NUMBER SYSTEM AND CONVERSION

NUMBER SYSTEM AND CONVERSION. BASIC COMPUTING INTRODUCTION TO COMPUTING.

NUMBER SYSTEM AND CONVERSION. As we know, the number system is a form of expressing the numbers. In number system conversion , we will study on how to convert a number of one base, to a number of another base. There are a variety of number system such as binary numbers, decimal numbers, hexadecimal numbers, octal numbers, which can be exercised..

NUMBER SYSTEM CONVERSION TABLE. Binary Numbers Octal Numbers Decimal Numbers Hexadecimal Numbers 0000 0 0 0 0001 1 1 1 0010 2 2 2 0011 3 3 3 0100 4 4 4 0101 5 5 5 0110 6 6 6 0111 7 7 7 1000 10 8 8 1001 11 9 9 1010 12 10 A 1011 13 11 B 1100 14 12 C 1101 15 13 D 1110 16 14 E 1111 17 15 F.

NUMBER SYSTEM CONVERSION METHODS. Number system conversions deal with the operations to change the base of the numbers. For example, to change a decimal number with base 10 to binary number with base 2. We can also perform the arithmetic operations like addition, subtraction, multiplication on the number system. Here, we will learn the methods to convert the number of one base to the number of another base starting with the decimal number system..

Operation Output Remainder 25 ÷ 2 12 1(MSB) 12 ÷ 2` 6 0 6 ÷ 2 3 0 3 ÷ 2 1 1 1 ÷ 2 0 1(LSB).

Decimal to Hexadecimal: Again in decimal to hex conversion , we have to divide the given decimal number by 16. Example 3: Convert 128 10 to hex. Solution: As per the method, we can create a table;.

OTHER BASE SYSTEM TO DECIMAL CONVERSION. Binary to Decimal: In this conversion, binary number to a decimal number, we use multiplication method, in such a way that, if a number with base n has to be converted into a number with base 10, then each digit of the given number is multiplied from MSB to LSB with reducing the power of the base. Let us understand this conversion with the help of an example. Example 1. Convert (1101) 2 into a decimal number. Solution: Given a binary number (1101) 2 . Now, multiplying each digit from MSB to LSB with reducing the power of the base number 2. 1 × 2 3 + 1 × 2 2 + 0 × 2 1 + 1 × 2 0 = 8 + 4 + 0 + 1 = 13 Therefore, (1101) 2 = (13) 10.

REFERENCES. https://byjus.com/maths/number-system-conversion/.

BASIC COMPUTING. MAGNITUDE SIGNED BIT.

MAGNITUDE SIGNED BIT. Signed Binary Numbers Signed Binary Numbers use the MSB as a sign bit to display a range of either positive numbers or negative numbers.

POSITIVE AND NEGATIVE SIGNED BINARY NUMBERS. Signed Binary Numbers - Electronics-Lab.com.

SIGNED BINARY NUMBERS EXAMPLE 1. -15 10 as a 6-bit number ⇒ 101111 2 +23 10 as a 6-bit number ⇒ 010111 2 -56 10 as a 8-bit number ⇒ 10111000 2 +85 10 as a 8-bit number ⇒ 01010101 2 -127 10 as a 8-bit number ⇒ 11111111 2.

ONE’S COMPLEMENT OF A SIGNED BINARY NUMBER. One’s Complement or 1’s Complement as it is also termed, is another method which we can use to represent negative binary numbers in a signed binary number system. In one’s complement, positive numbers (also known as non-complements) remain unchanged as before with the sign-magnitude numbers. N egative numbers however, are represented by taking the one’s complement (inversion, negation) of the unsigned positive number. Since positive numbers always start with a “0”, the complement will always start with a “1” to indicate a negative number..

1’s COMPLEMENT USING INVERTERS. 1's complement using inverters.

ADDITION AND SUBRACTION USING ONE’S COMPLEMENT. One of the main advantages of One’s Complement is in the addition and subtraction of two binary numbers. In mathematics, subtraction can be implemented in a variety of different ways as A – B , is the same as saying A + (-B) or -B + A etc. Therefore, the complication of subtracting two binary numbers can be performed by simply using addition..

SUBTRACTION OF TWO BINARY NUMBERS. An 8-bit digital system is required to subtract the following two numbers 115 and 27 from each other using one’s complement. So in decimal this would be: 115 – 27 = 88 . First we need to convert the two decimal numbers into binary and make sure that each number has the same number of bits by adding leading zero’s to produce an 8-bit number (byte). Therefore:.

TWO’S COMPLEMENT. Two’s Complement or 2’s Complement as it is also termed, is another method like the previous sign-magnitude and one’s complement form, which we can use to represent negative binary numbers in a signed binary number system. In two’s complement, the positive numbers are exactly the same as before for unsigned binary numbers. A negative number, however, is represented by a binary number, which when added to its corresponding positive equivalent results in zero. In two’s complement form, a negative number is the 2’s complement of its positive number with the subtraction of two numbers being A – B = A + ( 2’s complement of B ) using much the same process as before as basically, two’s complement is one’s complement + 1..

4 BIT SIGNED BINARY COMPARISON. Decimal Signed Magnitude Signed One’s Complement Signed Two’s Complement +7 0111 0111 0111 +6 0110 0110 0110 +5 0101 0101 0101 +4 0100 0100 0100 +3 0011 0011 0011 +2 0010 0010 0010 +1 0001 0001 0001 +0 0000 0000 0000 -0 1000 1111 – -1 1001 1110 1111 -2 1010 1101 1110 -3 1011 1100 1101 -4 1100 1011 1100 -5 1101 1010 1011 -6 1110 1001 1010 -7 1111 1000 1001.

REFERENCE. https://www.electronics-tutorials.ws/binary/signed-binary-numbers.html.

BASIC COMPUTING. 1’s AND 2’s COMPLEMENT.

COMPLEMENTS. Complements are used in digital computers in order to simply the subtraction operation and for the logical manipulations. For the Binary number (base-2) system, there are two types of complements: 1’s complement and 2’s complement..

1’s COMPLEMENT OF A BINARY NUMBER. . 1's complement of "0111" is "1000" 1's complement of "1100" is "0011".

2’s COMPLEMENT OF A BINARY NUMBER. is 1, added to the 1’s complement of the binary number. In the 2’s complement representation of binary numbers, the MSB represents the sign with a ‘0’ used for plus sign and a ‘1’ used for a minus sign. the remaining bits are used for representing magnitude. positive magnitudes are represented in the same way as in the case of sign-bit or 1’s complement representation. Negative magnitudes are represented by the 2’s complement of their positive counterparts..

SIGNED NUMBERS – 4 BIT EXAMPLE. DECIMAL 2’s COMPLEMENT SIGNED MAGNITUDE 7 0 111 0111 6 0 110 0110 5 0 101 0101 4 0 100 0100 3 0 011 0011 2 0 010 0010 1 0 001 0001 0 0 000 0000.

REFERENCES. https://www.geeksforgeeks.org/1s-2s-complement-binary-number/.

BASIC COMPUTING. ASCII NUMBERING SYSTEM.

ASCII. ASCII. Short for American Standard Code for Information Interexchange , ASCII is a standard that assigns letters , numbers , and other characters in the 256 slots available in the 8-bit code. The ASCII decimal (Dec) number is created from binary , which is the language of all computers. As shown in the table below, the lowercase "h" character (Char) has a decimal value of 104, which is "01101000" in binary. ASCII was first developed and published in 1963 by the X3 committee, a part of the ASA (American Standards Association). The ASCII standard was first published as ASA X3.4-1963, with ten revisions of the standard being published between 1967 and 1986 ..

STANDARD OR LOWER ASCII CHARACTERS AND CODES. Char Dec Binary Char Dec Binary Char Dec Binary ! 033 00100001 A 065 01000001 a 097 01100001 " 034 00100010 B 066 01000010 b 098 01100010 # 035 00100011 C 067 01000011 c 099 01100011 $ 036 00100100 D 068 01000100 d 100 01100100 % 037 00100101 E 069 01000101 e 101 01100101 & 038 00100110 F 070 01000110 f 102 01100110 ' 039 00100111 G 071 01000111 g 103 01100111 ( 040 00101000 H 072 01001000 h 104 01101000 ) 041 00101001 I 073 01001001 i 105 01101001 * 042 00101010 J 074 01001010 j 106 01101010 + 043 00101011 K 075 01001011 k 107 01101011 , 044 00101100 L 076 01001100 l 108 01101100 - 045 00101101 M 077 01001101 m 109 01101101 . 046 00101110 N 078 01001110 n 110 01101110 / 047 00101111 O 079 01001111 o 111 01101111 0 048 00110000 P 080 01010000 p 112 01110000.

EXTENDED ASCII CHARACTERS AND CODES. Extended ASCII uses eight instead of seven bits, which adds 128 additional characters. This gives extended ASCII the ability for extra characters, such as special symbols, foreign language letters, and drawing characters as shown below..

7-BIT ASCII CHARACTER CODES. The ASCII table contains letters, numbers, control characters, and other symbols. Each character is assigned a unique 7-bit code. ASCII is an acronym for American Standard Code for Information Interchange..

REFERENCES. https://www.sciencebuddies.org/science-fair-projects/references/ascii-table.

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