Binary Number Systems

Binary Numbers

• Digital systems use the binary (base-2) number system for efficient data processing, storage, and transmission
• This system assigns discrete values to each data point
• Binary numbers use only the digits 0 and 1, simplifying the assignment of numerical values to physical parameters
• A single binary digit (0 or 1) is termed a "bit"
• A "byte" comprises a group of eight bits
• A group of bits, commonly 8, 16, 32, or 64, is processed as a single unit by a CPU
• A binary number consists of bits, each having a weight that is a power of 2
• The weights increase from right to left
• The Most Significant Bit (MSB) is on the left, while the Least Significant Bit (LSB) is on the right
• For example, the binary number 10110010₂ is equal to (1x128 + 0x64 + 1x32 + 1x16 + 0x8 + 0x4 + 1x2 + 0x1) = 178₁₀

Decimal and Binary Representations

• Demonstrates the binary representation of the first ten decimal numbers
• Decimal 0 is Binary 0
• Decimal 1 is Binary 1
• Decimal 2 is Binary 10 (2+0)
• Decimal 3 is Binary 11 (2+1)
• Decimal 4 is Binary 100 (4+0+0)
• Decimal 5 is Binary 101 (4+0+1)
• Decimal 6 is Binary 110 (4+2+0)
• Decimal 7 is Binary 111 (4+2+1)
• Decimal 8 is Binary 1000 (8+0+0+0)
• Decimal 9 is Binary 1001 (8+0+0+1)

Converting Binary to Decimal

• To convert a binary number to decimal, the bits are multiplied by their corresponding weights (powers of two) and then added together
• Considering B = bn-1.....b1b0 for and n-bit number
• As an example, consider (101101)2 = (2^5 x 1) + (2^4 x 0) + (2^3 x 1) + (2^2 x 1) + (2^1 x 0) + (2^0 x 1) = 32 + 8 + 4 + 1 = (45)10

Converting Decimal to Binary

• One method used to convert a decimal number to binary is to divide the decimal number by 2 and record the remainder
• The process continues until the answer is zero
• The remainders form the equivalent binary number, beginning from the last remainder as the most significant bit (MSB)
• For example, converting 157 to binary involves the following steps:
• 157 ÷ 2 = 78 with a remainder of 1 (LSB)
• 78 ÷ 2 = 39 with a remainder of 0
• 39 ÷ 2 = 19 with a remainder of 1
• 19 ÷ 2 = 9 with a remainder of 1
• 9 ÷ 2 = 4 with a remainder of 1
• 4 ÷ 2 = 2 with a remainder of 0
• 2 ÷ 2 = 1 with a remainder of 0
• 1 ÷ 2 = 0 with a remainder of 1 (MSB)
• Hence, (157)10 = (10011101)2

Common Number Bases

• Decimal (base-10) is good for add/subtract/multiply
• Binary (base-2) is used by computers since it is simple to assign 2 values to a physical parameter
• Hexadecimal (base-16) combines binary digits in groups of four
  • This is more commonly used to represent memory addresses and operation codes
• Octal (base-8) combines three binary digits

Hexadecimal Representation

• Disadvantage of Binary Numbers: The binary equivalent of a large decimal number can be quite long, which makes it difficult to read and write
• Solution: Arrange the binary numbers into groups of four in Hexadecimal, starting with the least significant digit at the right hand side
• Two hexadecimal numbers are required for each byte, since 16 (in decimal) is the fourth power of 2 (or 2^4)
• One hex digit has a value equal to four binary digits and the numbers 10 to 15 are represented by capital letters of the alphabet from A to F inclusive
• 110111001111₂ = (2^11 + 2^10 + 2^8 + 2^7 + 2^6 + 2^3 + 2^2 + 2^1 + 2^0)₁₀ = (2048 + 1024 + 256 + 128 + 64 + 8 + 4 + 2 + 1) 10 = 353510
• Conversion to hexadecimal number system is simpler, it requires dividing the binary into four bit 'nibbles'
• 110111001111₂ = 1101 1100 1111₂ = DCF16
• To count in hexadecimal beyond the final letter F, just start over again with another set of 4 bits
• 0...to...9, A,B,C,D,E,F, 10...to...19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21....etc
• With two hexadecimal digits we can count up to FF which is equal to decimal 255

Converting to Hexadecimal

• To convert a binary number to hexadecimal if the number of bits are not a multiple of four, we can add zero's to the left of the most significant bit (MSB)
• The main advantage of a Hexadecimal Number is that it usually needs less (or otherwise an equal number of) digits than in binary or decimal to represent a number
• Conversion from/to binary is very simple and straightforward
• To covert 1110 10102 to its Hexadecimal number equivalent:
• Group the bits into four's starting from the right hand side
• 1110 = E
• 1010 = A
• Hexadecimal number is EA16

Hexadecimal to Binary

• To convert the Hexadecimal number 3FA716 to its Binary Number:
• 3 = 0011
• F = 1111
• A = 1010
• 7 = 0111
• Binary number = 0011 1111 1010 01112
• (3FA7)16 = (16^2 x 3) + (16^1 x 7) + (16^0 x 15) = (256 x 3) + (16 x 7) + (1 x 15) = 768 + 112 + 15 = (895)10
• For Decimal to Hexadecimal Conversion, convert decimal numbers to hex equivalent by dividing the number by 16 and recording the remainders
• Convert 1128 to Hex:
• 1128 ÷ 16 = 70 with a remainder of 8
• 70 ÷ 16 = 4 with a remainder of 6
• 4 ÷ 16 = 0 with a remainder of 4
• Hence we get (1128)10 = (468)16

Octal Number System

• Octal Numbers are very similar to hexadecimal numbering system except that an Octal digit is equivalent to 3 bits
• The main characteristic of an Octal Numbering System is that there are 8 distinct counting digits from 0 to 7 with each digit having a weight or value of just 8 starting from the least significant bit (LSB)
• As the base of an Octal Numbers system is 8, which also represents the number of individual numbers used in the system, the subscript 8 is used to identify a number expressed in octal, (237)8
• Like hexadecimal, the octal number system provides a convenient way of converting large binary numbers into smaller groups, however, octal numbering is used less frequently than the more common hexadecimal numbering system
• An octal digit can be represented by three bits and to count above 7 in octal, add another column and start over again in a similar way to hexadecimal
• 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21....etc
• 1101110011112 = 110 111 001 1112 = 67178 = 353510