PART1 CHAPTER 1: AN INTRODUCTION TO THE BINARY NUMBER SYSTEM 1.1 BINARY NUMBERS: NUMBERS IN EVERY DAY USE ARE WRITTEN IN THE DECIMAL SYSTEM, THAT IS, TO THE NUMBER BASE 10. A POSITIONAL NOTATION IS USED REPRESENTING ONE "I00's; TWO "I0's & EIGHT T s AS THE SYMBOL 128. THE RIGHTMOST (i.e. LEAST SIGNIFICANT) DIGIT IS IN THE "UNITS" COLUMN, THE 2 INTHE 'TENS" COLUMN, THE 1 INTHE "HUNDREDS" COLUMN, AND THE VALUE OF THE SYMBOL '128' IS EVALUATED AS 1x100+2x10 + 8x1 = 128.
SIMILARLY '1024' IS EVALUATED AS 1x1000 + 0x100 + 2x10 + 4x1 = 1024, WHICH IS MORE CONVENIENTLY WRITTEN AS 1x10 3 +0x10 2 + 2x10* 4x10° = 1024, USING THE MATHEMATICAL SHORTHAND FOR 1000= 10x10x10= 10 3 ,AND THE CONVENTION "ANY NUMBER TO THE POWER ZERO IS 1 " TO GIVE A CONSISTENT METHOD OF EVALUATING SUCH SYMBOLS. SO 1024 CAN BE WRITTEN IN COLUMNS 3 2 1 0 1 0 2 4 AND EVALUATED AS 1x10 3 +0x10 2 + 2x10* +4x10° TO THE BASE 10. TO THE BASE 8,1024 WOULD MEAN 1x8 3 + 0x8 2 + 2x8* +4x8° WHICH IS THE DECIMAL NUMBER 532. TO THE BASE 16,1024 WOULD MEAN 1x16 3 + 0x16 2 + 2x16 1 +4x16° WHICH IS THE DECIMAL NUMBER 4132.
TO DISTINGUISH THE BASE TO WHICH A NUMBER IS WRITTEN WE'LL WRITE ITS' BASE AFTER IT AS A SUBSCRIPT: 1024 1O AND NOW WE CAN WRITE 1024* = 532 1O | 1024 16 =4132 l 0 | 10000000 2 = 128! o JUST AS BASE TEN HAS THE NAME 'DECIMAL', BASE SIXTEEN HAS THE NAME 'HEXADECIMAL', BASE EIGHT HAS THE NAME 'OCTAL' AND BASE TWO 'BINARY'. THESE FOUR BASES ARE IN COMMON USE WITH MODERN COMPUTERS, ESPECIALLY HEXADECIMAL (HEX) AND BINARY. CONVERSION BETWEEN BINARY, OCTAL & HEX NUMBERS IS VERY SIMPLE. SINCE THEY ARE ALL POWERS OF TWO, NUMBERS JUST NEED DIVIDING UP:- 100000002= 1100011 00001! 6 = 80 i 6 = I010||000II000» 6 =200 8 - EACH HEX DIGIT IS FOUR BINARY DIGITS (BITS) & EACH OCTAL DIGIT IS 3 BITS.
OCTAL DIGITS ARE 0 , 1 , 2, 3, 4, 5, 6, 7. HEX DIGITSARE0,1,2,3 / 4,5,6,7,8,9,A,B,C,D,E / F A....F ARE USED INSTEAD OF 10....15 TO ALLOW UNRESTRICTED USE OF THE POSITIONAL SYSTEM.