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6The Quintessential PIC Microcontroller

cessor, the Intel 4004, handled its data four bits (a nybble) at a time. Many current processors cope with blocks of 8 bits (a byte), 16 bits (a word), or 32 bits (a long-word). 64-bit (a quad-word) devices are on the horizon. These groupings are shown in Table 1.2. The names illustrated are somewhat de-facto, and variations are sometimes encountered.

As in the decimal number system, large binary numbers are often expressed using the prefixes k (kilo), M (mega) and G (giga). A binary kilo is 210 = 1024; for example 64 kbyte of memory. In an analogous way, a binary mega is 220 = 1, 048, 576; thus a 1.44 Mbyte floppy disk. Similarly a 2 Gbyte hard disk has a storage capacity of 2 × 230 = 2, 147, 483, 648 bytes. The former representation is certainly preferable.

Table 1.2: Some common bit groupings.

(0000 0000 0000 0000−1111 1111 1111 1111)

Long-word (32 bits) 0−4,294,967,295

(0000 0000 0000 0000 0000 0000 0000 0000−1111 1111 1111 1111 1111 1111 1111 1111)

Long binary numbers are not very human friendly. In Table 1.2, binary numbers were zoned into fields of four digits to improve readability. Thus the address of a data unit stored in memory might be 1000 1100 0001 0100 0000 1010b. If each group of four can be given its own symbol, 0…9 and A…F, as shown in Table 1.3, then the address becomes 8C140Ah; a rather more manageable characterization. This code is called hexadecimal, as there are 16 symbols. Hexadecimal (base-16) numbers are a viable number base in their own right, rather than just being a convenient binary representation. Each column is worth 160, 161, 162 . . . 16n in the normal way.5

Binary Coded Decimal is a hybrid binary/decimal code extensively used at the input/output ports of a digital system (see Example 11.5 on page 298). Here each decimal digit is individually replaced by its 4-bit binary equivalent. Thus 1998 is coded as (0001 1001 1001 1000)BCD. This is very di erent from the equivalent natural binary code; even if it is represented by 0s and 1s. As might be expected, arithmetic in such

5Many scientific calculators, including that in the Accessories group under Windows 95, can do hexadecimal arithmetic.

1. Digital Representation 7

Table 1.3: Di erent ways of representing the quantities decimal 0…20.

a hybrid system is di cult, and BCD is normally converted to natural binary at the system input and processing is done in natural binary before being converted back (see Program 5.9 on page 131).

The rules of arithmetic are the same in natural binary6 as they are in the more familiar base 10 system, indeed any base-n radix scheme. The simplest of these is addition, which is a shorthand way of totalling quantities, as compared to the more primitive counting or incrementation process. Thus 2+4 = 6 is rather more e cient than 2+1 = 3; 3+1 = 4; 4 + 1 = 5; 5 + 1 = 6. However, it does involve memorizing the rules of addition.7 In decimal this involves 45 rules, assuming that order is irrelevant; from 0 +0 = 0 to 9 +9 = 18. Binary addition is much simpler as it is covered by only three rules:

Based on these rules, the least significant bit (LSB) is totalized first, passing a carry if necessary to the next left column. The process ends with

6Sometimes called 8-4-2-1 code after the weightings of the first four lowest columns. 7Which you had to do way back in the mists of time in primary/elementary school!

8The Quintessential PIC Microcontroller

the most significant bit (MSB) column, its carry being the new MSD of the sum. For example:

Just as addition implements an up count, subtraction corresponds to a down count, where units are removed from the total. Thus 8 − 5 = 3 is the equivalent of 8 − 1 = 7; 7 − 1 = 6; 6 − 1 = 5; 5 − 1 = 4; 4 − 1 = 3.

The technique of decimal subtraction you are familiar with applies the subtraction rules commencing from LSB and working to the MSB. In any given column were a larger quantity is to be taken away from a smaller quantity, a unit digit is borrowed from the next higher column and given back after the subtraction is completed. Based on this borrow principle, the subtraction rules are given by:

0 − 0 = 0

10 − 1 = 1 Borrowing 1 from the higher column

1 − 0 = 1

1 − 1 = 0

For example:

Although this familiar method works well, there are several problems implementing it in digital circuitry.

•How can we deal with situations where the minuend is larger than the subtrahend?

•How can we distinguish between positive and negative quantities?

•Can a digital system’s adder circuits be coerced into subtracting?

To illustrate these points, consider the following example:

10 The Quintessential PIC Microcontroller

From this example we see that if negative numbers are in a signed 2’s complement form, then we no longer have the requirement to implement hardware subtractors, as adding a negative number is equivalent to subtracting a positive number. Thus A − B = A + (−B). Furthermore, once numbers are in this form, the outcome of any subsequent processing will always remain 2’s complement signed throughout.

There are two di culties associated with signed 2’s complement arithmetic. The first of these is overflow. It is possible that adding two positive or two negative numbers will cause overflow into the sign bit; for instance:

In (a) the outcome of (+8) + (+11) is −13! The 24 numerical digit has overflowed into the sign position (actually, 10011b = 19 is the correct outcome). Example (b) shows a similar problem for the addition of two signed negative numbers. Overflow can only happen if both operands have the same sign bits. Detection is then a matter of determining this situation with an outcome that di ers. See Fig. 1.5 for a logic circuit to implement this overflow condition.

The final problem concerns arithmetic on signed operands with different sized fields. For instance: