2 Signs and  Numbers

2.2 Numbers

2.2.3 Rational Numbers

Whenever people have been forced to do division, they have noticed that integers are not enough. Mathematically speaking: to solve the equation for within a number set we are forced to extend the integers to rational numbers by adding the inverse numbers or . We use the notation for the set of integers without the zero. Then we have for each integer different from exactly one

We are familiar with this concept. The inverse to the number 3 is , the inverse number to  -7 is . This way the  fraction for solves our starting equation as desired. In general, a rational number is the quotient of two integers, consisting out of a numerator and a denominator (different  from). Rational numbers are therefore mathematically speaking, ordered pairs of integers: .

When they are divided out, the rational numbers become finite, meaning breaking off or periodic decimal fractions: for example , and , where the line over the last digits indicates the period.

With this definition of the inverse elements the rational numbers form a group not only relative to addition, but also, relative to multiplication (with the Associative Law, the one and the inverse elements). This group is, due to the Commutative Law of the  factors, Abelian.

The rational numbers lie densely on our number line, meaning  in every interval we can find countable infinity of them:

Figure 2.4: The rational numbers

Because of the finite accuracy of every physical measurement the rational numbers are in every practical aspect the working numbers of physics as well as in every other natural science. This is why we had  paid such an attention to their rules.

By stating results as rational numbers, mostly in the form of decimal fractions, scientists worldwide have agreed on indicating only as many decimal digits as they have measured. Along with every measured value the uncertainty should also be indicated. This for example is what we find in a  table for Planck's quantum of action . This statement can also be written in the following way: meaning that the value of ( with a probability of ) lies between the following two borders: .

As a first application of powers we mention the  Pythagoras Theorem: In a right-angled triangle the square over the hypotenuse equals the sum of the squares over both catheti and :

 

Figure 2.5:

very frequently we need the so-called

which can  be easily derived, but need to be memorized.

The binomial formulas are a special case (for ) of the more general formula

whereare the so-called binomial coefficients. We can calculate them either directly from the definition of the  factorial , e.g.

 

or find them in the Pascal Triangle. This triangle is constructed in the following way:

We start with the number in the line . In the next line  () we write two ones, one on each side. Then (for ) we add  two ones to the left and right side once again, and in the gap between them  a  as the sum of the left and right "front man" (in each case a ). In the framed box, we once again recognize the formation rule. The required binomial coefficient is then found in line on position 3.