Binary arithmetic

Note The sequence of columns in binary runs 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 and so on, each column being two times greater than the previous one. Compare this with the decimal system where each column is 10 times greater than the previous one. 

To convert 1001 into a decimal number we have to take 1 x 8 plus 0 x 4 plus 0x2 plus 1 X 1 which is 9 (decimal). Conversion of 3265 into binary is a little more difficult the largest binary number near 3265 is 2048 (or 2 ) and there is only one of them with 1217 left over the next largest binary number is 1024 (or 2 °), leaving 193 the next binary number is 128 (or 2 ) and then 64 (2 ) and 1 (2°). Thus, the binary equivalent of 3265 is 110011000001. 

What is the binary equivalent of decimal 5 The largest power of 2 which fits 5 is 2 = 4. Therefore there is 1 x 2 in 5 with 1 left over 2 = 2 which is too big and so we write 0x21 finally we see that 1 x 2° = 1. The number 5 is made up from a 1 x 2 and a 1 X 2° = 1 but in binary we must not forget to put 0 x 2 in decimal we may write 304 meaning three hundred and four and if the 0 had been omitted then the number would have been thirty four (34). Thus, the binary equivalent of 5 is 101 and not 11. Other numbers may be converted into binary in an exactly similar fashion. Decimal 39 is  

Addition of two numbers in binary code. Note the carryover in the left-most column caused by adding two ones. 

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The account begins with binary arithmetic, moves on to on-off (flip-flop) electronic switches, then to serial and parallel processing, and finally to computers/transputers. 

Recently, Teymour and coworkers developed an interesting computational technique called the digital encoding for copolymerization compositional modeling [20,21], Their method uses symbolic binary arithmetic to represent the architecture of a copolymer chain. Here, each binary number describes the exact monomer sequence on a specific polymer chain, and its decimal equivalent is a unique identifier for this chain. Teymour et al. claim that the... 

The Jesuit Joachim Bouvet... approached Leibniz with a complete philosophy built from Christian dogma, Hermetic magic, Chinese classics, and Leibniz s binary arithmetic. In... 

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... 

Life, as we will explain, runs on base 4 the DNA bases A, G, C, and T. Computers (except analog computers, now rarely used) run on base 2. That is, they are digital and use binary logic for digits, based on two symbols typically expressed as 0 and 1. Deep down, the work in binary logic, such as 1 = true, 0 = false, and binary arithmetic on binary numbers 0 = 0,1 = 1,10 = 2,11 = 3, 110 = 6, 111 = 7,..., blocks of 8 bits a time (e.g., a byte), could also represent a machine code instruction such as add or jump to another part of the calculation. ... 

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. 

All information handled or generated by the central processing unit (CPU) must be binary or binary-coded machine language. This includes instructions, memory addresses, and data. Thus, the small-computer user must quickly become familiar with this number system. It would be well to review here the binary number system and binary arithmetic. 

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... 

TABLE 1.5 Single Digit Binary Arithmetic Tables... 

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 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... 

What is a key A key is a series of bits that can be used by software or hardware to convert plain text into cipher text by binary arithmetic processes. The length of the key, and how it is used, determine how secure the encryption is against attack by another party. 

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