Prophets or Evolution - An LDS Perspective

 

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 when multiplying numbers which have 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 like 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 (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 10‑1.  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):

 

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 10‑2.  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 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 5th grade class," is not 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 exactly 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 Jefferson Grade School," then we could identify exactly which girls belonged to the set.  And we could identify which girls were not in the set.

 

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.

 

 

Subsets

 

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 some of the elements of the set, but not all of them.

 

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.

 

 

What is a Combination?

 

Let us consider the set, Set 5U, of all possible ways to pick 5 unique letters of the alphabet (duplicates are not allowed).  Here are some examples as shown above:

{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 not elements of "Set 5U" because each set has duplicates:

{a, b, c, e, e}

{a, b, c, c, f}

{a, a, c, d, g}

{a, a, a, a, h}

. . .

 

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 not important.

 

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.

 

 

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, 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!!

 

 

The Number of Elements of a Set of Permutations

 

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 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/subset in the listing.

 

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 526 elements of the set we just talked about is a unique "permutation."

 

Here is a key statement you need to understand.  Set 5A can be defined thusly: "Set 5A is the set of all possible permutations of 5 letters of the alphabet."

 

Thus every one of the 1,490,116,119,384,770,000 elements in the listing of Set 5A is a unique permutation.

 

 

A Simple Example

 

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 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

 

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 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 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 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 in more unique 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!!!

 

But human DNA does not contain 150 nucleotides, it includes:

3,000,000,000 pairs 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.