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1 Lecture Introduction, Numbers and Number System Contents.. Number Systems (Appendix B) Example. Converting to Base Number Representation Number Conversion To convert a number N expressed in base 0 to base B Example.2 Converting Decimal Number to base Example.3 Converting Decimal Number to base Example.4 Converting Decimal Number to base Example.5 Converting Decimal Number to base Binary-to-hexadecimal (4 bits = hex digit) Binary-to-octal (3 bits = octal digit) Hexadecimal-to-binary Octal-to-binary Example.6 Converting Number Signed Binary Number Two s Complement Numbers Example.7 Two s Complement Numbers Example.8 Two s Complement Numbers Find Decimal Value of Signed Binary Number Negating two s Complement Numbers Example.9 Negating in Two s Complement System Negating Octal and Hexadecimal Numbers... 6 a. Example.0 Negating Hexadecimal Numbers in Two s Complement System Addition in Two s Complement System Subtraction in Two s Complement System Examples on Two s Complements Example. Addition and Subtraction in Two s Complement Systems... 8 Lecture Introduction, Numbers, and Number System Page of 8

2 .. Number Systems (Appendix B) Representation of an integer number on base B is done by creating a set of coefficients a n a a 0 with weights specified by powers of the base B. The coefficients in base B are numbers that can take values 0 through B-. A number N can be written in base B as N 2 0 a... a aa n n 2 0 a B... a B a B a B B n 2 0 Fractional numbers can be written by introducing negative powers of the base. A fractional number N can be written in base B as N a... a aa. a a a... a m m m n B n... 2 B 0B B 2B 3B... nb a B a B a a a a a a There are four common number systems: decimal, hexadecimal, octal, and binary.. Example. Converting to Base 0 Find the equivalent numbers in the decimal base system of the following numbers: (204.2)0 = (204.2)8 = * * *8 + 4*8 0 + * *8-2 = (A2F.)6 = * * *6 + 5*6 0 + *6 - + * 6-2 = ( )2= * * * * * *2 2 + *2 + 0*2 0 + *2 - +0*2-2 = Number Representation Decimal Binary Octal Hex A B C D E F Lecture Introduction, Numbers, and Number System Page 2 of 8

3 .3. Number Conversion. To convert a number N expressed in base 0 to base B Separate integer part of number from fractional part Divide integer part by B continuously and save the remainder at each step. Multiply fractional part by B continuously and save integer part at each step. N a aa a a a B a B a B a B a B B Example.2 Converting Decimal Number to base 8 Converting the following decimal numbers into base 8: (75.32) (75.32)0 ( )8 Remainder Fraction Result * * * * * 8 3. Example.3 Converting Decimal Number to base 8 Converting the following decimal number into base 8: ( ) ( )0 ( ) Remainder Fraction Result * * * * * * 8 4. Example.4 Converting Decimal Number to base 6 Converting the following decimal number into base 6: ( ) ( )0 (A4.A) Remainder Fraction Result * A 5. Example.5 Converting Decimal Number to base 2 Converting the following decimal number into base 6: (53.375) (53.375)0 (00.0)2 Remainder Fraction Result * * * Lecture Introduction, Numbers, and Number System Page 3 of 8

4 6. Binary-to-hexadecimal (4 bits = hex digit) Group binary digits into groups of four and assign each group a hex digit B E Binary-to-octal (3 bits = octal digit) Hexadecimal-to-binary C 2 8 E Octal-to-binary Example.6 Converting Number (5CB.2F)6 = ( )2 (47.62)8 = ( )2 ( )2 = (DEA.A9)6 ( )2 = ( )8.4. Signed Binary Number For signed numbers, in the binary system, the sign of the number is denoted by the left-most bit: 0 = positive = negative For a n-bit number, the remaining n- bits represent the magnitude. For signed binary system, there is a duplicate representation of 0 such as +0 ( ) and 0 ( ). As a result, there is one less combination of bits to represent signed numbers..5. Two s Complement Numbers All modern computers use the two s complement number system to represent negative numbers. s Complement of N: complement individual bits (0 and 0) 2 s Complement of N: complement individual bit and add to it. In the two s complement system, positive numbers are identical to binary numbers with the sign bit is 0. Lecture Introduction, Numbers, and Number System Page 4 of 8

5 Range of number in a system of n bits: Unsigned number: 0 N 2 n s complement: 2 n + N 2 n 2 s complement: 2 n N 2 n The 2 s complement of N is used to represent N. It can represent integers in the range between 2 n- to 2 n- -, where n is the number of bits available for representing N (for unsigned numbers N, the range would be 0 to 2 n -).. Example.7 Two s Complement Numbers Find the two s complement of the following numbers in 8-bit system: 0, 8, 4, +7, and 4. 8-bit s Compl. + 2 s Compl Example.8 Two s Complement Numbers Find the equivalent decimal numbers of the following number (2 s complement) in 6-bit system: 6-bit s Compl. + Magnitude Numbers Find Decimal Value of Signed Binary Number Observe that the 2 s complement of a negative number is the corresponding positive number. For example, -50=02 and 2 s complement of 02 is 002 Procedure ) Find the 2 s complement of the number, i.e. its positive value 2) Convert the positive binary number to decimal 3) Add a negative sign to the decimal value Two s Complement Numbers in 6-bit system Signed System s Complement 2 s Complement Lecture Introduction, Numbers, and Number System Page 5 of 8

6 Negating two s Complement Numbers To negate a number in the Two s Complement system: Invert all bits of the number Add.. Example.9 Negating in Two s Complement System Find the opposite numbers of the followings numbers (2 s complement) in 6-bit system: 2 s Compl. s Compl. + 2 s Compl. Opposite Number Negating Octal and Hexadecimal Numbers Convert the octal or hex number to its binary equivalent with an appropriate number of bits (e.g. 8, 6, 32 ) Negate the binary equivalent Convert the two s complement number to its octal or hex equivalent Lecture Introduction, Numbers, and Number System Page 6 of 8

7 a. Example.0 Negating Hexadecimal Numbers in Two s Complement System Find the opposite number of the following hexadecimal numbers Binary Equivalent s Complement. + 2 s Complement Hexadecimal Equivalent A EF9 FEB A0CF F3 FFCA F0FA F06.7. Addition in Two s Complement System Any carry-out from the signed bit is ignored (+2) 000 +( 2) ignore (+2) 000 +( 2) Ignore.8. Subtraction in Two s Complement System The easiest way of performing subtraction is to negate the subtrahend and add it to the minuend. First, find the 2 s complement of the subtrahend and then perform addition. 2 s Compl 2 s Compl (+2) ( 2) ignore 2 s Compl 2 s Compl (+2) ( 2) ignore Lecture Introduction, Numbers, and Number System Page 7 of 8

8 .9. Examples on Two s Complements. Example. Addition and Subtraction in Two s Complement Systems Perform arithmetic operations of these numbers in 8-bit system: 2 s Compl (+6) (+2) s Compl 2 s Compl (+) ( ) Lecture Introduction, Numbers, and Number System Page 8 of 8

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