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Square root of two as an irrational number
by Br. Dunstan Robidoux OSB
edited by Mr. Michael Hernandez MA
When Lonergan discusses inverse insight in the first chapter of his Insight: A Study of Human
Understanding, he presents a mathematical example to illustrate the nature of inverse insight as an
act of understanding which realizes that an expected, desired intelligibility is not to be reasonably nor
rationally expected. (1) In some situations, in some inquiries, to anticipate in the type of intelligibility
sought is to perdure in "barking up the wrong tree" and to waste time by asking irrelevant questions.
However, since Lonergan's example pains readers who have never acquired any easy familiarity with
mathematics and who have lost what familiarity they once had, this paper will parse out the discussion
in ways which should help. Let us begin.
Lonergan's argument consists of the following sequence of numbered propositions:
Proposition 1: The square root of 2 is some magnitude greater than unity and less than two
Proposition 2: One would expect it to be some improper fraction,
say m/n, where m/n are positive integers and by the removal
of all common factors m may always be made prime to n.
Proposition 3: If this expectation correct, then the diagonal
and the side of a square would be respectively m times and n
times some common unit of length.
Proposition 4: So far from being correct, the expectation
leads to a contradiction.
Proposition 5: If sqrt(2) = m/n, then 2 = m2/n2
Proposition 6: But, if m is prime to n,
then m2 is prime to n2
Proposition 7: In that case, m2/n2
cannot be equal to two or, indeed, to any greater integer
Proposition 8: The argument is easily generalized, and so
it appears that a surd is a surd because it is not the rational fraction that
intelligence anticipates it to be
To understand the controversy about the square root of 2, let us look briefly at the historical
origins of the problem.
First, with respect to numbers, the square root of 2 is some sort of number. Numbers fall into
different types or species since the square root of a number is unlike the number whose square root
is sought. Numbers rank as human inventions since they do not exist as purely natural entities
apprehended by sense. They were invented as the human need for them arose. (2) Different needs, as
they emerged, formed new types of numbers. Hence, the first type of numbers invented were the
counting numbers, sometimes cited as natural numbers: 1, 2, 3, 4, 5.... (3) They arose as correlatives
to designate quantities: how many of this or how many of that. For example, "3" identifies three
sheep or three fish. The sequence of counting numbers is potentially infinite since the human mind
can keep adding units of 1 to form an ever greater number. Subsets are similarly infinite in their
sequences. The odd numbers, as in 1, 3, 5, 7..., are infinite as are the even numbers, 2, 4, 6, 8.... On
a straight line, in one vector, each natural number can be represented by one point on a line ad
infinitum. (4)
A second species of number emerges in whole numbers when counting proceeds in reverse:
toward and beyond 1. Nought or zero emerges as a number to signify the absence of some item. The
creation of this numerical designation signifies an "empty set" as in "the number of Eskimos living
in our house is 0." (5) The inclusion of 0 with the counting numbers thus creates a larger system of
numbers than the old quantitative counting numbers. Enumeration now begins from 0 which can also
be represented by a point on a line.
A third, more comprehensive set of numbers emerges when the reverse counting which had
led to 0 continues backwards to include numbers that are now less than zero. The result is a
potentially infinite set of negative whole numbers. When these numbers are then added to the
numbers that have already been generated by counting from zero upwards (the positive whole
numbers), the result is a set of numbers known as integers. An integer is defined as a positive or
negative whole number as in 0, ±1, ±2, ±3, ±4 . . . (6) The negative and positive signs indicate direction:
all these numbers are directed. On a number line, the negative numbers go to the left of 0 while the
positive go to the right. Each number has a point.
Rational numbers deriving from a ratio or fraction of integers or whole numbers emerged
when it became necessary to specify measurements which are parts of a number. How does one
express a length which is between 4 and 3 meters or 4 and 3 cubits? Is a loaf of bread, equally
divided among 5 persons, divided in a way where each piece has a numeric value of 1/5? Does the
addition of 1 piece to another not result in a union with a numeric value of 2/5? A number designating
parts thus consists of parts in its makeup. There are two halves: a numerator above a line and
denominator beneath. (7) The denominator indicates how many intervals exist between two possible
whole numbers while the numerator indicates how many of these intervals are pertinent in a given
measurement. The denominator cannot be 0 since, otherwise, one would be indicating that no
intervals or parts exist between two numbers. Why specify numerators for portions or parts that do
not exist? A rational number is commensurate with given lengths that are being measured. A number
which includes a fraction can be assigned a point on a line. The position is determinate.
In the 5th Century B.C., the Pythagoreans initially assumed that numbers measuring the sides
of a triangle are rational where each number can be expressed as the ratio or quotient of two integers
(or two whole numbers). (8) Divisors (or denominators) exactly divide into numerators as in ½, 1/10, and
1/100: a half (or .5), a tenth (or .10), and a hundredth (or .100). A ratio as the quotient of two numbers
or quantities indicates relative sizes. (9) The ratio of one number to another is expressed in terms of a/b
or a:b. It was assumed that a one-to-one correspondence joins straight-line segments of length with
rational (whole) numbers. (10) In attempting to measure the diagonal of a square by taking a small part
of one side as the measuring unit, one should be able to fit the measuring unit a fixed number of times
within both the side and the diagonal. (11) All lengths are measurable and commensurate in terms of
rational (whole) numbers. Two quantities are commensurable if their designating numbers are
multiples: both numbers arise as products of common factors (a factor being a number that divides
a given number exactly or completely (12)). For instance, 16 and 12 are commensurable since both exist
essentially as multiples of 1, 2, or 4: each exactly divides into 16 and 12 and no other number exactly
divides 16 and 12. By multiplying one or more of these numbers together, one arrives at numbers
16 and 12 (in conjunction with other possible numbers that are also commensurable). Similarly, 3 feet
and 2 inches designate commensurable quantities since 3 feet contains 2 inches an exact or integral
number of times. (13) Hence, according to Pythagorean assumptions and expectations, the length of a
square's diagonal whose side is represented by a rational number should be represented by another
rational number.
On the basis of this belief in rational numbers and the corresponding commensurability of
lengths, according to the Pythagoreans, "numbers are things" and "things are numbers." All things
are numerable in terms of whole numbers and their properties. (14) A cosmic harmony exists in the
universe given the interrelation of things based on whole numbers where the relation between two
related things can be expressed according to a numerical proportion or ratio. For example, in music,
ratios of concord exist between musical sounds (pitch) and whole numbers since by halving the length
of a string on a lyre, one can produce one note one octave higher. All harmonies can be represented
by ratios of whole numbers and, by extending this principle to all things, through geometry one can
explore the configurations of perfect solids in the belief that all lengths are measurable in terms of
rational whole numbers.
A crisis emerged for the Pythagoreans when, possibly prior to 410 B.C., they realized that
some numbers, though real (as existing), class as irrational because they cannot be written as whole
numbers, as integers or as quotients of two integers. (15) No assignable point of a line can be given
them. Some numbers do not exist thus as whole numbers as can be seen through a deduction from
Pythagoras' Theorem in geometry which describes the relation between the lengths of the sides of
a right-angled triangle in the following terms:
In a right-angled triangle, the square on the hypotenuse [the side of
right-angled triangle opposite the right angle] is equal to the sum of
the squares on the other two sides. (16)
Thus, if the hypotenuse has a length c and the other two sides, lengths a and b, then c2 = a2 + b2.
Now, if, in a square, the side length constitutes 1 unit, then
c2 = 1 + 1
Hence,
c2 = 2
Thence,
c = sqrt(2)
The diagonal is 2 units in length. (17) This number obviously designates some magnitude greater than
1 or unity but less than two where, initially, one naturally assumes that this number is an improper
fraction expressing a whole number (an improper fraction being defined as a fraction whose
numerator exceeds its denominator as in 4/3 versus 3/4, designating a proper fraction (18)). (19) However,
if the square root of 2 cannot be expressed as a whole number, its irrationality in terms of whole
number properties creates major problems given expectations which assume the adequacy of whole
numbers. After all, conversely, if only rational numbers exist, the hypotenuse of every right-angled
triangle will have a length that cannot be measured by any whole number. (20) It is incommensurable,
non-measurable: in the relation between the diagonal d and an adjoining side s, d cannot be divided
by any unit common to s an integral number of times. In trying to effect any measurements, the
Greeks found that however small or large would be their measuring unit, it failed to fit within both
the diagonal and the adjoining side a fixed number of times. (21) A measuring unit that would fit the
adjoining side a fixed number of times would not fit the length of the diagonal. It was either too short
or too long. Proofs demonstrating the irrationality of 2 came in a number of varieties.
Aristotle refers to a proof on the incommensurableness of a square's diagonal with respect
to a side that is based on the distinction between odd and even, an odd number being an integer that
is not divisible by 2 while an even number is divisible by 2. (22) To understand how this argument
works, a digression on prime numbers introduces the discussion.
A prime number is a whole number with exactly two whole-number divisors, itself and 1.
Some primes are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . , 101, . . . , 1093
Prime numbers are the building blocks of other whole numbers. For example,
18 = 233 40 = 2225 105 = 357
This type of factorization is possible for all nonprime whole numbers greater than 1 and it illustrates
the fundamental theorem in arithmetic known as the Unique Factorization Theorem (23) which says, as
follows, about the prime decomposition of a whole number:
Any nonprime whole number (greater than one) can be written as the
product of a unique set of prime numbers. (24)
Every prime integer shares the important property that if it divides a product of two integers, then
it must divide at least one of the factors (prime numbers being only divisible either by themselves or
by 1). This theorem is important in many parts of mathematics. In one simple consequence, when
the square of any whole number is written as a product of primes, each prime occurs as a factor an
even number of times. For example:
(18)2 = 1818 = 233233 = 223333
two 2's four 3's
(40)2 = 4040 = 22252225 = 22222255
six 2's two 5's
(105)2 = 105105 = 357357 = 335577
two 3's two 5's two 7's
To prove that the square root of 2 is irrational, let us suppose that 2 is a rational number;
that is, suppose that 2 = m/n, where m and n are whole numbers (necessarily greater than 1). Then:
2 = m2/n2
and so
2n2 = m2
Now, imagine that both n and m are written as products of primes where, for instance (using algebraic
notation), n = xy while m = zpt. But, as previously noticed, both n2 and m2 must then have either
an even number of 2's or no 2's. But, in the above equation, the prime 2 appears on the left an odd
number of times either once (if n2 has no 2's) or more than once (if n2 has an even number of 2's) but,
on the right, the prime 2 appears either an even number of times or no times. This is clearly
impossible since, given the nature of primes, m2 equates with a number or produces a number that has
either an even number of 2's or no 2's. A contradiction obtains despite the equals sign. Therefore,
what can be wrong? The only thing that can be wrong is our supposition that 2 is a rational number.
If this proof is applied to other primes in terms of square roots for 3, 5, 7, . . ., the same dilemma
results. (25) Odd clashes with even to demonstrate the irrationality of these numbers. Hence, could all
numbers be the kind of numbers that the Pythagoreans had postulated? Are they all rational?
In Boyer's version of the mathematical proof demonstrating the incommensurableness of the
square root of 2 through the contrast between even and odd, he argues as follows: (26)
1. Let d and s respectively signify the diagonal and side of a square and let us assume that they
are commensurable: the ratio d/s is rational and equal to p/q, where p and q are integers with no
common factors.
2. given the Pythagorean theorem d2 = s2 + s2 reconfigured as d2/s2 = 2 (since d2 = 2s2), if the
ratio d/s = p/q (p and q being integers with no common factor), then (d/s)2 = p2/q2 = 2 or p2 = 2q2
3. therefore, p2 must be even since its equivalent 2q2 is divisible by 2 (which corresponds to
the definition of an even number as a number divisible by 2).
4. hence, if p2 is even, p is even since p2 when decomposed into constituent prime numbers
necessarily includes at least two instances of 2 as both a prime number and a factor, and the presence
of 2 in p makes p an even number since it is divisible by 2 (which again corresponds to the definition
of an even number).
5. as a result, q must be odd (not divisible by 2) since, according to conditions stated in
aforementioned proposition 2, q is an integer with no factors common to p and so it cannot have 2
as a constituent prime factor.
However, letting p = 2r and substituting in the equation p2 = 2q2 with, hence, the result that
4r2 = 2q2, 4r2 = 2q2 as reconfigured becomes q2 = 2r2. Then q2 must be even; hence q must be even
(according to the argumentation which had explained why formerly p2 and p must both be even).
However, a contradiction follows if one argues that q is both odd and even. No integer can be both
odd and even. As a consequence, it thus follows that the numerical relation between d and s is
incommensurable. (27) The result is not a definitive whole number.
A third but second species of proof relying on a study and understanding of prime numbers
demonstrates the absence of an anticipated whole number by adverting to the relation between d and
s. If, indeed, d (a whole number) is decomposed into constituent prime numbers and s (a second
whole number) is similarly decomposed, and if no factor is common between them, the improper
fraction d/s can never be resolved into a whole number since, in every case, the denominator does not
perfectly divide into the numerator to produce an anticipated, desired whole number. The result is
always some sort of fraction which, by definition, is not an integer, a whole number.
A geometrical proof that evidences the existence of irrational numbers in general, and not 2
specifically, designates a third species of proof. (28) Its lesser abstractness suggests earlier origins
predating the construction of later proofs using other types of arguments. When examining the sides
and diagonals of a regular pentagon (defined as a five-sided polygon with all the sides possessing
equal length) and the respective relations between s and d, if the diagonals of this pentagon are all
drawn, they form a smaller regular pentagon whose diagonals can also be drawn to form a smaller
regular pentagon ad infinitum. Hence, pictorially, the relation or ratio of a diagonal to a side in a
regular pentagon is indeterminate because it is indefinite. It is irrational. Similarly, if a straight line
is divided into two parts and one part is divided into two smaller parts, it will be possible to keep
dividing lengths indefinitely. (29) No determinate end is reached. Our expectations meet with frustration
as our inquiry encounters mysteries that occasion questions about the adequacy of our intelligible
anticipations. What is to-be-known cannot be known too easily or simply.
1. Bernard Lonergan, Insight: A Study of Human Understanding, eds. Frederick E. Crowe
and Robert M. Doran 5th ed. (Toronto: University of Toronto Press, 1988), pp. 45-6.
2. Leslie Foster, Rainbow Mathematics Encyclopedia (London: Grisewood & Dempsey
Ltd., 1985), p. 43.
3. Foster, p. 43.
4. Foster, p. 43.
5. Foster, p. 43.
6. 6The Penguin Dictionary of Mathematics, 1989 ed. S.v. "integer."
7. Foster, p. 44.
8. 8E. T. Bell, The Development of Mathematics (New York: Dover Publications, Inc.,
1992), p. 61.
9. 9The Penguin Dictionary of Mathematics, 1989 ed. S.v. "ratio."
10. 10Bell, p. 61.
11. Joseph Flanagan, Quest for Self-Knowledge (Toronto: University of Toronto Press,
1997), p. 33.
12. The Penguin Dictionary of Mathematics, 1989 ed. S.v. "factor."
13. The Penguin Dictionary of Mathematics, 1989 ed. S.v. "commensurable."
14. Carl B. Boyer, A History of Mathematics, 2nd ed. (New York: John Wiley & Sons, Inc.,
1989), p. 72; Frederick Copleston, S.J., A History of Philosophy, volume 1: Greece & Rome part
1 (Garden City, New York: Image Books, 1962), pp. 49-50; A Concise Oxford Dictionary of
Mathematics, 1991 ed., s.v. "Pythagoras," by Christopher Clapham.
15. The Penguin Dictionary of Mathematics, 1989 ed., s.v. "irrational number."
16. A Concise Oxford Dictionary of Mathematics, 1991 ed., s.v. "Pythagoras' Theorem," by
Christopher Clapham.
17. Bell, p. 61.
18. 18Penguin Dictionary of Mathematics, 1989 ed., s.v. "improper fraction."
19. 19Lonergan, Insight, p. 45.
20. 20Euclid quoted by Walter Fleming and Dale Varberg, College Algebra: A Problem-Solving Approach (Englewood Cliffs, New Jersey: Prentice Hall, n.d.), p. 16.
21. Flanagan, p. 33.
22. 22Boyer, p. 72; Penguin Dictionary of Mathematics, 1989 ed., s.v. "odd number," and
"even number."
23. Clapham, p. 187.
24. Fleming and Varberg, p. 16.
25. Fleming and Varberg, p. 17.
26. Boyer, pp. 72-3.
27. Boyer, p. 73.
28. Boyer, p. 73.
29. Boyer, p. 51.
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