DCAM Part 66 B 1.1 Level 1Module 5.2 - Numbering SystemNumbering System : Binary, octal and hexadecimal; Demonstration of conversions between the decimal and binary, octal and hexadecimal systems and vice versa.

Level 1 :A familiarisation with the principal element of the subject.Objectives: the applicant should be able to give a simple description of the whole subject, using common words and examples.The applicant should be able to use typical terms.

5.2 Numbering SystemIntroduction:- knowledge of Numbering Systems is fundamental to understanding computers and their operation use to count objects or perform mathematical calculations- Each is a set of symbols and characters referred as digits.Positional Notation the standard shorthand form writing numbers. The value of the particular digit depends on; * the digit value.* the position of the digit within the number.e.g. 3721 standard shorthand.Three thousand seven hundred and twenty one standard longhand.The digit at far right Least Significant Digit (LSD).The digit at far left Most Significant Digit (MSD).

Numbering SystemBase Has a base which is equal to the number of digits. A subscript is added to a number to indicate its base.e.g.1012 - indicates the number 101 is a base of 2 or binary number.The value of largest digit of a numbering system is:- One less than the base The value of smallest digit of a numbering system is:- ZeroEach digit is multiplied by the base raised to the appropriate power for the digit position.e.g. decimal No. 3721 is equal to:(3 X 103) + (7 X 102) + (2 X 101) + (1 X 100) 3000 + 700 + 20 + 1 3721

1031021011003721thousandshundredstensunits

Binary Number System a simplest number system employing positional number. has a base of 2. two binary digits (BITS) used are 0 and 1.IN A DIGITAL COMPUTER:Only two distinct states.All inputs to a digital computer must be converted to a series of 1s and 0s (binary) before the computer can make use of the data.Conversion from binary to decimal is straight forward and easily performed using positional notation.EXAMPLE: 1.Positional notation16 + 4 + 2 + 1 =2310

2726252423222120Weight Value 1286432168421Base 10 Value00010111Binary Number to be Converted 16421Equivalent base 10 Number

EXAMPLE: 2.Positional notationBinary Number SystemIn example 2, a binary number representing a fraction is shown.Add together the base 10 values for each bit position containing a 18 + 4 + 0.5 + 0.125 + 0.0625 = 12.687510

232221202-12-22-32-4Weight Value 84210.50.250.130.06Base 10 Value11001011Binary Number to be Converted 84 0.5 0.13 0.06Equivalent base 10 Number

Decimal Number System has a base of 10. Most familiar, used for everyday counting. mathematical calculations. contains ten digits from 0 to 9, with 9 the largest digit.DECIMAL DIGITS987654321POSITION NOTATIONDecimal point600000 + 500000 + 8000 + 900 + 10 + 2 + 3/10 + 3/100658912.3310

10510410310210110010-110-2Weight Value 65891233Number

Decimal to Binary Conversion to repeatedly divide the decimal by the base number. by keeping track of the remainders.- The new numbering base equivalent is obtained. Case of DECIMAL to BINARY conversions, The decimal number is divided by the base number 2. The first remainder obtained is the least significant digit (LSD).- The last remainder is the most significant digit (MSD).

Decimal to Binary Conversion

Octal Number System has a base of 8 the weight value of each BIT position (80, 81, 82..) and the base 10 equivalent are shown.To convert 4522 (base 8) to base 10, multiply EACH total octal digit by its corresponding base 10 value, then add together the computed base 10 values.2048 + 320 + 16 + 2 = 238610 45228 = 238610

8483828180Weight Value 4,096 5126481Base 10 Value04522Octal Number to be Converted 2,048 320162Equivalent base 10 Number

Decimal to Octal Conversion Decimal to Octal conversions can also be accomplished by successive division. The decimal number is repeatedly divided by base 8 and again the remainders are used for decimal to octal equivalent number.

Binary to Octal Conversion - three-bit position represent eight combinations (000 thru 111). - octal no. can be substituted for a 3-bit binary numbers. - binary no. separated into groups of 3-bits from right (LSD) to (MSD) at the left. - Each group of 3-bits is replaced by an octal equivalent. - Octal to binary is the reverse procedure.

HEXADECIMAL NUMBER SYSTEM- another system often used in microcomputers. - has base 16 which requires 16 digits. - Digits used are 0 through 9 and A through F. - A thru F - equivalent decimal numbers of 10 thru 15 respectively.HEXADECIMAL DIGITS F E D C B A 9 8 7 6 5 4 3 2 1 015 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 This system is called alphanumeric number system since numbers and letters are used to represent the digits.

40,960 + 2,048 + 240 + 5 = 4325310 A8F516 = 4325310HEXADECIMAL NUMBER SYSTEM- Refer positional notation and convert A8F5 (base 16) to base 10 equivalent shown. Often hexadecimal numbers are written with an H following the hexadecimal number.

163162161160Weight Value 4,096 256161Base 10 ValueA8F5Hexadecimal Number to be Converted 40,9602,048 2405Equivalent base 10 Number

HEXADECIMAL CONVERSION- decimal to Hexadecimal conversion can be done by successive division. - the decimal number is divided by base number 16. if the remainder is greater than 9, it should be changed to the hexadecimal equivalent of the remainder.

e.g., if the remainder is 10 it should be changed to A, if the remainder is 11 it should be changed to B and so on, up to 15 which is F.HEXADECIMAL CONVERSION

BINARY TO HEXADECIMAL CONVERSION used as a shorthand notation for binary numbers. in binary, 4-bit positions are necessary to obtain 16 combination numbers (0000 thru 1111). the binary numbers is separated into groups of four beginning at LSD and preceding to the left. - Each group of four bits is replaced by hexadecimal equivalent.

BINARY TO HEXADECIMAL CONVERSIONIn forming the 4 - bit grouping, 0s may be required to complete the first (MSD) group.

BINARY CODED DECIMAL NUMBER SYSTEM A method of representing decimal numbers in digital computers is known as Binary Coded Decimal (BCD).DECIMAL TO BCD 7 3 8 10

0111 0011 1000 DECIMAL TO BCD 1001 0100 0110

9 4 6 10

232221202322212023222120WEIGHT VALUE011101101001BCD769DECIMAL102101100WEIGHT VALUE