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Number Sets

Number Sets

The number sets took an evolution during the centuries. In the beginning of mankind the numbers were fixed to the object. The abstraction from an object was the first step to develop our modern number sets. Our decimal system was not the only system in the world, but the best one for man. The best system for computers is the binary system.

The invention of a zero sign was not so easy. The old Indians developped it and the Arabs brought it to Europe.

Although there is some quarrel about zero, where it belongs to, many mathematicians say, it is a natural number. This thought is useful in higher mathematics, for instance when you need an index of natural numbers. Then zero is needed and you can write  without an index 0. In computer science the index of an array begins with 0. To begin with 0 has an advantage.

 = natural numbers {0, 1, 2, …}

The next number set is the integers which consist of natural numbers, also with the negative sign.

 = integers {…,-3, -2, -1, 0, 1, 2, 3, …}

If you broke an integer, you get a fraction. There are two sorts of fraction. The fraction which can be transformed into a number (with a decimal point) and which has a finite end or permanently repeats after the decimal point is called rational  number. The sign of repetition of the number after the decimal point is a short line above the repeated number .

 = rational numbers  consist of finite numbers or repeated numbers behind the decimal point.

n = integer, m = integer

  Examples

 (periodical)

There are numbers which can’t be expressed as fractions because they never stop behind the decimal point. They are infinite. These numbers are called irrational numbers. The term irrational means non-rational (“fake fraction”). There is no special sign for them. The square root of 2 is an irrational number.

 = irrational numbers (“fake fractions”)

  Examples

The set of rational and irrational numbers is called real numbers. With this set mankind reckons the daily arithmetics.

 = real numbers (fractions and more)

If you thought this was enough, I’ll show a further set of numbers, the imaginary numbers. They are the result of negative roots. Try to evaluate the root  . The mathematicians invented a new sign i (imaginary) for it.

This is the basic definition of an imaginary number.

 = imaginary numbers

Together with the real numbers the imaginary numbers form the complex numbers.

 = complex numbers

The complex numbers can only be described in a plane. They have two dimensions and consist of a real part  and an imaginary part .

It is possible to reckon with complex numbers. They have some nice effects, especially you sqare the imaginary i.

Russian dolls (Picture: Fanghong)

The number sets can be compared with Russians dolls. You can pack a small doll inside a bigger doll. This model suggests that the natural numbers fit inside the integers, the integers fit in the rational and irrational numbers, the real numbers with the imaginary numbers in the complex numbers.