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If our program must count, how large a count can an 8-bit computer maintain Going back to our discussion of the binary number system, this is the largest number we can represent with 8 bits: 11111111 This number is 128, plus 64, plus 32, plus 16, plus 8, plus 4, plus 2, plus 1 255. That s it for an 8-bit computer, unless we resort to some workaround. The first IBM PC used the Intel 8088 processor. It had an 8-bit data bus (meaning it read and wrote 8 bits at a time from/to peripheral devices), but internally it was a 16-bit computer. How large a count can a 16-bit computer maintain Here s the number, broken into two 8-bit chunks (bytes) for legibility: 1111111 11111111 This number is 32,768 (215), plus 16,384, plus 8192, plus 4096, plus 2048, plus 1024, plus 256, plus 255 (the lower 8 bits we already computed above) 65,535. That s a much bigger number than the maximum number an 8-bit computer can work with, but it s still pretty small for some jobs. You d never be able to use a 16-bit computer for census work, for instance, without some workaround. Today, most computers we re familiar with use a 32-bit word size. The maximum count possible with 32 bits is over 4 billion. The next generation computers will likely use a 64-bit word size, and the maximum count possible with 64 bits is something like a trillion billions! The ability to represent a large number directly is nice, but it comes at a cost of bit efficiency. Here s what the number 6 looks like in a 32-bit word: 00000000000000000000000000000110 There are a lot of wasted bits (leading zeros) there! When memory was more expensive, engineers used to see bit-efficiency as a consideration, but memory is now so inexpensive that it usually is no longer a concern. INTEGER DATA FORMATS So far our discussion has been of whole numbers only, and even of positive whole numbers. Computers need to keep track of the sign of a number, and must also be able to represent fractional values (real numbers). As you might expect, if we need to keep track of the sign of a number, we can devote a bit of the computer word to maintaining the sign of the number. The leftmost bit, also known as the most significant bit ( msb in contrast to the least significant bit, lsb, at the right end of the word), will be zero if the number is positive, and 1 if the number is negative. Here is a positive 6 for an 8-bit computer: 00000110 The msb is 0, so this is a positive number, and we can inspect the remaining 7 bits and see that the value is 6. Now here s a counter-intuitive observation. How do we represent 6 You might think it would be like this: 10000110 That would be incorrect, however. What happens if we add 1 to that representation We get 10000111, which would be 7, not 5! This representation does not work correctly, even in simple arithmetic computations. Let s take another tack. What number would represent 1 We can test our idea by adding 1 to 1. We should get 0 as a result. How about this for negative 1: 11111111 That actually works. If we add 1 to that number, we get all zeros in the sum (and we discard the final carry).
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In fact, the correct representation of a negative number is called the two s complement of the positive value. To compute the two s complement of a number, simply change all the zeros to ones and all the ones to zeros, and then add one. Here is the two s complement of 6: 11111001 +00000001 11111010 All the bits of +6 are complemented (reversed) Add one The two s complement of 6 = 6
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You can check to see that this is correct by adding 1 to this representation 6 times. You will find that the number becomes 0, as it should (ignoring the extra carry off the msb). You can also verify that taking the two s complement of 6 correctly represents +6. Larger word sizes work the same way; there are simply more bits with which to represent the magnitude of the number. These representations are called integer or integral number representations. They provide a means of representing whole numbers for computation. REAL NUMBER FORMATS Numbers containing fractions are more difficult to represent. Real numbers consist of a mantissa and an exponent. Computer designers decide how to allocate the bits of the computer word so that some can be used for the mantissa and some for the exponent. In addition, the mantissa can be positive or negative, and the exponent can be positive or negative. You might imagine that different designers could create different definitions for real number formats. A larger mantissa will provide greater precision; a larger exponent will provide for larger and smaller magnitudes (scale). As recently as the 1980s, different computer manufacturers used different representations, and those differences made it difficult to move data between computers, and difficult to move ( port ) programs from one make of computer to another. Since then, the IEEE has created a standard for binary floating-point number representation using 32 and 64 bits. The 32-bit format looks like this: SEEEEEEEEmmmmmmmmmmmmmmmmmmmmmmm The msb is the sign of the number, the 8-bit field is the exponent of 2, and the 23-bit field is the mantissa. The sign of the exponent is incorporated into the exponent field, but the IEEE standard does not use simple two s complement for representing a negative exponent. For technical reasons, which we touch on below, it uses a different approach. How would we represent 8.5 First we convert 8.5 to binary, and for the first time we will show a binary fractional value: 1000.1 To the left of the binary point (analogous to the decimal point we re familiar with) we have 8. To the right of the binary point, we have 1/2. Just as the first place to the right of the decimal point in base 10 is a tenth, the first place to the right of the binary point in base 2 is a half. In a manner akin to using scientific notation in base 10, we normalize binary 1000.1 by moving the binary point left until we have only the 1 at the left, and then adding a factor of 2 with an exponent: 1.0001 * 23 From this form we can recognize the exponent in base 2, which in this case is 3, and the mantissa, which is 0001. The IEEE 32-bit specification uses a bias of 127 on the exponent (this is a way of doing without a separate sign bit for the exponent, and making comparisons of exponents easier than would be the case with two s complements trust us, or read about it on-line), which means that the exponent field will have the binary value of 127 + 3, or 130. After all this, the binary representation of 8.5 is: 01000001000010000000000000000000
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