Irrational numbers belong to material, but are not rational, that is their exact value is unknown. But if there is a description of a way which received irrational number, then it is considered known. In other words, its value can be calculated with a necessary accuracy.
According to concepts of geometry if two pieces contain any amount of identical values, then they are commensurable. For example, the different parties of a rectangle are commensurable. But here the party of a square and its diagonal are not commensurable. They have no general measure in which they could be expressed. Irrational numbers belong to implicitly expressed. They are incommensurable to rational numbers. The whole, fractional and also final and periodic decimal numbers belong to rational. They are commensurable to unit. Recurring decimal acyclic decimals are called irrational, they are incommensurable to unit. But the way of receiving such number can be specified, then it is considered set precisely. By means of such way it is possible to find any number of signs after a comma at irrational number, it is called to calculate number with a certain accuracy which just and is set by the number of the signs demanded to calculation. Properties of irrational numbers are in many respects similar to properties of rational. For example, they are compared equally, over them it is possible to make the same arithmetic actions, they can be positive or negative. Multiplication of irrational number by zero, likewise as well as rational, gives zero. If operation is made over two numbers, one of which rational, and another irrational, then it is accepted not to use whenever possible approximate value, and to take precisely the set number (for example, in the form of not decimal fraction). It is considered that for the first the concept of irrational numbers was opened by Gippas from Metapont living approximately in the 6th century BC. He was a follower of Pythagorean school. Gippas made the discovery during a sea campaign, being by the ship. According to a legend when he told other Pythagoreans about irrational numbers, having provided the proof of their existence, those listened to him and recognized its calculations as correct. Nevertheless, Gippas's opening so shocked them that he was thrown out overboard for the fact that created something, disproving the central Pythagorean doctrine that everything in the Universe can be reduced to integers and their relations.