The Real Numbers

Back to Contents

If I have a number P of sets and each of those sets contains a number Q of elements (and none of the sets have elements in common with any of the other sets) and if I combine all of the elements from all of those sets into a single set, then that single set contains R = PQ elements. In writing P and Q together like that I have indicated that we must combine the numbers that they represent through the process of multiplication, the process of multiple addition. Multiplication offers us a convenience over the tedium of actual multiple addition, because our place-notation representation of numbers allows us to use the process of summing partial products to multiply large numbers together.

Because multiplication comprises nothing more than repeated additions, we can infer some rules immediately. First we have the commutative rule: if we want to multiply P by Q, then changing the order in which we write the numbers down does not change the product; that is,

PQ = QP.

(Eq'n 1)

Whether we have five trucks carrying seven barrels each or seven trucks carrying five barrels each, we still have thirty-five barrels.

And second we have the associative rule: if we want to add together numbers Q and T and then multiply their sum by P, then changing the order in which we carry out the addition and the multiplication does not change the result; that is,

P(Q+T) = PQ + PT.

(Eq'n 2)

If a truck carries eight pallets of bricks, we can have eleven trucks carrying bricks from one supplier to our worksite and nineteen trucks carrying bricks from another supplier or we can send all thirty trucks directly to the brickworks: in either case we will have two hundred and forty pallets of bricks delivered to our worksite.

But we have a new kind of number to consider and we must ask whether we can carry out multiplication with negative numbers and ask what such multiplications mean. Since negative numbers represent deficiencies or debts, we can think of their multiples as multiple debts. But can we multiply them?

If I owe each of seven people five dollars, then I owe thirty-five dollars. I represent the number of people with a positive seven and the debts I owe them with a negative five and I then say that I have negative thirty-five dollars; that is, if someone gave me thirty-five dollars, I would end up with zero dollars because I would have to pay the debts -- the positive cancels the negative. So multiplying a positive number by a negative number yields a negative number. What do we get when we multiply a negative number by a negative number?

Let's start with some number N and write N-N=0. If we then multiply that equation by another number -M, we get

-M(N-N)=0.

(Eq'n 3)

We apply the associative rule to that equation to convert it into

(-M)(+N)+(-M)(-N)=0.

(Eq'n 4)

But the product of minus M and plus N yields a negative number. We know that if we add a negative number to something and get zero, then the something must be a positive number; therefore, from Equation 4 we infer that multiplying a negative number by a negative number yields a positive number.

Thus we have the rules of multiplication.

We can reverse the basic process of multiplication and divide a set containing R elements into S identical sets, each containing T = R÷S elements. We model the process by sequential subtraction: remove S elements from the set R and put one element into each of the new sets and repeat the procedure until the original set contains zero elements.

Suppose that we have carried out that procedure T times, putting T elements into each of the new sets, and we find that we have V elements left in the original set with V<S. We have only one way in which we can distribute those leftover elements among the new sets without violating the criterion that we make those sets all identical: we must break each element into S identical pieces and then repeat the procedure above V times. Thus, each of the new sets contains T whole elements and V broken elements. In order to denumerate that fact we need to indicate into how many pieces we broke each of the leftover elements so that we can devise a proper broken number (fraction) to represent those pieces in each set. We have, then, T+V/S elements in each set.

The fractions now become a new kind of number that we include with the integers. We now have an expanded set comprising the integers, the fractions, and the mixed numbers (each comprising an integer plus a fraction): we name that set the set of the real numbers. If we now lay out the integers on a line and insert the fractions between them to represent the mixed numbers, we have a linear representation of the set of all of the real numbers.

Just as addition and subtraction allow us to solve the equation X+M=N, multiplication and division allow us to solve the equation

AX+B = C.

(Eq'n 5)

In fact, we can use that equation to prove and verify the proposition that the real numbers comprise a set large enough to contain a guaranteed solution of that equation, regardless of the values we assign to A, B, and C.

Appendix:

The Fundamental Theorem of Arithmetic

We have with the processes of multiplication and division the definition of a new set of numbers. We can take the numbers in sequence and multiply the other numbers by them. So, for example, we take the products of multiplying the numbers by two and get 4, 6, 8, 10, 12, and so on. From three we get 6, 9, 12, 15, 18, 21, and so on. Combining all of those products gives us the set of composite numbers, the numbers that we can represent as the products of two or more integers.

If we compare that set with the set of natural numbers (the positive integers), we will see that it has gaps in its membership. Missing from that set are 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on. We cannot represent these numbers as products of two or more integers. These are the uncuttables, the atoms, of arithmetic. We call them the prime numbers.

We can now state

The Fundamental Theorem of Arithmetic

"The canonical factorization of a natural number is unique."

or

"Every positive integer is either prime or can be

uniquely factored as a product of primes."

We can prove and verify that theorem as true to mathematics through a standard three stage process. First we state, prove, and verify Euclid's First Theorem, then we demonstrate that we can factor any composite number N into a set of factors that contains only prime numbers, and finally we show that the set that we have so obtained must be unique, that no other possible set of prime factors will multiply out to the same number N.

Euclid's First Theorem

Proposition: an integer P>1 is prime if and only if it satisfies the following statement; for all integers M and N the statement that P evenly divides the product MN necessarily implies that P evenly divides M or that P evenly divides N.

Proof of the If part: Take the proposition as true to mathematics and assume that P is a composite number, P=AB for integers A<P and B<P. P certainly divides itself evenly, so, by the proposition, it must evenly divide A or B. But it is impossible to divide a number evenly into a smaller number, so we must discharge on of the premises upon which we reached that result; thus, we dismiss the assumption that P is composite. P must be prime, which necessitates for any integer N that the greatest common divisor of N and P must equal one or P (gcd(N, P)=1 or P), because P has no divisors.

Proof of the Only If part: Let M and N represent non-zero integers in the proposition above. Because P is prime we know that gcd(M, P) = 1 or P. We now have two possibilities;

a. If gcd(M, P) = P, then P evenly divides M.

b. If gcd(M, P) = 1, then we must carry out a calculation. We know that we can always find integers (positive or negative) X and Y that allow us to express the greatest common divisor of M and P as

gcd(M, P) = MX+PY.

(Eq'n 6)

We have assumed that P evenly divides the product MN, so we know that we have a natural number K such that MN=PK. Now let's take the case gcd(M, P)=1 and multiply the gcd by N; we get

MNX+PNY=N=PKX+PNY.

(Eq'n 7)

We can see that P evenly divides the right side of that equation, so now we know that P evenly divides N.

Thus Euclid's First Theorem stands proven and verified as true to mathematics.

Proof of Existence

To prove and verify the proposition that we can represent any natural number N as a product of prime numbers only, we assume the existence of a number G that we cannot so represent. We also assume that G is the smallest member of the set of such numbers; that is, the set of all numbers that we cannot represent as products of prime factors only. It must be a composite number (G=RS), because if it were a prime number itself, it would constitute a product of prime numbers, albeit the simplest one. By our assumption that G is not a product of primes we infer that R and S must also be composite numbers, but, because they are both smaller than G, they are no elements of the set of numbers that cannot be represented as the products of prime numbers only and we can represent them as products of prime numbers only. But that makes their product a product of primes only, which contradicts our assumption about G. We must infer either that G is not the smallest number that we cannot represent as a product of primes only or that no number cannot be expressed as a product of primes only.

No product of positive integers can have a value less than four, so we know that we have a smallest composite number of the natural numbers. Thus in any subset of the composite natural numbers we must find a smallest member and we have assumed that we have done just that with the set of all composite natural numbers that do not have all-prime factorizations. We then eliminated that member from the set, thereby rendering the set empty (After eliminating the smallest member, we find that another member has become the smallest and we eliminate it in the same way. We continue that process until the set has no more members in it.). So now we know that an all-prime factorization exists for any natural number N.

Proof of Uniqueness

We now represent the natural number N as a product of a set of prime numbers {P} and assume that we can also represent that same number as the product of a different set of prime numbers {Q}, so we have NP=NQ. Each and every element Pi in {P} divides NP evenly and so must also divide NQ evenly. That fact necessitates that Pi exist as an element of {Q}, because no prime number divides another prime number evenly. Because every prime that exists in {P} divides N evenly, we infer that all of the primes in {P} also exist as elements in {Q}. Likewise, from the fact that each and every element Qj in {Q} divides NQ evenly we infer that all of the primes in {Q} also exist as elements in {P}. Thus the sets {P} and {Q} contain the same prime numbers and neither set contains primes that the other does not contain.

But each prime may occur as an element of its set more than once. For example, eleven occurs three times in the prime factorization of 9,195,879 (=3x72x113x47). Because no prime can evenly divide another prime, we know that every occurrence of a prime in {P} must be matched by an occurrence of the same prime in {Q} and that every occurrence of a prime in {Q} must be matched by an occurrence of the same prime in {P}. Thus {P} and {Q} must be identical to each other; that is, we must have {P}={Q} as true to mathematics, which means that {P} represents a unique all-prime factorization of N, which verifies the theorem.

This brings us to Euclid's wording of the theorem: "If a number is the least that is divided by certain prime numbers, it will not be divisible by any other prime numbers."

Yes, that's a neat little theorem, but why would we consider it fundamental to arithmetic? And why THE fundamental theorem? We had to wait until we had defined multiplication and division before we could state it. Shouldn't some theorem pertaining to addition and subtraction be even more fundamental?

Let me note here that the fundamental theorem of arithmetic stands on the threshold between the simple counting processes of addition and subtraction and the processes of multiplication and division, which take us into the realm of arithmetic in which we use the manipulation of symbols alone to carry out calculations (think of the partial product method of multiplication as an example). The theorem marks the transition of arithmetic from the concrete to the abstract.

We must note further that the fundamental theorem applies only to the natural numbers (that is, the positive integers), the ordered sequence of names that comprises the most fundamental set of numbers, from which all arithmetic and the mathematics emanating from it flows. We have learned that if we take those numbers in pairs, in all possible pairs, and multiply them together, the products will comprise a set of numbers identical to the set of all natural numbers with some gaps: the numbers in those gaps comprise the set of the prime numbers. The fundamental theorem of arithmetic tells us that we can represent each and every composite number in the set of the natural numbers as a unique product of prime numbers. We know already that we can represent any natural number as a sum or difference of other natural numbers. And we also know that multiplying natural numbers together gives us other natural numbers, which we call composite numbers. We get the surprise that makes our theorem worthwhile when we discover that we can generate all of the composite natural numbers by multiplication of prime numbers only and then discover that such all-prime multiplications are unique to each and every composite number.

efefefaaabbbefefef

Back to Contents