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Introduction
to the Mathematics of Evolution Chapter
13 Basic
Mathematics "If you want to make an apple pie from scratch, you
must first create the Universe." Carl Sagan, astronomer What is an Exponent? An exponent
is simply a way to represent a series of multiplications. For
example, suppose we wanted to multiply 10 by itself 12
times. We could represent this as: 10 x 10 x
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 This is
cumbersome to write down, especially if we were to multiply 10 by itself a thousand times.
Exponents are simply a shorthand way of expressing a number being
multiplied by itself. For
example, 10, multiplied by itself 12 times, is
represented as: 1012. 1012 has a "base," the 10,
which is the number being multiplied by itself. 1012
also has an "exponent," the 12, which is the number of times 10 is
multiplied by itself. Thus,
listing the number 10, being multiplied by itself 12
times, is written 1012. The
"base" does not have to be 10.
For example, how would you write out 47? The answer
is: 4 x 4 x 4 x 4 x 4 x 4 x 4 Note that
the number '7' is not in the above line.
The '7' is the exponent in 47 and represents how many times 4
is multiplied by itself. Remember,
exponential notation is a way of writing a multiplication problem in a very
short and simple way. Exponential
notation was not designed to complicate things, but rather to simplify things. Multiplying Exponents When you
multiply exponents, the numbers must have the same base!! For
example, this is legal: 105 x 106 x 108 It is legal
because all three exponents have the same base: 10 But this is
illegal: 510 x 610 x 810 It is illegal
because the three bases are not the same number. 5, 6 and 8 are not the same number. The rule of
multiplying exponents is that when you multiply exponents, you add their
exponents. For
example: 106 x 107 = 10(6+7) = 1013 Does this
make sense? Let us do this longhand: (10 x 10 x
10 x 10 x 10 x 10) x (10 x 10 x 10 x 10 x 10 x 10 x 10) is equal
to: 10 x 10 x
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 Thus, it
does make sense to add exponents which multiplying exponents. It is
always important to remember that when multiplying exponents the base must
be the same!! Dividing Exponents When
dividing exponents, the same rule applies: when dividing exponents the bases
must be the same!! When
dividing exponents we subtract the exponents. The '/' symbol represents division. Thus, 107
/ 106 is equal to 10(7-6) equals
101 equals 10. Is this
logical? Consider the above problem
written longhand: (10 x 10 x
10 x 10 x 10 x 10 x 10) / (10 x 10 x 10 x 10 x 10 x 10) Six of the
10s cancel each other out (the six 10s which cancel each other out are
underlined in the next line): (10 x 10
x 10 x 10 x 10 x 10 x 10) / (10 x 10 x 10 x 10 x 10 x 10) Only one 10
is not underlined. Thus, the answer is: 107
/ 106 = 10(7-6) = 101 = 10. Again, our
method leads to a logical answer. Also
remember, the bases must be the same!! Negative Exponents What does a
number list 10‑5 mean? Actually, this is a way to write small
numbers. While 105 is a big
number, 10‑5 is a small number. Actually,
10‑5 is equal to: 1 /
105 105
equals 100,000, but 10‑5 equals 1 / 100,000. Another way
to write 10‑5 is: .00001 We can look
this chart to better understand negative exponents: 104
= 10 x 10 x 10 x 10 = 10,000 103
= 10 x 10 x 10 = 1,000 102
= 10 x 10 = 100 101
= 10 100
= 1 (by definition any number to the zero power is 1) 10‑1
= .1 (which is 1 / 10) 10‑2
= .01 (which is 1 / 100) 10‑3
= .001 (which is 1 / 1,000) 10‑4
= .0001 (which is 1 / 10,000) 10‑5
= .00001 (which is 1 / 100,000) Thus, 105
/ 108 = 10(5‑8) = 10‑3 = 1 / 1,000
= .001 What is a Probability? Suppose you
had a die or dice with 10 sides. What is
the "probability;" if you rolled this dice; you would get a '1'? The term
"probability" means: "what is your chance?" Thus, "what is your chance;" or
"what is the chance" you will roll a '1'? There are
10 sides of the dice, and each side has the same chance of landing on top. The probability of rolling a '1' is 1 / 101
or 1 / 10 or 10‑1. In
other words, the probability is 1 in 10 or 10%. What is the
probability you will roll a '1' two consecutive times? In order to
calculate this we need to multiply 10‑1 x 10‑1. Remember, when we multiply two numbers with exponents
we add their exponents, thus 10‑1 x 10‑1
equals 10(‑1 + ‑1) = 10‑2. In other words, 1 in a
hundred or .01 or 1%. Is this
logical? Let us think about all the
different orderings of rolling a ten-sided dice twice (there are 102
unique orderings): 1 & 1 1 & 2 1 & 3 1 & 4 . . . 2 & 1 . . . 3 & 1 . . . 10 & 1 10 & 2 . . . 10 & 10 There are
100 different possibilities of rolling a ten-sided dice twice. Rolling a '1' and '1' represents one of these
100 possibilities. This order of rolls
has an equal chance as any other ordering of rolls. Thus the logical probability of rolling a '1'
twice in a row is 1 in a hundred possibilities or 1 / 100 or .01 or
10‑2. So the answer is
logical. What is a Set? A
"set" in mathematics is a collection of objects. They can be physical objects, such as people;
or abstract objects, such as numbers. For
example, the set of books in a library is a "set of books." A collection of marbles in a marble collection
is a "set of marbles." The students
in a particular school class are a "set of students." Likewise,
we could talk about more refined "sets." For example, the collection of students who
have brown hair, in Mrs. Smith's class; is a "set of students with brown
hair in Mrs. Smith's class." Sets can
also relate to mathematics. For example,
the set of even numbers (i.e. numbers divisible evenly by 2), less than 10, is
a set. This set can be represented as: {x | x is an even number less than 10} The symbol
"{x |" means the following: "x, such that." Thus, we could write the above set as this: {x, such that x is an even number less than 10} Or this set
can be represented as: {x | 0, 2, 4, 6, 8} Or this set
can simply be represented as: {0, 2, 4,
6, 8} Or this set
can be represented as: 02468 The
"members" of a set (e.g. 0, 2, 4, 6, and 8 in this case) are called
the "elements" of the set.
There are 5 elements. The key
concept when discussing sets is that we can determine exactly what elements are
in the set and which elements are not in the set. For
example, if we said "the girls in the 5th grade class," this would
not be a set until we refined the definition of set membership so we could
determine exactly which girls were in the set. If we said:
"the girls in Mrs. Jones 5th grade class at Thus, a
"set" is merely a defined set of objects, such that set membership
can be exactly determined. Sets can
also be defined by abstract methods. For
example, we could say: the set of 4 letters of the alphabet, such that the
first three letters are: ABC. Before
reading on, look away from this book and try to figure out how many elements
there are in this set, and what those elements are. The answer
is there are 26 members or elements in this set. They are: 1) ABCA 2) ABCB 3) ABCC 4) ABCD ... 26) ABCZ Note that
we did not list all 26 elements; rather we listed a pattern of set membership
which the reader is expected to fill in.
For example, the first three members of the set which are not listed above
are: 5) ABCE 6) ABCF 7) ABCG Can you
tell the last element of the set which is not listed above? The answer is: 25) ABCY Many times
all of the elements of the set are not listed, but only a pattern is given. Sets are
very important to understand when discussing key mathematical concepts because
in many cases it is impractical or impossible to list all of the elements of a
set. What is a Combination? To
understand what a "combination" is let us consider the set of 26 letters
in the English alphabet: {a, b, c, d, e, ..., y, z} Now let us
consider the set of all possible ways to pick 5 letters of the alphabet. Here are some examples: {a, b, c, d, e} {a, b, c, d, f} {a, b, c, d, g} {a, b, c, d, h} . . . Each set of
5 items just listed represents a "subset" of the set of all letters
of the alphabet. A "subset"
means "part of a set."
Actually, in some cases a "subset" can include all of
the elements of a "set." Thus, when
considering "combinations" of the letters of the English alphabet, we
will only consider 5 of them at a time, meaning a "subset" of all the
letters of the English alphabet, which has 5 elements. What
happens when two of these subsets have the same elements, but the elements are
listed in a different order? For
example, let us consider these three possible subsets of the alphabet: {a, b, c, d, e} {e, d, c, b, a} {a, c, b, d, e} These three
subsets have the same elements, but the elements are listed in a different
order. When discussing
"combinations" only one of these subsets would be listed as a
"combination" subset. It
doesn't matter which one is listed, or even if a different subset was listed
(e.g. {d, c, b, e, a}). The order of the
elements is not important. It is the
"combination" of 5 different elements which must be unique, but the
order which is chosen to be listed is irrelevant. In some
cases elements can appear more than once in a combination. For example, this is a combination where
"redundancy" (i.e. an element is allowed more than once) is allowed: (a, a, c,
c, d} Again,
order is not important, only the rules for defining set membership is
important. What is a Permutation? A
"permutation" is the same thing as a "combination" except
that a "permutation" is concerned about the "order" of the
elements in each subset. Thus, when
listing subsets of 5 letters of the alphabet, and when considering
permutations, all four of these subsets, and many others, would be in the list
of permutations: {a, b, c, d, e} {e, d, c, b, a} {a, c, b, d, e} {d, c, b, e, a} . . . Even though
they have the same letters, all of them are counted as a different and unique
permutation because the letters are listed in a different and unique order. When
dealing with permutations in this book, redundancy is always allowed, meaning
the same letter can appear any number of times.
In other words, this would be allowed: {a, a, a, a, a} Here is
another way to list permutations of 5 letters of the alphabet: abcde edcba acbde dcbea abbbc abbcb tpzat qqqmq turew zpzpz wxdee Note that
each "element" of this listing is itself a set, or technically a
subset of letters of the alphabet. Thus, the
set of 26 letters of the alphabet is a "set." The listing of all possible permutations of 5
letters of the alphabet is a "set."
And each item listed (e.g. "abcde")
is also a "set," or it could be called a "subset." In this
book the focus will be on permutations because this book will be concerned with
DNA, and the order of nucleotides on DNA is very important!! Thus, the "order" of things will be
critical and redundancy is always allowed. So how many
different ways can we uniquely order 5 letters of the English alphabet? Try to guess before reading any further and
write down your guess. First, let
us clarify the rules. Rule 1) The elements (i.e. elements of the listing); meaning the "subsets"
in the listing; must consist of 5 letters of the English alphabet. Rule 2) The order of the letters (in each element) is important,
meaning each element (i.e. each subset) must be unique (i.e. the ordering of 5
letters cannot be found anywhere else in the listing). Thus, aaaab and baaaa are two distinct and different elements of the
listing. But if aaaab
is the 50th element of the listing, it cannot also be the 1 millionth element in the listing. Rule 3)
Redundancy is allowed (i.e. the same letter can be used more than once in a
single element). Thus, 'mmmmm' is an element of this set. Rule 4) Every possible unique ordering of 5 letters, with
redundancy, is required to be in the set. With these
four rules, the number of "elements" or "subsets" or
"items" in the listing (i.e. the set of permutations) turns out to be
526. The exponent, 26,
represents the number of letters in the English alphabet; and the base,
5, represents the number of letters in each element of the listing. This is
equal to: 1,490,116,119,384,770,000
permutations (i.e. items in the listing) That is
more than one quintillion, so don't try to order them at home, you will run out
of paper and time. Let us go
through the process of finding some of these elements or subsets: First, we
list the 26 letters of the English alphabet: {a, b, c, d, c, ..., x, y, z} Then let us
pick all possible "combinations" of 5 of these letters. Redundancy of individual letters is allowed,
thus here are some "combinations" which include the same letter more
than once: {a, b, c, e, e} {a, a, a, e, f} {a, b, b, b, b} {m, n, p, p, z} {n, n, p, y, y} and so
on. Now, after
we have every possible "combination" of 5 elements, with and without
redundancy, we will go back through the list and "order" each
possible "combination" every possible way (i.e. we will take each
"combination" and consider every permutation (i.e. ordering) of these
5 elements. This is one
way of finding every possible "permutation" of 5 elements of the
English alphabet. So, what is
a "permutation?" Every one of
the 526 elements of the set we just talked about is a
"permutation." See if you
can easily understand this next sentence: A
"permutation" is every one of the 1,490,116,119,384,770,000 elements
in the listing of: 5 elements of the alphabet, where every possible ordering,
of every possible combination of 5 letters, is important, and redundancy is
allowed within the 5 letters of an element. In other
words, given every combination of 5 letters of the English alphabet, every
unique way of ordering these combinations of 5 letters is a
"permutation." When talking
about a "permutation," the order of the letters is important and
redundancy is allowed. Let us see
an example which is so simple we can list every possible permutation. Let us
consider three people: Bob, Bill and Mary.
How many different ways can we "order" these three names? This is exactly the same question as this:
how many different permutations are there when listing the names of three
people: Bob, Bill and Mary? They are the
same question. The original
"set" is the names of three people: Bob, Bill and Mary. Let us
think for a moment about "combinations" again. If we considered all three of these names, as
"combinations," taken three names at a time, how many
combinations would there be? These are
the combinations: (Each
person is listed once) (order is not important) Bob, Bill,
Mary (Bob is listed twice) (order
is not important) Bob, Bob,
Bill Bob, Bob,
Mary (Bill is
listed twice) (order is not important) Bill, Bill,
Bob Bill, Bill,
Mary (Mary is
listed twice) (order is not important) Mary, Mary,
Bob Mary, Mary,
Bill (Each
person is listed three times) Bob, Bob,
Bob Bill, Bill,
Bill Mary, Mary,
Mary That's
it. There are 10 combinations. How many
permutations are there of these three names?
When considering permutations of all three names, there is more than one
possible permutation for most of the combinations listed above. There are in fact, 27 different permutations. Try to list these 27 different ways before
reading any further. (Each
person is listed three times) Bob, Bob,
Bob Bill, Bill,
Bill Mary, Mary,
Mary (Bob is
listed twice) Bob, Bob,
Bill Bob, Bill,
Bob Bill, Bob,
Bob Bob, Bob,
Mary Bob, Mary,
Bob Mary, Bob,
Bob (Bill is
listed twice) Left to the
reader - should be 6 items or elements in list (Mary is
listed twice) Left to the
reader - should be 6 items or elements in list (Each
person is listed once) Bob, Bill,
Mary Bob, Mary,
Bill Bill, Bob,
Mary Bill, Mary,
Bob Mary, Bob,
Bill Mary, Bill
Bob In total,
there are 27 permutations. This is 33. Each is a
"permutation" of three names and each is a "unique
ordering" of three names. The term
"permutation" and the term "unique ordering" mean exactly
the same thing. Permutations and DNA How many
different ways can we uniquely order (i.e. how many different permutations)
four of these "letters:" A, C, G, T? These 4 letters represent the four different
types of nucleotides, which are the key molecules which make up DNA. The answer, of course, is 44. Here are some examples: ACCT GGGG TGTA AACT ACTG GTCA The study
of permutations of nucleotides is at the heart and soul of the evolution
debate. Let us ask,
how many permutations are there in a string of 150 nucleotides? There are 4150 permutations. This looks like a small number. Do you think you could list all of the
different permutations? Just how big is
this small-looking number? A galaxy in
our Universe consists of about 100 billion stars. Our sun, for example, is really a star. If you were several light-years away (a
"light-year" is the distance the speed of light would travel in one
year), and you looked at our sun from far away; our sun would look like any
other star. So how many
galaxies are there in our Universe? About 100 billion galaxies, which have an average size of
about 100 billion stars. Comparing
the size of our earth to the size of the average star would be like comparing
the size of a beach ball to the size of a Ferris wheel. Stars are huge in comparison to our little,
puny earth. Yet, there
are 100 billion galaxies and 100 billion stars, on average, in each
galaxy. This is a total of about
10,000,000,000,000,000,000,000 stars in our Universe. This is 1022 stars. Now that we
have talked about big things, let's talk about little things - atoms. Atoms are very small. They are so small it would take about 5
million million hydrogen atoms to fill an area the
size of the head of a pin. This is 5 x
1012 or 5,000,000,000,000 atoms in an area the size of the head of a
pin!! Yet, in
spite of these huge and small numbers, there are only about 1080 atoms
in our entire Universe. In other
words, there are about: 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 atoms in our Universe. The number
we just talked about (4150) is BIGGER than the number 1080. In fact, 4150 is approximately
equal to 1090. In other
words, we could order 150 nucleotides more uniquely different ways than
there are atoms in our Universe!! Are you
beginning to see the power of permutations?
They look small, but in fact they are huge!!! The "First Living
Cell" Now let us
assume the "first living cell" of evolution had 900,000
nucleotides. How many permutations of
900,000 nucleotides are there? The
answer is 4900,000. How much
bigger is 4900,000 than 4150,
and remember that the number 4150 is bigger than the number of atoms
in our Universe? Try to
calculate it before reading on. If you said
6,000 times bigger, you would be wrong.
The correct answer is 4899,850 times bigger!!! Remember
from above, when you are dividing exponents, which have a common base,
you subtract their exponents, you do not
divide their exponents. Thus, 4900,000
divided by 4150 is equal to 4(900,000-150) = 4899,850. And this is
just the "first living cell."
Human DNA has 3,000,000,000 pairs of nucleotides!! There are 43,000,000,000 unique
permutations of 3 billion nucleotides. This is
just an introduction to the subject of permutations of nucleotides. Comment If you are
lost at this point, you would be wise to seek out someone who can explain these
things to you before going on because the use of exponents and an understanding
of permutations will be very important in the rest of this book. As a
minimum read these same concepts from another source to make sure you
understand them. |