Kavita Hatwal Fall The decimal system, also called the base 10 number system is based on ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. All the numbers are formed by combination of these 10 numbers -> to form a 2- digit number, the second digit is multiplied by is equal to (1 X 10) + 5. To form a 3-digit number, the third digit is multiplied by 100. This concept continues with the next digit multiplied by 1000, the next by 10,000, and so on. A number like 8679 would be calculated as (8 X 1000) + (6 X 100) + (7 X 10) + (9 X 1). Or More generally, decimal or base 10 notation is based on the fact that any positive integer can be written uniquely as sum of products of the form Where each n is a nonnegative integer (0 and above) and each d is one of the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 This concept is easy to grasp, because we use the decimal system in every day applications.

Kavita Hatwal Fall But the computers use base 2 or binary notation. The signals used in modern electronics are always in one of only two states->0,1. Just like in decimal any any positive integer can be written uniquely as sum of products of the form where 10 is the base Similarly in binary or base 2 notation, any positive integer can be written uniquely as sum of products of the form where 2 is the base Example

Kavita Hatwal Fall Converting binary to decimal = mutiply Add these to get the equivalent base 10 number

Kavita Hatwal Fall Converting decimal to binary To convert a base 10 number into base 2, repeatedly divide the number by 2 till the quotient is 0 and keep collecting the remainders at each step Divisor QuotientRemainder Collect these starting from the bottom So the binary equivalent of 35 is

Kavita Hatwal Fall Divisor QuotientRemainder Or reverse the position of quotient And then collect the remainders from top to bottom to get the binary equivalent of 35 which is

Kavita Hatwal Fall Binary addition Binary addition is similar to decimal addition in the sense that you start adding the addends proceeding from right to left. But the numbers that you’re allowed to use are only 0 and 1. Rules for binary addition 0+0 = = = = 0, carry over = 1, carry over 1 Add If the carry goes on beyond the maximum number of bits, then an overflow has occurred. Page 73, # 14

Kavita Hatwal Fall Negative number representation in binary notation Earlier it was decided to make the leftmost bit the sign bit ->1 denotes a negative number and 0 in that bit position indicates a positive number. So 6 = 0110 (in binary) and -6 = 1110 But this led to complications like if we add 6 and –6, we should get 0, but what do we get? Try it out To avoid this and many other complications like non unique representation of 0, adding a negative and a positive number etc, an easier way called the two’s complement was adopted in which leading 0’s means a positive number and leading 1’s means a negative number. So in two’s complement land, for a fixed 4-bit space, we read for negative integers Page 73, #25

Kavita Hatwal Fall To find the 8-bit two’s complement of a positive integer a that is at most 255 Write the 8-bit binary representation for a. Switch all the 1’s to 0’s and 0’s to 1’s Add 1 in binary notation. Page 74, # 21 Two’s complement is extra handy when representing negative integers and performing addition and subtraction on signed integers. The 8-bit binary representation for a This knowledge comes handy when we’ve to do subtraction or addition of two different signed numbers Convert both integers to their 8 bit representations. Add using binary addition Truncate any leading 1’s that occur in the rightmost (2^8) position. Convert the sum back to decimal form interpreting leading o’s as +ve and leading 1’s as -ve Page 73, # 31

Kavita Hatwal Fall Hexadecimal notation The cool thing about Hexadecimal (base 16) is that we save space writing long binary numbers and also it is very easy to convert from hexadecimal to binary, since almost all computer data are multiples of 4. Similarly in hexadecimal or base 16 notation, any positive integer can be written uniquely as sum of products of the form To convert from hexadecimal to binary and vice versa just remember table on page 71. Use that table to convert E0D into binary into hexadecimal E 0 D C 5 Juxtapose the results