4 Gödel, Hawking and the Foundations of Physics
65
Fig. 4.1 A right-angled triangle: the length of its hypotenuse is expressed as an
irrational number
patterns in the order of the digits. Such a number cannot be expressed as a
fraction, 2 and is called an irrational number .
In other words, to find the solution of the equation x 2 = 2, one must go
into a still wider number world, the one of “real ” numbers, which includes
both the rational and the irrational numbers. By the way, the word “irrational” conveys a deep apprehension of these numbers, which were not
wanted by their discoverers (the school of Pythagoras) because they contradicted their faith in a numerological “purity”. According to legend, the
discoverer, a student of Pythagoras, was forced to commit suicide, lest his
sacrilegious discovery be propagated to others.
Surely, now at last, our number world is sufficiently broad to allow us
to carry out all the numerical operations we are ever likely to encounter.
Not quite. For instance, take up your calculator and ask it to calculate the
square root of −1 (or any other negative number). You will obtain a fairly
dismissive error message from your machine, as if you had done something
rather stupid. However, negative numbers are a legitimate part of our number
world, and taking the square root is a legitimate mathematical operation.
Surely one might expect a valid result from such an endeavour.
Mathematicians certainly thought so, and introduced into the number
world a quantity i which they defined to be the square root of −1(so that
i * i, or i 2 = −1). The square root of −4 then became 2i. Numbers of this
type were deemed imaginary, because mathematicians thought that i was only
a “trick” to solve the equations. The number world would now seem to have
2 The converse is not true; some fractions can indeed be expressed as an infinite number of decimal
places, e.g. 1/3 = 0.333333… and 1/7 = 0.142857142857142857…, but in this case there is a
pattern of repetition in the string of digits.
65
Fig. 4.1 A right-angled triangle: the length of its hypotenuse is expressed as an
irrational number
patterns in the order of the digits. Such a number cannot be expressed as a
fraction, 2 and is called an irrational number .
In other words, to find the solution of the equation x 2 = 2, one must go
into a still wider number world, the one of “real ” numbers, which includes
both the rational and the irrational numbers. By the way, the word “irrational” conveys a deep apprehension of these numbers, which were not
wanted by their discoverers (the school of Pythagoras) because they contradicted their faith in a numerological “purity”. According to legend, the
discoverer, a student of Pythagoras, was forced to commit suicide, lest his
sacrilegious discovery be propagated to others.
Surely, now at last, our number world is sufficiently broad to allow us
to carry out all the numerical operations we are ever likely to encounter.
Not quite. For instance, take up your calculator and ask it to calculate the
square root of −1 (or any other negative number). You will obtain a fairly
dismissive error message from your machine, as if you had done something
rather stupid. However, negative numbers are a legitimate part of our number
world, and taking the square root is a legitimate mathematical operation.
Surely one might expect a valid result from such an endeavour.
Mathematicians certainly thought so, and introduced into the number
world a quantity i which they defined to be the square root of −1(so that
i * i, or i 2 = −1). The square root of −4 then became 2i. Numbers of this
type were deemed imaginary, because mathematicians thought that i was only
a “trick” to solve the equations. The number world would now seem to have
2 The converse is not true; some fractions can indeed be expressed as an infinite number of decimal
places, e.g. 1/3 = 0.333333… and 1/7 = 0.142857142857142857…, but in this case there is a
pattern of repetition in the string of digits.