Binary Arithmetic Operations

The implicit loop and summation conventions have the effect of extending the meanings of all PL/I unary and binary arithmetic operators. Some compromise of the principles set forth earlier is necessary in that, while the elementwise applications of the operators +,, unary-, REAL, IMAG, COMPLEX, CONJG, etc. to... 

This particular function is called the exclusive OR function (XOR) and is the basis for binary arithmetic operations. It says that X is true if A or B, but not both, are true. It has a defining symbol and is written as... 

In addition to the four binary arithmetic operations, there are some important mathematical operations that involve only one number unary operations). The magnitude, or absolute value, of a scalar quantity is a nonnegative number that gives the size of the number irrespective of its sign. It is... 

The previous discussion concentrated on arithmetical operations by computing in binary numbers represented as bits and bytes. However, other computer functions also use bytes of information. 

An example of clause coding for a specific problem is given in Table 15.1, which contains the problem text for a multistep problem, the binary vector that serves as input for the connectionist network, and the clause encodings used by the production systems. Each clause contains three types of information owner, object, and time. Owner contains two fields name and type. Object contains four fields name, type, value, and action. The action contains information necessary to determine which arithmetic operation to use. For example, an action might be increase, decrease, more, or less. The final type of clause information is time, which contains just one field that indicates a relative time of occurrence within the problem. A clause can contain multiple owners and multiple objects, and it can omit time. 

The arithmetic registers are high-speed electronic accumulators (ACs). That is, each is a set of n electronic two-state devices (like flip-flops—see Sec. 23.2), which can be used to accumulate intermediate results of binary arithmetic involving -bit data. Nearly all the arithmetic and logical operations of the CPU are carried out in the arithmetic registers. Binary information can be transferred to or from memory and the arithmetic registers by the execution of appropriate instructions. 

How are calculations done using binary numbers Arithmetic operations are similar but simpler than those for decimal numbers. In addition, for example, four combinations are feasible ... 

Apart from arithmetic operations in the computer, logical reasoning is necessary too. This might be in the course of an algorithm or in connection with an expert system. Logical operations with binary numbers are summarized in Table 1.2. 

Burks, Goldstine, and von Neumann first identified the principal components of the general-purpose computer as the arithmetic, memory, control, and input-output organs, and then proceeded to formulate the structure and essential characteristics of each unit for the IAS machine. Alternatives were considered and the rationale behind the choice selected presented. Adoption of the binary, rather than decimal, number system was justified by its simplicity and speed in elementary arithmetic operations, its applicability to logical instructions, and the inherent binary nature of electronie components. Built-in floating-point hardware was ruled out, for the prototype at least, as a waste of the critical memory resource, and because of the increased complexity of the circuitry consideration was given to software implementation of such a facility. 

Table 1.6 gives binary examples for each of the basic arithmetic operations. 

The arithmetic logic unit, or Al,l,i, of a CPU is made up of a series of registers, or accumulators, in which the intermediate results of binary arithmetic and logic operations are accumulated. The Intel Pentium 4 processor coniains nearly 50 million transistors and is capable of operating al clock speeds greater than 5.5 GHz. The Intel tianium processor contains 22 million transistors (the Itanium 2 processor has410 million transistors I. The fastest computers can execute nearly 1 billion instructions per second. 

In one study, Steiglitz, Irfan Kamal, and Arthur Watson (1988) designed a particular class of CAs—the one-dimensional, binary-state, parity-rule filter automata — to perform arithmetic. This class of automata has the property that propagating periodic structures often act as solitons —that is, they can pass through each other in space-time without destroying eax h other, but only shifting each others phase. It turns out that such a feature can be useful for implementing arithmetic operations in CAs via particle interactions. 

Operations such as the above are carried out very rapidly by the computer through voltage switching, each switch lasting only a few nanoseconds. Therefore, although it is clumsier to represent numbers in binary for the human mind, and instead we use ten symbols (0, 1,. .., 9) to help us with complicated arithmetic, the speed with which we can do this arithmetic is nothing like the speed of the computer. Computer addition seems instantaneous, whereas human response to addition takes a finite time. 

OPERATING-LINE DIAGRAM. For a binary system, the compositions of the two phases in a cascade can be shown on an arithmetic graph where x is the abscissa and y the ordinate. As shown by Eq. (17.2), the material balance at an intermediate point in the column involves the concentration of the L phase leaving stage , and the concentration of the V phase entering that stage. Equation (17.2) can be written to show the relationship more clearly ... 

Input-output devices planned for the prototype include a cathode-ray-tube display unit and an operator s console typewriter. Machine subroutines or separate special-purpose hardware to handle the decimal-to-binary conversion upon input and binary-to-decimal at output are discussed with the subroutines getting the nod for the IAS-type machine on the basis that it is intended for problems with a large ratio of compute to input-output time. The second of these historic reports introduces the programming concepts of flow diagrams, storage tables, coding, subroutines, and loaders, and illustrates their application with examples such as decimal-binary and binary-decimal conversion, double-precision arithmetic, and sorting problems. 

Example 3 The roman arithmetic Mathcad works with decimal, binary, hexadecimal, octal numbers. However, we may need to make Mathcad work with forms more exotic, for example, with Roman numbers. For Roman numbers operations, we insert the function with the invisible name that returns a Roman number if its argument is an Arabic and conversely the Arabic number if the argument is Roman (Fig. 6.57). 

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