UNIT 2 : NUMBER SYSTEMS

Transcription

1 UNIT 2 : NUMBER SYSTEMS page 2.0 Introduction Decimal Numbers The Binary System The Hexadecimal System Number Base Conversion Decimal To Binary Decimal to Hex Binary to Decimal Hex to Decimal Binary to Hexadecimal Hexadecimal to Binary Binary Arithmetic Hexadecimal Arithmetic 13 Answers to SAQs Introduction The purpose of the unit is to introduce number systems in general, and working with the binary and hexadecimal systems in particular. This material will provide an essential background for the representation of information within a digital computer, which is the content of unit 4. We will start by establishing important facts about the number system we use, the decimal or base ten system, which apply to number systems in general. This will lead on to the systems used by digital computers: binary and hexadecimal.. Methods of conversion between the different systems will then be covered, followed by the simple addition and subtraction of binary and hexadecimal numbers. LCS, Edinburgh Page 1 of 17

2 2.1 Decimal Numbers This is the number system with which we are all familiar. It is based on counting in tens (i.e. base 10) and is referred to as the denary or decimal system. If there were only one symbol to represent numeric quantities then numbers would be most cumbersome. For example, arithmetic might look like + = Luckily this has been avoided and we use the symbols 1 to 9 to represent the first nine values and the symbol 0 to represent absence of a numeric quantity, as shown below. quantity symbol The denary system uses ten different symbols. The symbols used to represent quantities in a number system are known as its digits. Quantities greater than nine in value are notated by using combinations of the existing digits, arranged in columns. This is a positional notation and as a digit moves left by a column, its value increases by a factor of ten. For example the number 567 consists of five 1 s plus five 10 s plus five 100 s x 100 = x 10 = 60 7 x 1 = 7 Each column has a value or weight associated with it. The first column is always units, the next column to the left indicates the number of 10 s, the column to the left of this is the number of 100 s etc. A digit appearing in any column is multiplied by the column s weight to give the numeric value being referred to. The positional notation is usually illustrated as follows: s 1000 s 100 s 10 s 1 s LCS, Edinburgh Page 2 of 17

3 The value of each column can be expressed as a power of ten. By definition, any number raised to the power of zero is 1, therefore the units column may be written as The pattern is obvious, and the next column to appear in the above will have the weight 10 5 or The notation may also be extended to the right of the units column to indicate fractional values, and the first will have the value 10-1, or 1/10ths, then 10-2 or 1/100ths etc. The fractional part is separated from the whole part with a point, as in This is illustrated below s 1000 s 100 s 10 s 1 s. 1/10 s 1/100 s 1/1000 s We do not need to know about fractional numbers We have now enough information to state two facts about number systems: The base of a number system equals the number of digits available. Numbers are represented using a positional notation. Any number has at least one digit, which is always positioned in the least significant or units column. This column has a weight given by the base of the system raised to the power of 0. The next column has a weight given by the base raised to the power of 1. Each subsequent column has a weight raised by one more power of the base. 2.2 The Binary System The binary number system has a base of 2. That is, there are only two symbols, 0 and 1. Any number larger than 1 must be represented using combinations of the two binary digits. The value of each position in a binary number is a power of 2. This is illustrated for the first six positions, as follows: 32 s 16 s 8 s 4 s 2 s 1 s As an example, consider the binary number Starting at the rightmost position, there is one unit, zero twos, one four, zero eights, one sixteen and one thirty-two. LCS, Edinburgh Page 3 of 17

4 x 1 = 1 0 x 2 = 0 1 x 4 = 4 0 x 8 = 0 1 x 16 = 16 1 x 32 = Adding together the products from each position gives the corresponding decimal value of 53. We may write which states that the binary number is the same as the decimal value 53. The subscript 2 is to indicate that the number is base 2, and the subscript 10 indicates the number is decimal. This is important since a number, such as 101 may be interpreted as the binary value 101 (which is 5 in decimal), or as the decimal value 101 (one hundred and one). When referring to a binary number such as 101 2, it should be pronounced as one zero one, and not as one hundred and one. As another example we shall determine the equivalent decimal value of Examining the contents of each column, we find (1 x 8) + (1 x 4) + (1 x 2) + (1 x 1) = = The decimal value of a binary number is simply given by summing the values contained in each position of the number. The formal methods of converting between binary and decimal systems are covered in sections and SAQ 2.1 Why is the symbol 2 never used in a number which is represented in base 2? LCS, Edinburgh Page 4 of 17

5 2.3 The Hexadecimal System All data in a computer is stored and moved around in binary format. This comprises of long sequences of 0 s and 1 s, which is actually very difficult for people to recognise and work with. We shall find that the hexadecimal format is very closely related to binary and offers a much more compact way of notating binary values. For example the binary string is written as B3DF in hexadecimal this is much easier to remember and to compare with other hex values (note that hexadecimal is often shortened to hex). The hexadecimal system has a base of 16, which implies that there are 16 hexadecimal digits. The first ten digits in the hex system are the same as in the decimal system, and the next six are represented using the letters from A to F. That is, the numbers from 0 to 9 are represented by the digits 0 to 9, and the numbers from 10 to 15 are represented by the digits A to F. denary hex A B C D E F Numbers larger than the decimal value 15 are represented using combinations of the 15 digits. The weight of each position in a hex number is a power of 16. This is illustrated for the first six positions, as follows: s 4096 s 256 s 16 s 1 s As an example, we shall determine the value of the hex number 3A in decimal. 3A 16 (3 x 16) + (10 x 1) = = The units column contains the hex digit A 16 which is 10 in decimal, the second column contains 3 and with a weight of 16 the value of this column is 3 x 16 which is 48. Adding the values of each column gives 58 which is the value of the hex number 3A. LCS, Edinburgh Page 5 of 17

6 As a second example, consider the hex number 11FF. 11FF 16 (1 x 16 3 ) + (1 x 16 2 ) + (15 x 16 1 ) + (15 x 1) = = Number Base Conversion It is sometimes necessary to convert numbers from one base to another. This is particularly true when working with microprocessors at the machine or system level. A computer user need not concern themselves with the conversion between bases as the computer has software to convert decimal input into the computer s own internal binary representation. Similarly the computer converts the binary results into decimal form before displaying them Decimal To Binary To convert a decimal number to binary, divide the number repeatedly by 2, and after each division record the remainder. The last division will always be a 1 divided by 2 (with result 0 remainder 1). The remainders will give the corresponding binary number. For example, convert 75 to binary format: r r r r r r r The result is read from the most significant bit (the last remainder) upwards to give LCS, Edinburgh Page 6 of 17

7 2.4.2 Decimal to Hex The same method as converting decimal to binary is used, except that the division is now by 16, not 2. That is, divide the number repeatedly by 16, and after each division record the remainder. The division stops after the division of the units digit. The remainders give the digits for the corresponding hex number. As an example, convert decimal 78 to a hex representation r 14 E in hex 0 r 4 4E Therefore E 16 Example 2. Convert decimal 827 to a hex representation r 11 ( B 16 ) 16 3 r 3 0 r 3 33B Therefore E Binary to Decimal As we have seen it is possible to convert a binary number to decimal by adding the values in each column. For example, is given by = = 106 There is a quicker method which requires some mental arithmetic. It proceeds as follows. Take the leftmost bit, double it and add it to the bit on its right. Now take this result, double it and add it to the next bit on its right. Repeat this process until the least significant bit has been added to the number. To illustrate we shall convert into denary. Write down the binary number and, starting at the most significant position (the leftmost column), carry its value right one position and double it, giving 2. LCS, Edinburgh Page 7 of 17

8 Add this to the value in this position (2 added to 1 giving 3). Now carry the 3 right one position and double it (giving 6), then add it to the value in this position (6 added to 1 giving 7). Again carry the result so far, 7, right one position and double it (giving 14), then add it to the value in this position (14 added to 0 gives 14). This is the units position, and so the number has been converted. Therefore Another example, to convert , Therefore Hex to Decimal This method is identical to the process for binary to decimal except that 16 is used as the multiplier. As an example, convert 3EF 16 to decimal. Therefore 3EF E F Of course, multiplication by 16 is not so easy, and you may wish to use the alternative method of calculating the value of each column in decimal and summing the results. As was previously stated, computers use the binary and hexadecimal systems and so conversion between these two systems is a useful skill to have. Fortunately these conversions are much simpler than the conversions to and from decimal Binary to Hexadecimal The method is to start at the unit column and convert each group of four bits into a corresponding hexadecimal digit As an example take LCS, Edinburgh Page 8 of 17

9 5 A 9 Therefore A9 16. As another example take E B Therefore EB 16. The ease of conversion is because four binary digits represent the same range of values as a single hexadecimal digit Hexadecimal to Binary This is the reverse process of converting from binary to hexadecimal. Take each hexadecimal digit and convert it to the corresponding binary representation. As an example take Therefore As another example take CBA 16 C B A Therefore CBA These last conversions highlight the reason for the use of the hexadecimal system: it is simply as an aid to human memory. A long sequence of binary digits, such as is almost impossible to remember and compare with another binary number. It is much easier to work with the equivalent hexadecimal value 44EB 16, The hexadecimal representation is much more compact and the conversion is an easy task. Because of this, hexadecimal is the preferred way of referring to memory addresses and their contents. Utilities, such as those used by computer engineers and programmers, to allow examination of memory locations etc. usually provide a display in hexadecimal rather than binary. Numbers in binary, may be made easier to read if they are written in groups of four bits. For example rather than dealing with , write it as We will stick to this particular format for binary from now on. LCS, Edinburgh Page 9 of 17

10 Consider the representation of some numbers in all three number systems: binary denary hexadecimal F FE E D2 A number in its binary representation will always contain at least the same or more digits than the decimal representation. Similarly the decimal representation will always contain at least the same or more digits than the hexadecimal representation. When the decimal and hexadecimal representation contain the same number of digits, the most significant digit in the hexadecimal representation will always be less than the most significant digit in the decimal representation (except for decimal 16,17, 18, 19). This may be used as a rough check when performing number system conversions. SAQ 2.3 Complete the following table. Binary Decimal Hex F 256 ABC 2.5 Binary Arithmetic Binary arithmetic follows exactly the same rules as decimal arithmetic. Since the binary digits are 0 and 1, the following binary tables contain all the possible combinations when adding or subtracting binary numbers. Addition Subtraction = = = = 1 borrow = = 1 LCS, Edinburgh Page 10 of 17

11 1 + 1 = 0 carry 1 (or 10) 1-1 = 0 With addition, all that needs to be remembered is that when adding two 1 s there will be a carry into the next column. Following are a few examples of addition. 1. Add Add space for carries 11 space for carries 100 (3+1=4) 110 (3+3=6) 3. Add Add space for carries 11 space for carries (15+1=16) (14+29=43) In examples 2 and 4, a carry into a column results in three 1 s to be added. The sum is 11, and this means, write down 1 as the result for the column, and carry 1 to the next column. It is very useful to be able to add binary numbers together, but it can take some practice. The only alternative is to convert the numbers to decimal, perform the addition, and convert back to decimal which involves much more work. LCS, Edinburgh Page 11 of 17

12 SAQ 2.4 Add the following binary numbers. 1) 2) 3) ) 5) 6) Subtraction can also be carried out as for decimal numbers, although, as we shall see later, a computer does not subtract numbers in this manner. However, it is still a useful skill to have. The method is covered in class. SAQ 2.5 Subtract the following binary numbers. 1) 2) 3) ) 5) 6) LCS, Edinburgh Page 12 of 17

13 2.5 Hexadecimal Arithmetic The rules are as for decimal arithmetic, except that any position in a hex number may contain a digit from 0 to F. Here are some examples of addition in hexadecimal. 1. Calculate 6E + D7 2. Calculate E 999 +D space for carries 11 space for carries 145 ( =325) 1221 (4-1=3) When adding hex numbers I find it easier to convert the digits being added into decimal, do the addition, and convert back to hex, taking any carry into the next position. For example in the first example above. First add E and 7, in decimal this is = in hex. Write down the 5 and carry 1. In the second column we now have D, which is = in hex. Write down the 4 and carry 1 to give a final result of LCS, Edinburgh Page 13 of 17

14 3. Calculate DEF Calculate B34 + 1A78 AEF 1A B34 11 space for carries 1 space for carries D78 ( =3448) 25AC ( =9644) We have already noted that hexadecimal is the preferred method of referring to memory addresses in computers. For example, the software necessary to drive a monitor may be described as using the memory from 2F4E00h to 3377F0h (the postfix h is used to indicate hexadecimal format). To find out how much memory the program requires, the hexadecimal figures must be subtracted. The subtraction would proceed as follows: F 0-2 F 4 E B F 0 Therefore the memory required is 42BF1h. Converting this to decimal, the program will occupy 273,393 bytes of memory, that is k. When subtracting hex numbers I would use the same technique as when adding. That is, convert the digits in the current position to decimal, perform the subtraction, and convert back to hex. Here are more examples of subtraction in hexadecimal. 1. Calculate D7-6E 2. Calculate 3E24 - FED D7 3E24-6E - FED ( =105) 2E37 ( =11831) LCS, Edinburgh Page 14 of 17

15 SAQ 2.6 Perform the following hexadecimal calculations as indicated. 1) 2) 3) 728 F466 AAAB +F8 +A22 +BBA 4) 5) 6) 728 F466 AAAB -F8 -A22 -BBA SAQ 2.7 Calculate the amount of memory taken up by a program starting at memory address F and ending at address AAAAA 16. Answers to SAQs SAQ 2.1 The base of a number system refers to the number of different symbols used to notate numbers. In base 2, these are 0 and 1 and are known as the binary digits. Numbers with values of 2 or higher are notated by using combinations of 0 and 1. LCS, Edinburgh Page 15 of 17

16 SAQ 2.2 The hex system has a base of 16. This means that there must be 16 digits available for representing numeric quantities. The first ten numbers are represented by using the symbols 0 to 9 (just as in the decimal system). The next 6 numbers (corresponding to the numbers 10 to 15 in decimal) must be assigned single digits. These are chosen to be the first six alphabetics A to F. The digits 0 to F are therefore the hex digits used to represent the first 16 numbers (0 to 15 in decimal), and all higher values are notated using combinations of these digits. SAQ 2.3 Binary Decimal Hex AA DD F F ABC SAQ 2.4 Binary additions. 1) 2) 3) ) 5) 6) LCS, Edinburgh Page 16 of 17

17 SAQ 2.5 Binary subtractions. 1) 2) 3) ) 5) 6) SAQ 2.6 Hexadecimal arithmetic. 1) 2) 3) 728 F466 AAAB +F8 +A22 +BBA 820 FE88 B665 4) 5) 6) 728 F466 AAAB -F8 -A22 -BBA 630 EA44 9EF1 SAQ 2.7 The calculation is AAAAA - F Note that 1 is added to the subtraction in order to include the memory start address in the calculation. This gives a result of 9B889 bytes. LCS, Edinburgh Page 17 of 17