C++ for Scientists - Technische Universität Dresden

Finite World of Computers
Chapter 8
8.1 Mathematical Objects inside the Computer
First natural numbers N are introduced and the data types available in a programming language
used to represent them. The difference between a single digit, and their connection to the used
base is an important concept in computer science.
A number is represented by several single digits, with each digit being a factor for a corresponding
power of the base. The number is only complete when both the base and all of the digits are
known. To use an example, the digit sequence 123 is calculated with the corresponding base,
e.g. base = 10:
12310 = 1 · 10 2 + 2 · 10 1 + 3 · 10 0
If the base is switched, e.g. base = 4, then the following numbers are derived:
1234 = 1 · 4 2 + 2 · 4 1 + 3 · 4 0 = 2710
One of the drawbacks of the representation of numbers within the computer is the fact that the
built-in types such as int and long can only use a finite number of bits and are hence limited
in their range (the int can be omitted), e.g.:
short int: -32768 +32767
long int: -2147483648 +2147483647
unsigned long int: 0 +4 294 967 295
As can be seen the maximum number of countable items is restricted. If, as an example, a
program has to count the living humans on earth we have to switch to another number concept,
either floating point or a decimal data type. A plain and simple arbitrary digit number container
can be implemented by:
class big number{
long base;
std::vector digits;
public:
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