DigitValue Denotation NumericValue Denotation 2thousands1,000 0hundreds100 4tens10 8ones1 Computer Organization Basic Computer Concepts Number Systems Concepts  “Deka” – Greek word denoting a value equal to ten Table 1: Decimal Example 2,048 TABLE 2: The value denotation of a number “place” or position  Given a base-n number system, the value denotation of a position m is: m Value denotation x n 3 1 x n x n 4 1 x n x n x n M 1 x n x n... x n (n multiplied M-1 times)0

Basic Computer Concepts Number Systems Concepts Example A:  Finding the value of a number in any number system Given a base-n system and a number PQRST, the value is given by: T x 1 + S x n + R x n x n + Q x n x n x n + P x n x n x n x n Example B:  Given a base-8 system and a number 56472, the value is given by: Value 2x1= 2 7 x 8 = 56 4 x 64 = x 512 = 3,072 5 x 4,096 = 20,480 Step 2x1 +7x8 +4x8x8 +6x8x8x8 +5x8x8x8x8 Total: 23,866 Computer Organization

Basic Computer Concepts Number Systems Concepts TABLE 3: General method for obtaining the value denotation of a base-n numeral Value denotation V m The value denotation of a position m in a base-n number system is: value denotation V m = n (m-1) Decimal value N of a base-n numeral The decimal value N of a number p k p k-1 p k-2,..,p 2 p 1 in a base-n number system is given by: numerical value N = i=1 ? i=k p i V i = i=1 ? i=k p i n (i-1)

Computer Organization Basic Computer Concepts Number Systems Concept Examples: 1. Derive the decimal value of Solution: From Table 3: = 0 x x x 9 2 = 0 x1+ 4 x9+ 6 x9x9 = = Derive the decimal value of Solution: From Table 3: = 0 x x x 12 2 = 0 x x x 12 x 12 = =

Computer Organization Basic Computer Concepts Number Systems Concept Using the notation of Table 3: 3. Derive the decimal value of Solution: From Table 3: = i=1 ? i=3 pi 16 (i-1) = 0 x X X 16 2 Value = = 1600

Computer Organization Basic Computer Concepts Hexadecimal, Octal, & Binary Systems 1. Derive the decimal value of Solution: = 1 x x x x x x x x 2 7 Value = = Derive the decimal value of Solution: = 3 x x x x x x x x 8 7 Value = =

Computer Organization Basic Computer Concepts Hexadecimal, Octal, & Binary Systems 3. Derive the decimal value of 24CF 16 Solution: 24CF 16 = F x C x x x 16 3 Value = =

, , Computer Organization Basic Computer Concepts Converting: Decimal to Another Number System  Given a decimal number M, to convert to a number system with base - n: As an example, let M = 2,654 and n = 8

Computer Organization Basic Computer Concepts Converting: Decimal to Another Number System 2,65410 = 51368

Computer Organization Basic Computer Concepts Converting: Decimal to Another Number System Derive the octal value of Solution: a. Convert to decimal: = 1 x x x x 2 3 decimal value = = b. Convert to octal: : 11/8 = 1 remainder 3 : 3/1 = 3 remainder 0 octal value = 138

Computer Organization Basic Computer Concepts Converting: Decimal to Another Number System Shortcut Method Derive the octal value of Solution: Group the binary digits into three : = = Now010= = = 3 Therefore = 273 8

Computer Organization Basic Computer Concepts Converting: Decimal to Another Number System Shortcut Method Derive the hexadecimal value of Solution: Group the binary digits into four : = Now = = 11 But = B; therefore = BB 16