3.5 Real Numbers

Rational and irrational numbers together constitute the set of real numbers R. They fill the number line densely (every arbitrarily small neighborhood of a real number on the number line contains at least another real number)


Figure 3.5: Number line with two transcendental irrational numbers

Taking the power and root taking with rational exponents is possible in the domain of real numbers, if the root exponent is odd ( - 13 = -1; - 13 = 1 or if for even exponents the argument of the root is positive via the following definitions:

rational numberq = n m;n,minteger  for realnumbersx : xq = x n m = xnm = xmn

The real numbers constitute the largest possible ordered set of numbers. For two real numbers a and b it is clear whether a is larger, equal or smaller than b:

a > bora = bora < b

For the application in physics the distinction between rational, irrational and transcendental numbers plays an important role, as their symbols express relationships in a formula that has been derived via a model. If the number π appears, circular symmetry or periodicity plays a role, while the appearance of e points to a problem involving growth or damping.

As soon as computations take place, irrational numbers are always approximated with finite accuracy via rational numbers. The formally excluded division by zero in the domain of real numbers looses its exceptional position, since it will always be the division by a very small, but finite real number.

The arithmetic operations can be interpreted as transformations or mappings on the number line. Addition and subtraction are translations where all numbers are shifted by the absolute value of the summand. Multiplication and division lead to stretching or compression of the number line by a factor n.

Division corresponds to a transformation of the range of numbers outside of the divisor to the range of numbers between the divisor and zero.

For the example 1a a = 1 is mapped to itself, numbers a > 1 are mapped to the range 0 1, the closer to zero, the larger a is. Numbers a < 1 are mapped to the domain larger than 1 and the further away from 1 the closer a is to zero.