THE FOUNDATION OF ALL CALCULATION
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If we have a Map of Physics, we must necessarily have a Map of Mathematics to go with it. We can make that claim because mathematics serves as a sine qua non of physics; we simply cannot properly express the laws of physics without it. The idea that we can deduce all of the laws of physics from a small set of axioms properly fills us with astonishment, but the idea that we can do the same with mathematics elicits only yawns; mathematics, inherently axiomatic-deductive, embodies logic.
Geometry notwithstanding, numbers stand out in our minds as the fundamental entities of mathematical calculation. They stand before us as the sine qua non of the subject; without numbers, mathematics simply does not exist. So fundamental do we conceive them that we seldom give them any thought beyond the uses to which we put them. But what does the word number actually denote? Whence do we get numbers?
As the most primitive numbers of all, we have the counting numbers. While we conceive them as indicators of quantity of things taken in a jumble (things, in other words, put all together in no particular arrangement), they are, in fact, simply the names (and associated pictures) that we ascribe to the places in a sequence. At their most fundamental, numbers are ordinal. Thus the set of the counting numbers (which mathematicians call the natural numbers or the positive integers) consists, as a definition, simply of an ordered sequence of names.
We can devise a system of numbers with a purely arbitrary set of names, but once we have stated the names and put them into a sequence, we cannot change any of those names or their place in the sequence. With that sequence established, we can count a collection of objects by calling out the names in their proper order as we move the objects, one by one, from one pile to another, calling one name for every object we move. The name that we call out when we move the last object stands as the number of the objects in the pile. As we count, we say that we call out progressively greater (or bigger) numbers and we say that we count up the sequence. If we count the sequence backward from some number, we say that we call out progressively lesser (or smaller) numbers and we say that we count down the sequence.
That definition of numbers makes the well-ordering principle seem utterly trivial. The principle states that every non-empty subset of the natural numbers has a least (or smallest) element. If we call out the names of the elements of such a subset in the order in which they appear in the sequence of the natural numbers, we must call out one name first and that name denotes the least element of our subset. Can we prove such a statement or must we accept it as an axiom? We know that we can draw the set of the natural numbers on a line that has a left end, so we know that if we draw each element of a subset below its position on the number line, we have a farthest left element necessarily, so we have the principle in a different form, but that doesn=t look like a proper proof. We can also say that if we have a subset of the natural numbers and if we begin counting the numbers in that subset in their defining order, we will necessarily call out the name of one certain member of the subset before we call out the name of any of the other members of the subset, thereby marking that one element as the least element of the subset.
The definition of numbers as an ordered sequence of names would seem to limit the usefulness of numbers; after all, even if we spent a lifetime at it, we can count only so far by following the sequence step by step. But we have developed tricks that enable us to leap over the sequence to a specific number or range of numbers in it, obtaining a count without counting. How effectively we can apply those tricks depends, it turns out, on how we devise the names of the numbers and how we draw them.
The idea of naming numbers doesn=t impress us as anything significant. When we read of Archimedes= boast that he had devised a number that would allow him to enumerate all of the grains of sand in the universe, we wonder why people of his time regarded it as such a big achievement. (He actually devised a system similar to ours, but with a base of 100 million instead of ten.) We have forgotten, culturally, what a struggle people must have waged to create the sophisticated number naming system that we have now and that we take for granted because we began learning it in early childhood.
We must necessarily give a unique name to each place in the counting sequence. That would seem to impose upon us a strictly limiting requirement, somewhat akin to the Chinese language=s use of a unique ideogram for every word. But we have a trick that gets us around the limitation. On the number line we set up signposts that we place increasingly far apart, always in the same proportion, and use their names as labels appended to the numbers in their realms. Within those realms we simply repeat the fundamental count. Thus, for example, we name the number after ninety-nine Aone hundred@ and then begin counting anew beyond it B one hundred and one, one hundred and two, and so on. And, except for the numbers between ten and twenty, the tens work the same way, each decade having its own label and its own count from one to nine. Thus we need only nine counting numbers and the names of the signposts designating successive powers of ten: we have the tens (nine of them), the hundreds (nine of them), the thousands, the tens of the thousands, the hundreds of the thousands, and then the millions. Thence we name the major signposts by successive powers of one thousand B billion, trillion, quadrillion, and so on.
We not only name numbers; we draw pictures of them. Because we have conceived numbers as pure abstractions, we don=t have an obvious candidate for the pictures we want to use. But that means that we can use an entirely arbitrary set of pictures B whatever we choose. What pictures might we use to represent places in a sequence? The Greeks and the Hebrews used pictures from a different, but entirely obvious, sequence B their respective alphabets: the first nine letters represented one through nine, the next nine letters represented the decades, ten through ninety, and they could use the leftover letters for hundreds. Such a system provided a sufficient set of numbers for drawing accounts in a Bronze Age society. Later the Romans used a modified version of that system using their own alphabet, a version that would have allowed them to extend the use of the alphabet in drawing numbers much farther had they felt the need to do so. However, alphabet-based number-drawing systems have a severe limitation, the size of the alphabet: we will find rather quickly a biggest number that one can reasonably draw with the available symbols.
We must look to India and to the Hindus to discover who devised an unlimited system of drawing numbers, the one using Ghobar numerals (also called Arabic numerals because the version that we use evolved from the Arabic versions of the Hindu pictures) that we use today. To represent any number at all, the Hindus used only ten pictures, one each for the numbers one through nine and one picture for a new number that we call zero (well, the Europeans saw it as new when Arab scholars brought it to them in the Thirteenth Century). By conceiving a number to represent nothing (which exists in our minds as a fairly aetherial concept, so we feel no astonishment at the fact that European and Middle-Eastern cultures missed discovering it), they created the possibility of a place-based numbering system that would use the same ten symbols over and over again. Thus, where the Romans would write I, X, C, M, we would write 1, 10, 100, 1000 and then go on to write 10,000, 100,000, and so on beyond the point where their limited alphabet would have compelled the Romans to stop.
With a suitable naming convention and the associated pictures, we can represent any number at all; we merely require enough space to draw the Ghobar numerals or to write out the name. Thus we have available to us all of the numbers that we will ever need for counting things. But most of those numbers look very big and the prospect of counting anything that would need such numbers daunts us to say the least. Can we devise some way in which we can make the counting of many objects easier than the obvious method makes it?
I know that if I have counted a number of pebbles into a bowl and that if I come back later and want to put more pebbles into the bowl, I don=t have to begin the count over again from one. As I put the new pebbles into the bowl, I start my count with the next number beyond the number of the pebbles already in the bowl and continue counting up the sequence of number names as I put each new pebble into the bowl. If I had seventeen pebbles in the bowl and I wanted to put in five more, I would count, Aeighteen, nineteen, twenty, twenty-one, twenty-two@ and finish by marking twenty-two as the new number of the pebbles in the bowl. That fact provides us with the basis for the process of addition. It tells me that I have no need to recount objects that I have already counted when I combine them with other objects. I will also note that this description of counting pebbles does not give us as frivolous an example as we may think on first impression: people made the first coins, back when they first invented money, merely by stamping simple symbols onto pebbles of gold and silver.
If I use Ghobar numerals (and from here on I will assume just that), then I can use addition to make my counting more efficient. If someone gives me a very large pile of pebbles to count, I know that I will grow tired and make miscounts if I try to count the pebbles in one sitting. To make my job easier, then, I split the pile up into small batches of pebbles, count the number of pebbles in each batch, and draw the appropriate pictures of these numbers on a list with the numerals aligned in columns. Now I have only to devise a procedure by which I can add the numbers together to obtain a single number. Our place-notation representation of numbers makes that procedure simple and straightforward. I need to know how to add the numbers one through nine in any combination: I can determine those simplest of sums by direct counting and display the results in a table like this:
To us, accustomed to years of carrying out additions, this seems terribly simplistic, but this table displays what we use every time we carry out an addition. Now I treat each column of digits on my list as a set of units and add them in accordance with that table, remembering to set aside the tens produced by such additions and add them to the next column to the left as units. I then add, column by column, from right to left, to generate the total number of pebbles in the pile. We now have addition as the most fundamental of the arithmetic procedures that we use to manipulate numbers.
Our definition of numbers and our method of devising new names for the sequence of what we call the natural numbers tell us that, in concept, we can go on devising new numbers endlessly. If I claim that I have found a last number in the sequence, you need only apply this new procedure of addition to add one or some other number to it and continue to extend the realm of the numbers far beyond my Alast number@. We can carry out that procedure endlessly, so we see readily that the set of the natural numbers exists without a limit; thus, we refer to its lack of an upper end with the Latin word for without limit B infinite. We must take care to note and to emphasize that infinity does not denote an actual number or place on the number line: it simply denotes the mathematical concept of endlessness.
In the 1880's Georg Cantor gave us a simple test that would tell us whether we have an infinite set: if a proper subset has a magnitude equal to that of the set, we have an infinite set. That may seem stupid, but Cantor also gave us a procedure for carrying out such a test: if we can produce a perfectly unique one-to-one matching of the elements of the subset with the elements of the set, then we have an infinite set. Cantor also gave us a simple example that proves and verifies the infinity of the set of the natural numbers:
Imagine that we have laid out the conventional number line, displaying the natural numbers as locations to the right of the zero point, and imagine drawing beneath each number its double (which we obtain by adding each number to itself). In so doing we have matched each and every element of the set of the natural numbers with one and only one of the elements of the set of the positive even integers. In order to have enough even integers to match uniquely with the natural numbers, both the set and the subset must have infinite magnitude.
I must also note here that Cantor denoted the infinity of the natural numbers with the name Aleph-Null and he claimed that it stood as the first and smallest in a hierarchy of ever-larger infinities, an endless series of Alephs. But the concept of endlessnesses more endless than endlessness must give us pause. Such things simply cannot exist and when we get into the real numbers I will prove and verify that statement.
If I can do an addition, I can also undo it. If, after I have counted my pebbles, I must give a certain number of them to the tax collector, I can calculate (the word comes to us from the Latin Acalculus@, which means, weirdly enough, pebble) how many I have left by reversing the process of addition and subtracting the tax from my pile. For subtraction we can use the addition table above in a column-by-column procedure. For each pair of digits in the column (we can only subtract one number at a time from some other number) we find the minuend on the table below the subtrahend and find the difference in the leftmost column of the table. If I have a minuend smaller than my subtrahend, I borrow a one from the column immediately to the left of the one I have in process (remembering to decrement the digit in that column by one) and add a ten to my minuend, then carry out the subtraction. As with addition, I proceed column by column, right to left, withdrawing a ten from a column if I need it to support the subtraction in the column to its immediate right.
But the concept of subtraction creates a problem. Suppose that the tax collector demands from me a number of pebbles greater than the number of pebbles in my pile. Does the subtraction of a larger number from a smaller number make any sense? We can certainly calculate a difference. We merely subtract the smaller number from the larger number and prefix a minus sign to the result to mark the fact that it comes from subtracting a larger number from a smaller one: in essence, we have calculated the number that we would obtain from counting down the tax collector=s demand as I give him pebbles from my pile. If I have seven pebbles in my pile and if the tax collector demands that I give him ten pebbles, then the calculation tells me that I will have minus three (or negative three) pebbles left over after I give the tax collector all of the pebbles in my pile. The tax collector, counting down as I give him my pebbles, will tell me that I owe him three more pebbles. Thus the negative number represents a debt. It can also represent any other kind of deficiency that we can conceive: negative altitude, for example, represents distance below sea level. The concept of subtraction thus necessitates the concept of negative numbers.
If I wish to combine the pebbles in a large number of piles and if each of the piles contains the same number of pebbles, then I can carry out a fancy kind of multiple addition and call it multiplication. As we have in the case of simple addition, I must have a procedure that lets me carry out the simplest versions of the procedure on the numbers from one to nine: to determine the product of seven multiplied by six, for example, I merely add six sevens together. Those simple, fundamental multiplications comprise the standard multiplication table, which we have as
With that table I can multiply two numbers, place by place, to create a number of partial products (one for each digit in the multiplier), which I can then add together to yield the final product.
But in addition to the counting numbers, I have a new kind of number that I must learn how to multiply in order to ensure that I have a complete system of arithmetic: I must figure out how to multiply with negative numbers. The procedure seems straightforward enough: all I need to do is to carry out a multiple addition and see what I get as a result.
If I owe the butcher two pebbles, then I have negative two pebbles as a consequence. If I also owe the baker two pebbles and the candlemaker two pebbles, then I have a total debt of six pebbles; in other words, I have negative six pebbles. Thus I can say that three times negative two equals negative six. Multiplying a negative number by a positive number represents a consolidation of equal debts.
Suppose now that I must multiply a negative number by a negative number. What kind of product do I get from that multiplication and what does it represent? In order to determine what I get for the product I can exploit the associative law of arithmetic.
So far I have implicitly carried out all of my calculations on columns of numbers. But we can also carry out those calculations on horizontal arrays of numbers: we call those horizontal arrays equations, because they consist simply of an equality sign drawn between two different representations of the same number. We have, as two examples,
That seems fairly trivial, but we can make our equations as complex as we like; we need only ensure that the expressions on both sides do, indeed, represent the same number. We achieve that happy assurance by requiring that any arithmetic procedure that we apply to one expression in an equation we must also apply to the other expression in that same equation. We can also use letters in an equation to represent numbers whose values we will specify later, thereby turning the expressions into algebraic formulae, general recipes for carrying out arithmetic.
To find out what I get when I multiply a negative number by a negative number, I want to exploit the associative law of multiplication. If I want to multiply together the numbers A and D and I have D=B+C, then I know that I have as true to mathematics the statement that
Because multiplication consists of multiple additions, it makes no difference in the product whether we add B and C before we multiply the sum by A or whether we multiply B and C each by A before we add the partial products. Indeed, our procedure of multiplication by summing partial products exploits this rule.
I know that multiplying any number by zero equals zero; specifically, in my case here, -N x 0 = 0. And I also know that I can represent zero as a number subtracted from itself; that is, 0 = M - M. So I know that
The first term in the formula to the right of the zero gives us a negative number, so the second term must give us the positive equivalent and we must, therefore, have as true to arithmetic the statement that (-N)(-M) = NM.
And what does that positive product of two negative numbers represent? I know that the -NM in the formula above represents my consolidated debt of N pebbles owed to each of M tradesmen. And I know that anything that reduces that number to zero represents a payment of that debt, so the product of two negative numbers must represent the payment of a debt or something analogous to the payment of a debt.
Now I know that I can represent any integer as the product of two integers plus some other integer; in symbolic form we have
standing true to mathematics. We specify that in that equation r represents the least integer remainder, which necessitates that r<p and r<q. We call the statement that equation represents the division algorithm. But suppose that we relax our restriction a bit and let r represent a negative number smaller than the product pq and let n=0. In that case we have no integers p and q that will solve Equation 4. Again we must look to a new arithmetic process and the new kind of number that it creates.
Just as addition necessitates subtraction, so multiplication necessitates division, the reverse of multiplication. And just as multiplication represents a kind of repeated addition, so division represents a kind of repeated subtraction. If I have a certain number of pebbles and I wish to organize them into a certain number of piles, each pile containing the same number of pebbles as do the others, then division will tell me how many pebbles to put into each pile. In essence, when I divide one number by another, I must subtract the second number from the first number as many times as will reduce the first number to zero and then I must count the number of subtractions that I needed to carry out to achieve that end: if I have twenty pebbles and I want to put them into five identical piles, then I must subtract five pebbles (one into each pile) from twenty pebbles four times before I have no pebbles left over; which means, I will have four pebbles in each pile.
Again I have a problem with my numbers. What can I do if I carry out the procedure described above and the subtractions don= t yield a zero? If I had tried to put twenty-three pebbles into five identical piles, my subtractions would have ended with three pebbles left over. How can I distribute those three pebbles over my five piles? What does it mean to divide a number by a larger number? I answer that question and solve my problem by breaking each of the three pebbles into five equal pieces and then continuing the process of division on the resulting fifteen fifth parts. Each pile then contains four whole pebbles and three fifth parts of a pebble. The three fifth parts (or, simply, three fifths) represent what we call a broken number or fraction (from the same Latin root that also gives us Afracture@). The concept of division thus necessitates the concept of fractions.
Now I can work out the arithmetic of fractions B adding, subtracting, multiplying, and dividing. To aid us in that effort we must devise a convenient way to draw a picture of a fraction using numerals instead of words. We refer to the number of pieces into which I break my pebbles as the denominator and the number of those pieces that I put into each pile as the numerator and then describe the fraction by saying that we draw the numerator over the denominator with a horizontal bar separating the two numbers. In some cases we can draw the bar slanted from upper right to lower left and thus draw my fraction above as 3/5.
In adding and subtracting fractions we must ensure that the terms all have the same denominator; otherwise, we commit the mathematical analogue of adding apples and oranges. We know that if we multiply both the numerator and the denominator of a fraction by the same number, we do not change the value of the fraction: we exploit that fact to reduce fractions by factoring common factors out of the numerator and denominator and canceling them (so, for example, we reduce 8/24 to 1/3). So to carry out the addition or subtraction of two fractions we give the fractions a common denominator by multiplying both the numerator and the denominator of one fraction by the denominator of the other, carry out the addition or subtraction on the enhanced numerators, and then, if possible, reduce the resulting fraction.
We multiply two fractions together by multiplying their numerators and denominators separately. So if we want to multiply M/N by P/Q, we get (MP)/(NQ). In this case we originally have a pebble broken into Q pieces and we have taken P of those pieces and broken each of them into N pieces, of which we take M. So, having broken our pebble into NQ pieces, we have taken MP of them for what we require.
In dividing one fraction by another we need only remember that, dividing one number by another, we get the same result if we multiply the first number by the reciprocal of the second number. So in dividing A/B by C/D we get the same result if we multiply A/B by D/C.
Now we can extend our powers of ten representation of numbers into the fractional realm. We tacitly imagine, in our representation, that we have an infinite series of places to the left of the fundamental digit, each place representing a power of ten. So we can mark a spot to the right of the fundamental digit (with a decimal point) and imagine an infinite series of places to the right of that point, each place representing a power of one-tenth. We can then convert a fraction into its decimal equivalent by dividing the denominator into the numerator in the process that we call long division. Thus we can represent all possible fractions with strings of digits arrayed to the right of the decimal point, just as we represent all possible integers with strings of digits to the left of the decimal point.
Here we need to pause to discuss the mathematical concept of a function, a thing that we have already used implicitly in Equation 4. At its simplest a function tells us how to replace one number with another in a consistent way. We can think of a function as a recipe for carrying out arithmetic, taking numbers from some domain on the complete set of the numbers and replacing them with numbers from some range on the same set. We commonly depict functions on graph paper by drawing two mutually perpendicular axes on the graph (labeled x for the horizontal axis and y for the vertical axis), calculating for each value of x a corresponding value of y through the functional statement y=f(x), and then mark on the graph each point with the coordinates thus determined. If we mark enough such points, we connect them into a line that curves across the graph. Such curves possess two properties that have great importance for us; they have the properties of continuity (or discontinuity, continuity= s opposite) and of smoothness (or kinkiness).
Continuous means that adjacent values of x on the graph yield adjacent values of y: the difference between the values of y has a value only a finite amount larger or smaller than the difference between close values of x. If the ratio between the difference between the values of y and the difference between the close values of x were to grow endlessly larger as the difference between the values of x becomes progressively smaller, then we have a discontinuity in the function. The graph of the function has a gap in it.
Smooth means that the graph of the function has no kinks in it. If we suspect that we have a kinky function, we pick the value of x where we believe the function has a kink and pick values of x on either side of it. If we think of the curve as a road, thinking of the x-value as a mile-post and thinking of the y-value as an altitude, then we suspect a kink in the ascent of the road at milepost 2. We test that proposition by calculating the change in altitude between mileposts 1 and 2 and between mileposts 2 and 3, divide each change by the distances between the relevant mileposts, and subtract those ratios, one from the other. We then divide that difference between the ratios by the distance between the midpoint between mileposts 1 and 2 and the midpoint between mileposts 2 and 3 and call the number that we get the kink ratio. If we then let the distances between the mileposts shrink, mileposts becoming yardposts, footposts, inchposts, and so on, and if the kink ratio grows endlessly, then we have a kink at milepost 2; if the kink ratio tends toward some finite number, then we do not have a kink at milepost 2.
This distinction of smooth and continuous has great importance for us because the statement that we call Newton=s zeroth law tells us that we can represent the laws of physics mathematically with smooth, continuous functions of the relevant variables.
Now we can define another new kind of number, albeit one that belongs to a subset of the numbers that we have already defined, and in so doing take a good look at how we apply logic in the manipulation of our knowledge of numbers. We notice that we can represent most of the natural numbers as the products of other natural numbers (e.g. 6=2x3 or 15=3x5), but some of the numbers in the set do not have such compositions (e.g. 7 or 13). We call those latter numbers prime and regard them as the atoms of arithmetic. Here I must note that mathematicians, for technical reasons, do not include one as a prime number.
We now have two theorems attributed to Euclid concerning prime numbers. Euclid=s first theorem states that if a prime number p divides the product ab evenly, then p divides a or b evenly. Euclid=s second theorem asserts that the prime numbers comprise an infinite set.
To prove and verify Euclid=s first theorem we begin with Bézout=s lemma, which states that if a and b represent nonzero integers with the integer d as their greatest common divisor, then there exist integers x and y such that the statement
stands true to mathematics. We also state that d represents the smallest possible positive integer for which there exist integer solutions x and y for that equation.
Because I want to use that latter statement in my proof, I want to prove and verify it here. Assume that a=qd and that b=sd. Substituting those equations into Equation 5 and dividing the result by d gives us qx+sy=1. We can only have integer values for x and y in that equation because we require that q and s have a relative primality between them; in other words, neither q nor s can divide the other evenly. If we did not meet that criterion, if q and s had some common divisor between them, then we could divide the equation above by that common divisor to obtain q=x+s=y= some fraction. Since q= and s= must represent integers, x and y would have to represent fractions or mixed numbers in order that the sum of the products yield Asome fraction@. Thus, none of the common divisors of a and b smaller than d will satisfy Equation 5 if we require that x and y represent integers. Q.E.D.
Consider the set S containing all positive integers of the form am+bn, in which m and n represent integers. That gives us a non-empty set, for which the well-ordering principle necessitates a least member. We declare that d, as defined in Equation 5, represents that least member. The division algorithm tells us that we can find integers q and r such that
In that equation we must have r<d. Rearranging that equation gives us
which has the form of an element of the set S. But if r lies in S, then d does not represent the least member, contradicting our assumption; therefore, r cannot represent an element of S and must take the value r=0. Thus, we have a=qd, which means that d divides a. Applying the division algorithm to b by way of Equation 6 gives us the same result, confirming that d represents a divisor of b, so d must represent a common divisor of a and b.
Now suppose that we have a prime number p as a factor of the product ab and assume that p does not evenly divide a. That statement necessitates that p and a stand relatively prime to each other, so there must exist (by Bézout= s lemma) integers x and y such that the statement
stands true to mathematics. We multiply that equation by b to obtain
Clearly p divides the first term of that equation and, by hypothesis, p also divides the second term, so p must also perforce divide b. We can then repeat that proof by making p and b relatively prime to each other and prove and verify that p divides a, thereby proving and verifying that p always divides either a or b, Q.E.D.
We state the fundamental theorem of arithmetic as: The prime factors of each and every positive integer greater than one comprise a unique, finite set. Note that the prime terms of each and every positive integer do not comprise a unique set (e.g. 6=1+5 and 3+3).
We begin the proof of the theorem by stating that in each and every positive integer we have either a prime number or a composite number. The prime numbers automatically conform to the theorem, so we need only consider the composite numbers. We first prove and verify that the prime decomposition of the composite number exists and comprises a finite number of factors, then we prove and verify that the set of factors comprising the prime decomposition of any composite number is unique.
Every composite number, by definition, stands before us as the product of at least two integer factors, so we can write N=ab to represent an arbitrary composite number. If the factor a represents a composite number, then, by Euclid=s first theorem, some prime number p1 divides it. So we get N=p1cb, in which c represents a positive integer smaller than a. We then apply the procedure to c and repeat the process on subsequent quotients until we come to a quotient that itself represents a prime number. We then apply the procedure to b until we have
In that equation we have K as a finite number. We know that statement stands true to mathematics as a solid fact because we began with two finite numbers and divided them by finite numbers, a process that we can only carry out a finite number of times when all of the numbers involved can only have integer values.
To prove and verify that Equation 10 represents a unique decomposition of N we assume that N has a second decomposition different from the first. So we have by assumption
We assume, for convenience, that K<J and that we have arranged the factors in each decomposition in non-decreasing order. We know that p1 divides N, so it must divide the product Q=q1q2...qK. Because we have only prime numbers in that equation, we must find that p1 divides some factor qk and does so in the only way possible; that is, p1=qk. Thus the statement p1=qk$ q1 stands true to mathematics. Likewise we can infer that there exists some value of j for which q1=pj$ p1. Those two statements necessarily entail that p1=q1, so we can cancel those factors out of Equation 11 to obtain p2p3...pJ=q2q3...qK. We then repeat the process J-1 more times to get 1=qJ+1...qK. Because one does not represent a prime number, the factors on the right side of that equation cannot exist and we must, therefore, infer that K=J and that the prime factors of N comprise a perfectly unique set. Thus we prove and verify the fundamental theorem of arithmetic.
For his second theorem Euclid gave us one of the most elegant proofs in all of mathematics. Consider the ordered set of the consecutive prime numbers, remove the first n members of that set, and multiply them together to generate the n-th Euclid number, En. We then calculate the number N=En+1 and use it to prove the proposition that the prime numbers comprise an infinite set.
We claim that pn, the n-th prime number represents the last member of the set of the prime numbers, that there exists no prime number larger than pn. In N we have two possibilities; either N represents a prime number, which necessarily has a larger value than does pn, or N represents a product of prime numbers. If N represents a composite number and there exists no prime larger than pn, then N must come from the product of some set of the primes equal to or smaller than pn. But a number that divides N and also divides En must also divide their difference, which difference equals one. No prime in the given sequence could possibly satisfy that criterion, so if N represents a composite number, then it must have at least one of its factors as a prime number not in the set of primes that comprise the factors of En. Because those factors comprise consecutive elements of the set of primes, at least one of the factors of N must give us a prime number not in that subset; specifically, a prime number necessarily larger than pn.
Thus either N or one of its factors gives us a prime number larger than pn, so we know that we can extend the set of prime numbers up to that new largest prime. But then we can repeat that process to find yet another largest prime and we can repeat the process over and over again, endlessly finding ever larger prime numbers. That fact necessarily implies, by induction, that the prime numbers comprise an infinite set.
Here we have a good place to review the process of induction, which we have just used in the above inference. We take as a fundamental principle the statement that, with respect to any specific properties, the elements of the set defined by those properties do not differ from one another in their applications to mathematical reasoning. Thus, for example, if we have a process that distinguishes even and odd numbers, it does not distinguish among the even numbers. In the above example the process Euclid gave us implicitly generates a new prime number from consideration of a set of consecutive prime numbers. We assume, reasonably, that we can fill in the prime numbers lying between the original largest prime and the new largest prime to create an extended set of consecutive primes and then repeat Euclid=s process. We also assume, also reasonably, that no prime number has a magic property that makes it different from other prime numbers in a way that invalidates Euclid=s process. We know that we can deduce the first assumption, but we don=t know that we can deduce the second: we have perfect confidence in the first assumption, but with regard to the second assumption we must have less than absolute confidence because we cannot assert without reservation the impossibility of some prime number Away out there@ on the number line having some special property that interferes with Euclid=s process and thereby stops the chain of applications of that process. Thus we cannot say that we have a proper deduction of the infinity of the primes, but that we have an induction.
But now the process of division has given us the fractions, which we can add to the integers to generate the real numbers. How many elements do we have in the complete set of the real numbers? We begin to answer that question by determining how many elements we have in the set of the decimal fractions.
We assume that the decimal fractions lying between zero and one comprise an infinite set. Indeed, we can say immediately that the decimal fractions comprise a set that has exactly as many elements as does the set of the natural numbers. Our basic description of the numbers as drawn tells us that fact: the decimal fractions consist of all possible permutations of ten symbols on a line of spaces extending infinitely far to the right of the decimal point and the natural numbers consist of all possible permutations of ten symbols on a line of spaces extending infinitely far to the left of the decimal point. Those descriptions only differ from each other in the words Aleft@ and Aright@ and that fact, in its turn, tells us how to prove and verify the proposition.
Take an arbitrary decimal fraction, such as 0.54, and reflect it through the decimal point. We get a natural number, 45, and only one natural number by that procedure. There exists no decimal fraction to which we cannot apply that reflection procedure and there exists no decimal fraction that will not yield a perfectly unique natural number under that procedure.
Likewise, take an arbitrary natural number, such as 381, and reflect it through the decimal point. We get a decimal fraction, 0.183, and only one decimal fraction by that procedure. There exists no natural number to which we cannot apply that reflection procedure and there exists no natural number that will not yield a perfectly unique decimal fraction under that procedure.
Thus we infer that there exists one and only one decimal fraction for each and every natural number and that there exists one and only one natural number for each and every decimal fraction. We have thus matched the set of the decimal fractions uniquely one-to-one with the set of the natural numbers; therefore, the set of the decimal fractions has exactly the same magnitude as does the set of the natural numbers and thus exists as an infinite set.
The set of the real numbers consists of all possible combinations of an integer with a decimal fraction. If we want to know how many elements that set has, we might guess Ainfinity squared@, whatever that might mean. Georg Cantor put the size of the real numbers as Aleph-One, the next infinity larger than Aleph-Null, the infinity of the natural numbers. Actually Cantor made an error in his reasoning, as I demonstrate in the appropriate essays on the Platonic Dream page on this website. In fact, the set of the real numbers contains exactly as many elements as does the set of the natural numbers.
To prove and verify that statement I need only offer a slight revision of the reflection process that I described above. Take any real number and interleave the digits of the decimal fraction among the digits of the natural number or vice versa; if, for example, we take 65.37, we get either 7635 or 0.3576. In this way we get a unique one-to-one match between the elements of the set of the real numbers and the elements of the set of the natural numbers or the set of the decimal fractions. In either case we have a proof that the set of the real numbers contains exactly as many elements as does the set of the natural numbers. That proof suffices to show that Cantor=s hierarchy of infinities, the Alephs, does not exist.
I derived multiplication from an addition in which all of the addends stood equal to one another. Now I can devise a special kind of multiplication in which the multipliers stand equal to each other. Imagine that someone has given me a flat surface that they have marked by fine lines into an array of squares in the manner of a checkerboard and that someone tells me that I must put one pebble onto each square in such a way that the number of rows and the number of columns occupied by pebbles stand equal to each other. How many pebbles will I need? I can obtain the answer to that question easily: I merely multiply the number of rows (or of columns) by itself. Because I have laid out the pebbles in a square array, I call that product a square number. I can do the same trick with pebbles that I have arrayed in a cubic lattice and thereby define and calculate a cubic number (or the cube of the number of rows). I can even define higher Apowers@ of the number of rows, even though they represent numbers of pebbles arrayed in figures that we cannot visualize.
As I did with multiplication, I can define the reverse process of calculating the power of a number. Because the power grows from a given number, we call the given number the root of the power. Thus, for example, 27 gives us the cubic number of 3, so 3 represents the cube root of 27.
As you might expect by now, the process of extracting roots gives us some new numbers. We find that we get irrational numbers, numbers that we cannot represent as the ratio of two natural numbers. But the irrational numbers, like the transcendental numbers, offer little of interest here, showing us nothing more than fancy fractions. The number that we really want to consider, the truly radical number, emerges from the square root.
A square number as defined above always comes to us as a positive number, regardless of whether it has a positive or negative root. But if the square root of a positive number always comes to us as either positive or negative, then how can we have the square root of a negative number? What could we possibly have, for specific example, for the square root of negative one? We seem to have no possibility of an answer. Heretofore all of our new numbers could find a place on the number line, either on its extension into the realm of negative numbers or into the spaces between numbers. None of the numbers on that line has a negative square.
In one possible ploy that will tell us what we want to know, we recapitulate the procedure I employed in working out the product of two negative numbers. In this case let=s use the Pythagorean Theorem in the form N2 - N2 = 0. We can see that equation as the sum of a positive number and a negative number and we can also conceive it as representing the length of the hypotenuse of a right triangle as the usual sum of the squares of its sides. A right triangle whose hypotenuse has zero length may seem an absurdity, but we can use the equation nonetheless for what it can teach us. We will obtain what the Norwegian surveyor Caspar Wessel (1745-1818) devised and published in the Memoires of 1799 of the Royal Danish Academy of Sciences (if that seems a strange venue for a Norwegian to use, please recall that until 1814 Norway existed politically as a province of Denmark). Clearly the side that the positive number represents extends along the number line from zero to either +N or -N. In that case we must measure the side represented by the negative number along a line that stands perpendicular to the number line. That perpendicular line must, itself, have the character of a number line, but its numbers cannot combine with the real numbers in a way that lets the real numbers absorb them. We call the new numbers imaginary numbers and call their combinations with the real numbers complex numbers. For the typical complex number we have
in which the letter eye represents the square root of minus one. We also have the complex conjugate of that number,
which we will use in our calculations.
We refer to a map of the complex plane as an Argand diagram (after Jean-Robert Argand (1768 Jul 18 B 1822 Aug 13), who published his description of it in 1806). On that diagram we can represent a complex number as a point in two ways. We can use the obvious representation of the real component as the x-component (the distance of the point from the y-axis measured on a line parallel to the x-axis) and the imaginary component as the y-component of the vector extending from the origin of the plane to the point. Alternatively, we can represent the point by the length of the straight line extending from the origin to the point and the angle that the line makes with the x-axis, measured in the counterclockwise direction. Now we want to correlate those two descriptions.
What can we say about the number Miθ if M represents a positive real number and θ represents any real number (positive or negative)? Using Wessel=s description of the magnitude of a complex number,
we have standing true to mathematics the statement that
for all allowed values of M andθ. We know that Miθ cannot represent a purely real number, because, except for the cases M=1 and θ=0, it would not satisfy the magnitude equation, Equation 15. For the same reason, we know that Miθ cannot represent a purely imaginary number. Therefore, we must assert that Miθ represents a linear combination of real and imaginary numbers (a and ib) such that a2+b2=1; in other words, it represents a circle of unit radius centered on the origin of the complex plane.
For any positive real number N there exists a number n such that N=Mn, so we have as true to mathematics the statement that
Note that M can represent any positive real number, not only Euler=s number, the base of the Naperian (or natural) logarithms.
We can now ask whether the roots of higher powers oblige us to create other new kinds of numbers. The odd roots of -1 (the cube root, the fifth root, etc.) all equal -1, so they don=t necessitate anything new from the negative numbers. What can we say of the even-numbered roots of -1 (for example, the fourth root, the sixth root, etc.)?
The fourth root of -1 gives us the same number as does the square root of i, but how can we find that? We need to find the square of something and equate it to i. The quadratic formula will do what we need. Let=s start with 0=1-1. We add 2i and get 2i=1+2i-1=(1+i)2, which means that the square root of i equals 1+i divided by the square root of two.
To find the sixth root of -1 we need to find the cube root of i. Again we want to ask what function, when cubed, equals i? We found that the square root of i equals the sum of a real number and an imaginary number, so we guess that the cube root has the same form. We thus have
The real parts of the polynomial must add up to zero, so we calculate that A2=3B2. Making the appropriate substitution in what we have left gives us B=1/2 and thence
Again, we don=t have to invent any new numbers for this calculation.
If we plot those results on an Argand diagram, we find that they fall onto the unit circle, the circle of unit radius centered on the origin of the grid. We find that i itself lies on the circle ninety degrees counterclockwise from the real axis, its square root lies forty-five degrees counterclockwise from the real axis, and its cube root lies thirty degrees counterclockwise from the real axis. Thus we infer that on the Argand diagram the N-th root of i lies on the unit circle 90/N degrees counterclockwise from the real axis. In fact, the points that comprise the unit circle give us all of the roots (and powers) of one and of its roots, real, imaginary, and complex. Thus we have closed the set of the numbers. We have all that we need.
If we have a function that maps the elements of a numerical domain onto the elements of a numerical range, we can represent that function (by Taylor=s theorem) by a polynomial with complex coefficients. The polynomial may have infinite order and have negative integer powers, but it will properly represent the function. For y=f(z) and for arbitrary values of y we have f(z)-y0=P(z)=0. We can factor P(z)=(z-zn)P=(z) [after dividing out the coefficient an]. For z=zn that factored function equals zero; otherwise P=(z)=0 and we return to our starting point. At no value of the polynomial order n does that process change; therefore, by induction we factor the polynomial completely. Thus we obtain the n roots of the polynomial, which means that the complex numbers suffice to solve the equation, again verifying that we have closed the set of the numbers developed above and that we do not need to invent any more new kinds of numbers.
Now we need to prove and verify the proposition that some complex number will evenly divide any given polynomial. We can do it with complex analysis, but I want to do it algebraically.
We want to prove and verify the proposition that any complex polynomial,
with arbitrary complex coefficients ak, has a solution in the complete set of the complex numbers; in other words, that there exists some z0 such that P(z0)=0. We can express that proposition alternatively through the statement that
If we use the polar representation of complex numbers, we have
Thus we have the k-th term of the polynomial in the form CkCkexp(iθk+ikθ), which means that as θ goes from 0 to 2π the point thus described traces out k times a circle of radius CkCk, beginning at and returning to the angle θk.
Now we want to prove and verify that P(z) gives us a smooth and continuous function: in other words, the function must have no gaps (continuous) and no kinks (smooth or no gaps in its first derivative). The sum of continuous functions necessarily yields another continuous function and the sum of smooth functions necessarily yields another smooth function. We know that compound statement stands true to mathematics because in order for the sum to have a gap in its graphed curve at least one of the components would have to have a gap, a set of points where the function has no value; we cannot create such a gap by adding or subtracting functions that do not have gaps in their graphs. Likewise, in order for the sum to have a kink in its graphed curve at least one of the components would have to have a kink. Do we have any gaps or kinks in our general polynomial?
The line extending from the center of a circle to the point on the circle specified byθ corresponds to a vector, so the polynomial corresponds to a vector sum on the complex plane. As either θ (a real number) or C (a positive real number) increase or decrease, the endpoint of that chain of vectors moves across the complex plane and traces out a trajectory. Now we need only prove and verify that some combination of θ and C will put the endpoint of the vector chain on any arbitrarily chosen point on the complex plane (e.g. the point -a0=C0exp(iθ0).
Toward that proof assume the existence on the complex plane of a set of points, a dead zone, that P(z) does not reach for any values ofθ and C. Next we examine the behavior of P(z) in the region around the dead zone as the relevant values of θ and C change. The trajectories that we trace come to points at the boundary of the dead zone and must each either teleport to another point on that boundary or make a perfectly sharp turn: in either case we have a discontinuity (a gap or a kink). But P(z) has no discontinuities, so we infer that the assumed dead zone cannot exist.
But someone may object that the trajectories, continuous and smooth, might all manifest as tangents to the boundary, marking off the dead zone by their envelope. Assume so. Follow one trajectory to the point where it coincides with the tangent to the boundary. At that point manipulate the values ofθ and C to make P(z) trace out a trajectory that comes into the picture perpendicular to the first one (if we could not do this, the variables would not stand independent from each other). But that second trajectory must stop dead at the first trajectory, thereby creating a discontinuity; but the polynomial describing that trajectory has no discontinuities; therefore, the dead zone cannot exist; therefore, there necessarily exists a z0 such that P(z0)=-a0 for any a0; therefore, any P(z) that has the entire complex plane as its domain, also has the entire complex plane as its range.
Thus we have proven and verified, in sketch form, the fundamental theorem of algebra, which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root; which means, equivalently, that we have algebraically closed the complete set of the complex numbers. Mathematicians also state this theorem by saying that every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, though some of the roots may duplicate each other. We recognize the name today as something of a misnomer, because the theorem does not actually form the foundation of modern algebra; its name comes from a time in which mathematicians conceived algebra almost entirely as a means of solving polynomial equations with real or complex coefficients. We might better call it the fundamental theorem of arithmetic, but we already have a fundamental theorem of arithmetic, so we let the name stand.
Now we know that we need no more new numbers and no more new arithmetic processes. In the infinite set of the complex numbers we have all of the elements that we accept as both necessary and sufficient to all calculations. We have established the foundation of calculational mathematics, upon which we construct all the rest.
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