Esperanto for human and computer.
If an arbitrary binary number divided into groups of n bits, each group can be associated with number system with base b = 2n. In particular, any triple of bits corresponds to a unique eight-(8 = 23) figure, and, conversely, any octal digit can be associated with only three bits (the triad). This defines a simple relationship between the records of numbers in binary and octal number systems.
In order to transfer to the octal binary number is divided into triads (right). If, in the extreme left of the group are less than three binary digits, missing positions with zeros.
Representation of negative numbers, additional codes.
One of the reasons for the emergence of negative numbers in mathematics was the need to always feasible subtraction. Expression = 05/02, senseless on the set, takes a definite value of - 3, on the set of integers. The record itself negative, as well as any other numbers, in essence is a certain code, decoded by special rules.
Operations on negative numbers are defined so that they obey the same laws as the natural numbers.
Representation of fractions. Number of fixed-and floating-point.
To represent fractions programmer can mentally have the binary comma in any desired position to him. Naturally, when performing binary addition and subtraction, commas should be placed equally in both operands, so is the will and the point position as a result of. Representation of fractions in which the position of the binary point implicitly specified in a certain place the machine word is called a fixed-point representation.
The disadvantage of such a representation is manifested in problems with the quantities which can vary greatly in the direction of a very small and very large numbers.
Intermediate positions allow mixed-point shot, consisting of a decimal. Note that the dual economy of bits in a word at the cost of some loss in accuracy of representation is such fractions, because of their previous records could be meaningful to all 256-bit.
The real gain in the number of bits while large, but still less than it might seem. After all, if you allow point to "float" in all the numbers, then to manipulate them to write each number should have information about the location of the point. In cases where this information is expressed in writing numbers explicitly talk about the representation of floating point numbers.
The most convenient way of the status point is given if the numbers presented in the so-called. exponential notation. This means that the number is written as a product of the fraction with the sign (the mantissa) and the radix raised to the power with exponent (order).
Here the reader may ask, what makes sense to talk about the floating point mantissa if it is fixed? The fact that the normalized mantissa actually specifies only the number of significant digits. The true point position in particular after its translation into the usual form of no exponential explicitly specifies the order. And although for the mantissa and the order of the point position is obtained quite certain, for the same mantissa, but with a different order, comma, take a different position.
Instead of the term "floating point" is often used "floating point". This means the same thing. The fact that the record numbers at high-level language instead of the decimal (binary) decimal point character is used terms. The beauty of this approach, it is obvious to all who have faced the need to record several consecutive floating point numbers.
Arithmetic floating point numbers and its characteristics.
Perform arithmetic operations on floating point numbers in computers is different in many ways. Processors simple micro-computers do not have commands that allow directly manipulate the numbers in this format. To this end, have to be special programs that use teams of integer arithmetic. Complex instruction set computer commands include floating-point arithmetic, which, thanks to the hardware implementation is much faster. As a compromise the initial configuration of the micro-computer, designed for integral action can be replenished purchased separately floating point processor.
In any case, the algorithms of arithmetic operations on numbers in floating point format are the same; the difference is in their hardware or software implementation. Thus, to perform the multiplication to multiply the mantissa numbers and ads up their orders.