Download Real Numbers

On irrational numbers:
Natural numbers
Integers
are: 0, 1, 2, 3 . . .
are: . . . -2, -1, 0, 1, 2 . . .
Rational numbers ℚare all the quotients, or ratios, of integers: 1/2, 3/8, -2/3, etc.
Real numbers are all the numbers required to represent the points on a number line.
It is natural to think that there is a rational number for each point on the number line; if this is so, then the real numbers are the
same as the rationals. But there are points not represented by any rational number. So there are irrational numbers, and the reals
are distinct from the rationals. To see this, consider the following diagrams:
g
(1)
(2)
c
(3)
d
c
d
c
i
h
a
a
b
b
a
f
b
e
Suppose (1) is a right triangle and lengths ab and bc both = 1; we are interested in length ac. For this, consider square abcd as in
(2). The area of abcd is ab2 = 12 = 1. Now consider (3) which adds square efgh. The sides of efgh are the same as ac; so the area
of efgh = ac2. But notice that efgh contains eight equilateral triangles, abi, abe, etc. all with bases equal to ab and sides equal to ai.
So the eight triangles are the same size. Since abcd contains four of these triangles, and efgh all eight, the area of efgh is twice the
area of abcd. But the area of abcd = 1, so the area of efgh = 2. So ac2 = 2. So ac = 2. This is a special case of the Pythagorean
Theorem. Note, then, that 2 is a definite length, and so a point on the number line. To see this, one might place a compass at
point a, open it until its arm coincides with point c, and draw an arc. The point where the arc intercepts the number line is the
desired point, since the distance from the origin is the same as length ac = 2.
c
a
b
2
Granting now that 2 is a point on the number line, it is possible to show that this point is not represented by any rational number.
The proof is by reductio: Suppose 2 can be expressed as a rational number. Then there are some integers n and m such that n/m =
2 where n and m have no common factor; so n2/m2 = 2; so n2 = 2m2; so n2 is even; so n is even (the sum of an odd number of odd
numbers is odd; so the square of an odd number is odd; but the sum of any number of even numbers is even; so the square of an
even number is even). Since n is even, there is some number r such that n = 2r; so n2 = 4r2; and since n2 = 2m2, 4r2 = 2m2; so 2r2 =
m2; so m2 is even; so m is even. But if both n and m are even, 2 is a common factor of n and m, and this is impossible, for we are
given that n and m have no common factor. Hence the assumption, that 2 can be expressed as a rational number, leads to
contradiction; so the assumption is wrong: 2 cannot be expressed as a rational number.
These results were known to the Greeks as early as 500 BC. In particular, the Pythagoreans (named after their founder Pythagoras,
and after whom the Pythagorean Theorem is named) placed particular emphasis on observed harmonies and ratios--noting
numerical patterns in nature, and especially ratios associated with music. (If the ratio between the lengths of tuned strings is 2:1,
the associated interval is an octave; if the ratio is 3:2, the interval is a fifth, etc.) The Pythagoreans regarded the discovery of
irrational numbers as disastrous for their belief in universal harmony. Legend has it that they tried to suppress the discovery; but,
for whatever reasons, the news was leaked by one Hippasus.
Rational numbers have decimal expansions that terminate or repeat (a remainder for x/y is always < y; with this limited “space” for
remainders, within y digits some remainder must be zero or repeat--and corresponding expansions must terminate or repeat). And
expansions that terminate or repeat are of rational numbers (terminating expansions divide by some power of 10 and are rational;
expansions that repeat can be conceived as a sum of a terminating and a repeating part; for the repeating part, 0.aaa... expands
a/(101 - 1) = a/9, 0.abcabc... expands abc/(103 - 1), 0.00abcabc... expands abc/102(103 - 1), and similarly in other cases).
Expansions for irrational numbers then, are infinitely long and non-repeating, but nonetheless represent definite positions on the
number line.