Introduction to computer systems architecture and programming

More documents

Recommendations

Info

168 Introduction to computer systems architecture and programming Activity If the sampling frequency for a stereo CD recording is 44,100 samples per second, how much storage space is required for a 30-minute stereo recording? Representing numbers – the binary system When you look at an ASCII (or Unicode) table, you will find that it also includes codes for the representation of numbers. However, this is not an ideal way of representing numbers if they are used within a calculation. Also, within ASCII we would be using 8 bits to represent a single-digit number, and the largest number we could store would be 9. However, if we represent all numbers as binary numbers (rather than decimal) we can actually represent the numbers 0 to 127 with 8 bits. We also have the advantage that a number is translated into a different numeric system (i.e. the binary system) where it remains a number and where hence mathematical calculations can still be carried out (in ASCII a number is merely a symbol in the same way as a letter is a symbol). Binary numbers are represented using bit sequences. The representation follows the same simple principle as the decimal system. So you need to remind yourself of these underlying principles: The decimal system uses 10 as a base, and the 10 digits available are 0, 1, 2, 3, … , 9. Depending on its position in the whole number, each digit represents the amount of 1s, 10s, 100s, 1000s, etc. (i.e. 10 0 s, 10 1 s, 10 2 s, 10 3 s, etc.). For example, the decimal number 9348 represents 9×10 3 + 3×10 2 + 4×10 1 + 8×10 0 . The binary system uses 2 as its base with the digits 0 and 1. Let’s look at 110100 as an example of a binary number (in order to avoid confusion the base of a number may be indicated using the following convention: 110100 2 ): 110100 2 = 0×2 0 + 0×2 1 + 1×2 2 + 0×2 3 + 1×2 4 + 1×2 5 = 0 + 0 + 4 + 0 + 16 + 32 = 52 10 In summary, what you need to be able to understand is that the value that is represented by a digit depends on the digit’s position within the number and the base of the numeric system used. With every move of a digit to the left the value represented increases by a power of the base. For the decimal system this means digits – from right to left – need to be multiplied by 1, 10, 100, 1000, etc. For the binary system, digits – again from right (the least significant bit) to left (the most significant bit) – need to be multiplied by 1, 2, 4, 8, 16, etc. to calculate their value. As part of the explanation above you have been shown how a binary number can be converted into a decimal number. You also need to be able to convert a decimal number into a binary number. This can be done by: • dividing the decimal number by 2 and recording the quotient and the remainder • continuing to do this with every resultant quotient until the quotient is 0. The backwards sequence of the recorded remainders is then the binary representation of the original decimal number. 24
Chapter 2: Data representation Let us illustrate this with our example of 52: 52 : 2 = 26 remainder 0 26 : 2 = 13 remainder 0 13 : 2 = 6 remainder 1 6 : 2 = 3 remainder 0 3 : 2 = 1 remainder 1 1 : 2 = 0 remainder 1 Hence 52 10 = 110100 2 You should now have an understanding of the underlying ideas of the binary system and how to convert from decimal into binary and vice versa. Binary addition The next step is to learn how simple mathematical calculations are carried out within the binary system. We will start with the addition of two positive binary numbers, and again it is a matter of mapping the procedure we are familiar with within the decimal system onto the binary system. An example is probably the best way to explain how it works. Let us add 52 and 49. In the binary system we would calculate by adding the units, noting the overflow under the 10s, then adding the 10s and noting the overflow under the 100s, and finally adding the 100s (which in this case only consist of the overflow). 100s 10s 1s 5 2 + 4 9 Overflow: 1 1 Result: 1 0 1 We will now do the same addition in the binary system. We already know that 52 10 = 110100 2 , and you should be able to work out now (Try it!) that 49 10 = 110001 2 . We will follow the same approach as in the decimal system above. All you need to think about is what causes an overflow when adding binary digits: 0+0 = 0 1+0 = 0+1 = 1 1+1 = 10, i.e. there is an overflow of 1 1+1+1 = 11, i.e. there is an overflow of 1 64s 32s 16s 8s 4s 2s 1s 1 1 0 1 0 0 + 1 1 0 0 0 1 Overflow: 1 1 Result: 1 1 0 0 1 0 1 If you now check our result of 1100101 2 by converting it into a decimal number, you should find that 101 10 = 1100101 2 . Activities 1. Convert the following numbers into decimal numbers: • 111111 2 • 101001 2 25
Page 1 and 2: Introduction to computer systems ar
Page 3 and 4: Contents Contents Introduction ....
Page 5 and 6: Contents Programming paradigms ....
Page 7 and 8: Introduction Introduction Introduct
Page 9 and 10: Introduction At the beginning of ea
Page 11 and 12: Introduction Journals The topics in
Page 13 and 14: Introduction will be required to re
Page 15 and 16: Introduction Chapter Approximate pe
Page 17 and 18: Part 1 Part 1 11
Page 19 and 20: Chapter 1: Introduction to computer
Page 27 and 28: Chapter 2: Data representation Chap
Page 29: Chapter 2: Data representation Acti
Page 33 and 34: Chapter 2: Data representation Acti
Page 35 and 36: Chapter 2: Data representation inpu
Page 37 and 38: Chapter 3: Data manipulation Chapte
Page 39 and 40: Chapter 3: Data manipulation Our co
Page 41 and 42: Chapter 3: Data manipulation channe

Introduction to computer systems architecture and programming

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?