"If you want to make an apple pie from scratch, you
must first create the Universe." Carl Sagan, astronomer
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: 10
10 Thus,
listing the number 10, being multiplied by itself 12
times, is written 10 The
"base" does not have to be 10.
For example, how would you write out 4 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 4 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.
When you
multiply exponents, For
example, this is legal: 10 It is legal
because all three exponents have the same base: 10 But this is
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, For
example: 10 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 when multiplying numbers which have exponents. It is
always important to remember that when multiplying exponents
When
dividing exponents, the same rule applies: when dividing exponents When
dividing exponents we Thus, 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): ( Only one 10
is not underlined. Thus, the answer is: 10 Again, our
method leads to a logical answer. Also
remember,
What does a
number like 10 Actually,
10 10 Another way
to write 10 We can look
this chart to better understand negative exponents: 10 10 10 10 10 10 10 10 10 10 Thus, 10
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 (e.g. with numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), and
each side has the same chance of landing on top. The probability of rolling a '1' is 1 / 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 Is this
logical? Let us think about all the different
orderings of rolling a ten-sided dice twice (there are 10 The three
dots (. . .) mean that some of the items are missing and the reader is expected
to be able to figure out which pairs of numbers are missing. 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
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 collection of all 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 set 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} 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: 0, 2, 4, 6, 8. 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 same as: "all the girls in the 5th grade." The first statement would not be a set until we
refined the definition of set membership so we could determine notexactly
which girls were in the set (e.g. which 5th grade class the set refers to).If we said:
"the girls in Mrs. Jones 5th grade class at When we said
above: "all the girls in the 5th grade," this is an accurate enough
description of a set that we can determine the exact set membership (assuming
we knew which school we were talking about). Thus, a
"set" is merely a well-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.
A
"subset" of a set means "part of the set." In other words, you define the elements of a
"parent set," then a "subset" is For
example, suppose you defined a parent set (commonly called the Universal Set) to
be the following names: {fred, john, herman,
mary, ann, marilyn} This would
be a subset of the parent set: {fred, herman,
marilyn} This would
also be a subset of the parent set: {john} However, a
"subset" is sometimes defined so that all of the elements of the
parent set are elements of the subset.
For example, sometimes this would be a valid subset of the above parent
set: {fred, john, herman,
mary, ann, marilyn} In
mathematics, frequently we are interested in all possible subsets of a set which
follow a particular rule. For
example, suppose we defined the parent set to be all the letters of the
alphabet: {a, b, c, . . ., x, y, z} Here is a list
of "subsets" of that set which contain 5 unique elements of the set: {a, b, c, d, e} {a, b, c, d, f} {a, b, c, d, g} {a, b, c, d, h} . . . The three
dots (". . .") at the bottom of the listing indicates that we have
not listed every possible subset, but only a pattern or a sample of the
elements of the subset. In fact, each
element of the above set are themselves sets (i.e.
sets of five letters of the alphabet).
Thus, a set can have sets as members.
Let us consider
the set, letters of the
alphabet (duplicates are not allowed).
Here are some examples as shown above:unique{a, b, c, d, e} {a, b, c, d, f} {a, b, c, d, g} {a, b, c, d, h} . . . Here are a
few sets with 5 letters of the alphabet which are {a, b, c, }e{a, b, , f}c{ , c, d, g}a{ , a, a, h}a. . . The above
sets of 5 elements are not valid elements of Set 5U because they do not follow
the rules which defined the set. There are
two key rules when thinking about sets which are defined to be a
"combination." Rule #1 is
that duplicates are not allowed. Rule #2 is
that the order of the elements in the set is Now let us
think of Set 5U as a "combination."
We have already forbidden using the same letter more than once in each
set. But now we also have to exclude
sets which contain the same 5 letters, but the letters are not in the same
order. We have to exclude them because
the order of the elements in each set are not
important, and we don't want to repeat a set more than once. For
example, let us look at this proposed listing of elements of Set 5U: {a, b, c, d, e} {a, e, b, c, d} {e, a, b, c, d} {a, b, c, e, d} . . . Note that in
all 5 of these potential elements or subsets there are five letters, but in
each case the 5 letters are the same letters {a, b, c, d, e};
they are simply ordered differently. Are all 5
of these potential elements members of the Set 5U, now that we have defined it
to be a combination? When
talking about combinations, only one of these elements would be in the
set. And which of the elements is chosen
to be in the set is not important, because the order of the elements is not
important. In other words, any of the
elements could be in the set, but only one of them can be in Set 5U. Remember, when
defining a "combination" type of set, it doesn't matter which order
the elements in a row are listed. It is
the "combination" of 5 different elements which must be unique (i.e.
duplicates are not allowed), not the order of the letters in the element.
A
"permutation" is the same thing as a "combination" except
that a "permutation" is concerned about the "order" of the
elements in each subset, plus duplicates are allowed. Thus, a
permutation does away with the two main rules of a combination. Let us define
Set 5A to be the same as Set 5U, but in this case Set 5A is a permutation set. Each of
these sets would be an element of Set 5A if Set 5A were defined to be a
"permutation": {a, b, c, d, e} {e, d, c, b, a} {a, c, b, d, e} {d, d, d, e, a} . . . As noted in
the last element above, duplication of letters is also allowed, thus these
would also be elements of Set 5A: {a, a, a, d, z} {z, d, a, a, a} {a, d, a, a, z} {d, d, c, z, c} . . . Needless to
say, Set 5A would be much, much larger than Set 5U because it has more relaxed
rules!! The set of
26 letters of the alphabet is also a "set," but it is not a member of
Set 5A because each member or element or subset of Set 5A has exactly 5
elements. 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 and duplicates are
always allowed!!
So how many
different ways can we uniquely order 5 letters of the English alphabet? Make a wild guess before reading any further
and write down your guess. Do not try to
count them, you will see why in a moment. First, let
us clarify the rules. Rule 1) The elements (i.e. elements of the listing); meaning the "subsets"
in the listing; must each consist of 5 letters of the English alphabet. Rule 2) The order of the letters (in each element subset) 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 because the same element would appear in the listing more than
once. 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 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
5 This is
equal to: 1,490,116,119,384,770,000
permutations (i.e. items in the listing) This is
more than 1 quintillion. Now you know
why you shouldn't try to count them one at a time. So, what is
a "permutation?" Every one of
the 5 Here is a
key statement you need to understand.
Set 5A can be defined thusly: " Thus every
one of the 1,490,116,119,384,770,000 elements in the listing of Set 5A is a
unique permutation.
Since there
is not enough paper in the world to list all the elements of Set 5A, let us look
at a much smaller set so 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. 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 3 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.
While the
English alphabet has 26 letters, the DNA alphabet only has 4 letters: A, C, G,
and T 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 4 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 4 A galaxy in
our Universe consists of about So how many
galaxies are there in our Universe? About Comparing
the size of our earth to the size of the average star would be like comparing
the size of a tennis 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 10 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
10 Yet, in
spite of these huge and small numbers, there are only about 10 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 (4 In other
words, we could order 150 nucleotides in more unique ways Are you
beginning to see the power of permutations?
They look small, but in fact they are huge!!! But human
DNA does not contain 150 nucleotides, it includes: 3,000,000,000 pairs of nucleotides.
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. |