In mathematics, numbers are rigorously defined and separated into classes. The relationships between different systems is very well illustrated below:
N is the system for natural numbers. These include numbers, such as 1, 2 and 3 and do not include zero. For example 6 / 2 = 3 and 5 ^ 2 = 25 .
Z is the system for integers . The notation Z comes from the German word Zahlen , which means Numbers . These include positive and negative integers, such as -3, -2, -1, 0, 1, 2 and 3. For example -12 / 3 = -4 and 8 ^ 2 = 64 .
Q is the system of rational numbers. A rational number can be written as a fraction . The resulting decimal will be either repeating or terminating. The denominator of the fraction cannot be zero. The notation Q means Quotient . For example -13 / 9 = -1.444... and 8 ^ -2 = 0.015625 .
I is the system for any number that cannot be written as a fraction where the numerator and denominator are integers. Since irrational numbers cannot be expressed as a fraction they form decimals that are neither repeating nor terminating. For example Pi and Sqrt( 7 ) .
R is the real numbers system and includes the rational and irrational numbers.
The notation, such as Z* means Z \ { 0 } or Z excluding zero . Writing Z+ means only positive numbers of Z . Consequently, Z- means only negative numbers of Z .