Download Week 14: Whole Number Axioms

Week 14: Whole Number Axioms
I’m sorry to say that the subject I most disliked was mathematics. I have
thought about it. I think the reason was that mathematics leaves no room
for argument.
Malcolm X
The Natural Numbers
Many people would say that mathematics is “the science of numbers.” This is a common
misconception among those who are unfamiliar with the most modern parts of math – that
is, ideas that have been developed since the nineteenth century or so. But it is true that
most of mathematics does have some connection to numbers.
One of the central ideals of mathematics is that all mathematical truths can be proved from
other, more basic truths, which can be proved from even more basic truths, and so on.
Think about some of the mathematical statements for which we wrote proofs in the first
few weeks of the course. Many proofs used what you might think of as “algebra
knowledge.” For example, at one point we used the fact that 𝑥 2 + 2𝑥 + 1 can always be
rewritten as (𝑥 + 1)2 . But this fact itself can also be proved, from more basic facts about
numbers. And those facts might be able to be proved from even more basic facts…
We can keep thinking in this way until we reach a starting point, where we must simply
assume that certain statements about numbers are true, because they cannot be proved
from any others. The foundation upon which all of mathematics is built is the assumption of
certain statements – which are usually simple and very uncontroversial – from which we
move forward through proofs. The fact that we must take certain truths for granted at the
beginning should not be alarming; in a way, these foundational statements, or “axioms,”
define mathematics.
Definition: An axiom is a statement that is not proved from other statements, but is simply
assumed to be true. An axiom is also sometimes known as a “postulate.”
It is important to note that every mathematical fact is ultimately derived from
mathematical axioms. Mathematics is a work of deductive reasoning. This means that we
begin with certain assumptions and move forward by deducing other facts from these.
Some mathematicians have gone so far as to describe mathematics as a game we play, with
the axioms being the rules of the game – rules that are made up before we begin.
On the topic of numbers, the most basic kind of number are the elements of ℕ – that is, the
natural numbers 0, 1, 2, and so on. What are some of the most basic rules about these
numbers – rules that anyone would agree with? Let’s use the following.
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Some Axioms for the Natural Numbers:
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For any natural numbers 𝑥 and 𝑦, the numbers can be “added” to get a “sum,” or
“multiplied” to get a “product.” The sum is written as 𝑥 + 𝑦 and the product is
written as 𝑥 ⋅ 𝑦.
The sum of two natural numbers is always a natural number. (The technical way to
say this is that the addition of natural numbers is closed.)
The product of two natural numbers is always a natural number. (The technical way
to say this is that the multiplication of natural numbers is closed.)
The order in which two natural numbers are added does not matter. In other words,
for any natural numbers 𝑥 and 𝑦, 𝑥 + 𝑦 = 𝑦 + 𝑥. (The technical way to say this is
that addition is commutative.)1
The order in which two natural numbers are multiplied does not matter. In other
words, for any natural numbers 𝑥 and 𝑦, 𝑥 ⋅ 𝑦 = 𝑦 ⋅ 𝑥. (The technical way to say this
is that multiplication is commutative.)
Taking the sum of two numbers and adding a third is the same as adding the first
number to the sum of the last two. In other words, for any natural numbers 𝑥, 𝑦, and
𝑧, (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧). The technical way to say this is that addition is
associative.
The same goes for multiplication. In other words, for any natural numbers 𝑥, 𝑦, and
𝑧, (𝑥 ⋅ 𝑦) ⋅ 𝑧 = 𝑥 ⋅ (𝑦 ⋅ 𝑧). That is, multiplication is associative.
There is a definite relationship between addition and multiplication. For any natural
numbers 𝑥, 𝑦, and 𝑧, 𝑥 ⋅ (𝑦 + 𝑧) = 𝑥 ⋅ 𝑦 + 𝑥 ⋅ 𝑧. The technical way to say this is that
multiplication distributes over addition.
The sum of any natural number and 0 is the original natural number. In other
words, for any natural number 𝑥, 𝑥 + 0 = 𝑥. The technical way to say this is that 0 is
an additive identity.
The product of any natural number and 1 is the original natural number. In other
words, for any natural number 𝑥, 𝑥 ⋅ 1 = 𝑥. The technical way to say this is that 1 is a
multiplicative identity.
For any natural numbers 𝑥, 𝑦, and 𝑧, if 𝑥 + 𝑧 = 𝑦 + 𝑧, then 𝑥 must be the same as 𝑦.
The technical way to say this is that addition can be cancelled.
Sometimes, multiplication can be “cancelled.” See Exercise 6.
Hopefully, none of these twelve rules about natural numbers is controversial. In fact, as you
think about the axioms, you may find that there are other seemingly basic facts about
natural numbers that have been omitted. When writing axioms in mathematics, it is usually
considered bad form to write more than we need. If we do so, it is often said that the
axioms are not independent. As an analogy, think about the rules of chess, which dictate
how the pieces are placed and how they may move. It would be pointless to add a rule that
When reading these statements, it is a good idea to think of the “equals sign,” =, as if it stand for the words
“is the same as.” For example, given any natural numbers 𝑥 and 𝑦, 𝑥 + 𝑦 is the same as 𝑦 + 𝑥.
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states “Neither player may move his or her queen on the first turn,” because the existing
rules already take care of this. One might say that this fact of chess is a theorem, rather than
an axiom.
Getting back to the axioms above, one of them says that 1 is a multiplicative identity, which
means that when any natural number is multiplied by 1, the result is just the original
natural number. The axiom does not say that 1 is the only multiplicative identity, but we
can prove that this has to be true, based on what the axioms do explicitly say.
Proposition: There is only one multiplicative identity.
Proof: Suppose 𝑚 is any multiplicative identity. This means that when any natural number
is multiplied by 𝑚, the result is the original natural number. This guarantees that 1 ⋅ 𝑚
must be 1. Because multiplication is commutative, 1 ⋅ 𝑚 is the same as 𝑚 ⋅ 1, and so
𝑚 ⋅ 1 = 1. Because 1 is a multiplicative identity, 𝑚 ⋅ 1 is the same as 𝑚, and therefore 𝑚 =
1. This proves that if 𝑚 is a multiplicative identity, then 𝑚 must be 1. There is only one
multiplicative identity. ∎
For the rest of the course, we will be working at a level of mathematics very close to the
foundational axioms, and so our proofs must be written very carefully, with each logical
step explicitly justified by one of the axioms (or by a fact that we have already proved
directly from the axioms).
Another extremely basic fact that is noticeably absent from the axioms above is that when
any natural number is multiplied by 0, the result is 0. This rule is not needed, because it can
be proved from the others. Here is a detailed outline of the proof.
Proposition: If 𝑛 is any natural number, then 𝑛 ⋅ 0 = 0.
Proof: Let 𝑛 be any natural number. Then 𝑛 + 𝑛 ⋅ 0 is the same as 𝑛 ⋅ 1 + 𝑛 ⋅ 0 because 1 is
a multiplicative identity. And 𝑛 ⋅ 1 + 𝑛 ⋅ 0 is the same as 𝑛 ⋅ (1 + 0) because …. Next,
𝑛 ⋅ (1 + 0) is the same as 𝑛 ⋅ 1, because …, which is the same as 𝑛, because …. Lastly, 𝑛 is the
same as 𝑛 + 0 because …. In summary, we have 𝑛 + 𝑛 ⋅ 0 = 𝑛 + 0, so …… ∎
Exercise 1: Find any proof from Weeks 1 through 4 where we used the fact that the
addition of whole numbers is closed. Rewrite that proof so that this axiom is referenced
explicitly.
Exercise 2: Find any proof from Weeks 1 through 4 where we used the fact that the
multiplication of whole numbers is closed. Rewrite that proof so that this axiom is
referenced explicitly.
Exercise 3: Prove that there is only one additive identity.
Exercise 4: Let’s define 2 to be 1 + 1. Prove that if 𝑥 is a natural number, then
(𝑥 + 1) ⋅ (𝑥 + 1) = 𝑥 ⋅ 𝑥 + 2 ⋅ 𝑥 + 1.
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Exercise 5: Fill in the missing justifications in the proof given above that if 𝑛 is any natural
number, then 𝑛 ⋅ 0 = 0.
Exercise 6: Consider the following statement:
For any natural numbers 𝑥, 𝑦, and 𝑧, if 𝑥 ⋅ 𝑧 = 𝑦 ⋅ 𝑧, then 𝑥 must be the same
as 𝑦.
(a) Using only the first eleven axioms, prove that this statement cannot be true.
(As a side note, if we were to include this statement as an axiom, we would
have a set of axioms that contradict each other, which is called
inconsistent.)
(b) Give the “correct” axiom to govern when multiplication can be cancelled.
Some Definitions
Note that the axioms of the natural numbers do not mention the concept of “subtracting”
numbers. You should not have used the “minus sign” at any point in Exercises 1 through 6.
Definition: A natural number 𝑧 is called the difference of natural numbers 𝑥 and 𝑦 (written
as 𝑥 − 𝑦) if and only if 𝑥 = 𝑦 + 𝑧.
Definition: A natural number 𝑥 is greater than or equal to a natural number 𝑦 (or,
synonymously, 𝑦 is less than or equal to 𝑥) if and only if 𝑥 − 𝑦 is a natural number. This is
written “𝑥 ≥ 𝑦” or “𝑦 ≤ 𝑥.”
Exercise 7: Prove that if 𝑛 is any natural number, then 𝑛 − 𝑛 = 0.
Exercise 8: Write a properly-worded definition of the phrases “less than” and “greater
than” for natural numbers.
Exercise 9: Prove that every natural number is greater than or equal to 0.
Quite contrary to popular opinion, the main purpose all the definitions and symbols in
mathematics is to simplify the subject. When there is a concept or type of calculation that
comes up over and over again in thinking about mathematics, it saves us a lot of work to
create a new name and/or symbol for it. For example, there are many interesting facts that
are related to whether or not a given number is a multiple of 2, and therefore we have the
word “even.” Likewise, it is often useful to un-add numbers (so to speak), and therefore we
have the idea of “subtraction.”
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In a similar vein, early mathematicians often found themselves thinking about multiplying
a number by itself, and so they developed a shorthand notation for it: they began to write
𝑥 2 instead of 𝑥 ⋅ 𝑥, and 𝑥 3 instead of 𝑥 ⋅ 𝑥 ⋅ 𝑥, and 𝑥 4 instead of 𝑥 ⋅ 𝑥 ⋅ 𝑥 ⋅ 𝑥, etc. From this
pattern of examples, it should be clear to almost anyone how this new “exponential”
notation works. However, to move forward with absolute certainty, let’s write this down as
a more formal definition.
Definition: If 𝑥 is any natural number, then 𝑥 2 = 𝑥 ⋅ 𝑥, and if 𝑥 𝑛 is defined, then
𝑥 𝑛 ⋅ 𝑥 = 𝑥 𝑛+1 .
Exercise 10: When exponential notation was first used as a shortcut for repeated
multiplication, there would have been no reason to ever think about writing something like
“𝑥1 .” However, prove that if 𝑥1 is defined and 𝑥 is not 0, then 𝑥1 must be 𝑥.
Exercise 11: Prove that if 𝑥1 = 𝑥, and 𝑥 0 is defined, and 𝑥 is not 0, then 𝑥 0 must be 1.
Exercise 12: According to the axioms and definitions given so far (including the last two
exercises), can there be a definition for 00 ? Explain your answer.
(Optional) Closing Remarks
The statements about natural numbers that we have used here as axioms are, in the most
modern view of mathematics, commonly seen as theorems that can actually be proved,
with great difficulty, from a small set of even more fundamental rules, which are known as
Peano’s axioms. However, our assumptions, which we will continue to use, are the
immediate consequences of Peano’s axioms, and serve the purposes of this course more
than adequately.
A tour of Peano’s axioms, and how our assumptions can be derived directly from them, is
given in possibly the least technical form in a book called One, Two, Three: Absolutely
Elementary Mathematics by David Berlinski.