~ 1.26^5 ~ 3.1758
etc.
Discrepancies creep in do to rounding errors, but can be ironed out by finer calculations. In this way we can compute Ex where x is some terminating decimal. The integer part of "x" becomes the ordinary exponent in Scientific notation, where as the "decimal part" becomes a description of the multiplier (known as the "mantissa"). To convert from scientific notation to real-valued E-Notation simply compute the logarithm of the mantissa and add it to the exponent. ie:
m E n = E(n+log(m))
To compute a logarithm simply define it as follows:
log(Y) = x
if and only if 10^x = Y
Since we now have a rough way of computing decimal exponents, we could, at least in principle, approximate logarithms by this method. To convert from real-valued E-notation back to Scientific notation simply compute:
Ex = 10^x
Once you compute 10^x as a decimal number you can convert it back to Scientific Notation. Alternatively, if we split up x into it's integer part "z" and it's decimal part "d", then:
E(z+d) = (10^d)x10^z
STACKED EXPONENTS & STACK NOTATION
One final note. Later in this chapter the numbers will get so big that standard scientific notation won't even be sufficient. Normally scientists deal with numbers which are safely within the range of 10^-100 to 10^100 and have no use for larger numbers. It can be argued however that the number of ways something can be arranged is just as concrete a concept as counting those objects. These leads to combinatorics, and numbers so large that is literally overflows scientific notation. For that purpose we will learn how to deal with stacked exponents, and how these can be notated in a way analogous to scientific notation.
Basically the exponent eventually becomes so large that even scientific notation becomes impractical. While there is no official ruling against it, it is generally understood that the exponent should be expressed in decimal form. So what happens when the exponent itself is colossal ? For example ...
1. x 10^1,000,000,000,000,000,000,000,000,000,000,000
We can use scientific notation itself to express the exponent. We are feeding the notation into itself (a recursive concept which we will develop much further in Section III) We can call this "notation packing". The exponent above is actually 1 decillion or 1. x 10^33 . So we can write ...
1. x 10 ^ ( 1. x 10 ^ 33 )
We can also write it as a simple exponential tower:
10^(10^33)
The parenthesis are added for clarity, however it is convention to carry exponents out from right to left. We can then write more compactly:
10^10^33
In E Notation we can write:
1E(1E33) = EE33
Remarkably, we've taken an unimaginably vast number and compacted into the tiny expression "EE33"!
We can of coarse pack the notation into itself as many times as we like, and this will actually start to happen as we approach the largest number in science! Keep in mind that this notation packing is extremely powerful, especially when the initial notation is already quite powerful. The resulting numbers are in a completely different league, and it takes a great deal of effort to really understand their size ( in fact, on some levels it is really impossible for a human to imagine ).
To make stacked exponents, especially larger ones, even easier to express, we can establish the following extension of scientific notation, which I call "Stack Notation":
Let Ex#n = 10^10^...^10^x with "n" 10s in the stack
"x" can be any real number, without running into any trouble, and n can be any positive integer. By limiting x within the range:
1 =< x < 10
We get an extended form of scientific notation which can express numbers much larger than those in ordinary scientific notation. The example 10^10^33 can be written in Stack Notation as E33#2.
Even the largest numbers in science don't require a stack more than a few terms high. This suggests that our stack notation is already powerful enough to easily surpass anything in science. We'll explore the consequences of this line of inquiry as we continue through the rest of Section II and enter Section III. But even the numbers in science are mind-bogglingly huge. In the next article you'll see just what I mean...
CONCLUSION
This article was not meant to be completely thorough, so you may want to read some other articles on scientific notation to get a better grasp of it [4][5] . An understanding of logarithms is also important for working in scientific and stack notation, and we'll become indispensable in our study of stacks later in Section III. In the future I will probably provide a more thorough explanation of these topics.
The article was only a primer merely to provide enough information so that the reader could understand the basic concepts used throughout Chapter 2.1. We will be studying operations and number sets in greater detail in later chapters, as these will concepts become instrumental in the understanding of very large numbers later on.
Now that you have a basic understanding of scientific notation, you can go read the rest of the articles in this chapter where I discuss large numbers in cosmology and beyond...
NEXT>> 2.1.3 - Scales of the Macrocosm
Source Material, Footnotes & Recommended reading :
[1] http://www.madsci.org/posts/archives/2005-02/1109547573.Sh.r.html : This post of Mark Hubers discusses the origins of the term scientific notation while simultaneously discussing the difficultly of pinning down the exact place and time that a new term is coined except in rare cases.
[2] For those with some knowledge in Set Theory, I mean larger in the sense of set membership, not one-to-one correspondence in which it can be shown that the set of Naturals can be paired up with the set of Rationals.
[3] "Z" stands for "Zahl" in this case, which is the german word for number.
[4] http://en.wikipedia.org/wiki/Scientific_notation : Wikipedia article on scientific notation.
[5] http://janus.astro.umd.edu/astro/scinote/ : You can practice converting decimals to scientific notation and visa versa at this link.