Introduction to Number Systems

Introduction to Number Systems

• A number system provides a mathematical way to represent numbers using symbols and rules.
• It forms the basis for arithmetic and computing operations.
• Number systems are categorized by their base (or radix).
• The base indicates the number of unique digits in the system, including zero.
• In a number system with base 'r', numbers range from 0 to r-1.
• The total count of present numbers in such a system is 'r'.
• To define a number system, the base must be specified; common bases are 2, 8, 10, and 16.
• The base determines the count of available digits.
• The initial digit in any number system is zero, with the last digit being base-1.

Types of Number Systems

• A variety of number systems exist, including:
• Decimal (Base-10)
• Binary (Base-2)
• Octal (Base-8)
• Hexadecimal (Base-16)
• Binary Coded Decimal (BCD)

Decimal Number Systems

• The decimal system uses base/radix 10.
• Digits 0 through 9 are used.
• The integer part is the portion to the left of the decimal.
• The fractional part is the portion to the right of the decimal.
• Positions to the left of the decimal have weights of 100, 101, 102, 103, and so on.
• Positions to the right have weights of 10-1, 10-2, 10-3, and so on.
• Positions have specific weights as powers of base 10.
• Each digit has place value based on powers of 10.
• The decimal number 2735 breaks down to (2 × 10³) + (7 x 10²) + (3 × 10¹) + (5 × 10º).
• Decimal system consists of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 and is commonly used in everyday life.
• The decimal number 1358.246 breaks down to (1 × 103) + (3 × 102) + (5 × 101) + (8 × 100) + (2 × 10-1) + (4 × 10-2) + (6 × 10-3).

Binary Number System (Base-2)

• The binary system is fundamental in digital electronics and computing.
• The base/radix is 2.
• It uses only two digits: 0 and 1.
• All digital circuits and systems use the binary system, which radix is 2.
• Positions to the left of the binary point form the integer part.
• Positions to the right form the fractional part.
• Weights to the left of the binary point are successive powers of 2: 20, 21, 22, 23, and so on.
• Weights to the right are 2-1, 2-2, 2-3, and so on.
• Positions have specific weights as powers of base 2.
• Each digit (bit) represents a power of 2.
• The binary number 1011 can be expanded as (1 x 2³) + (0 x 2²) + (1 x 2¹) + (1 × 2º) which equates to 11 in decimal.
• The leftmost bit with the greatest weight is the Most Significant Bit (MSB).
• The rightmost bit with the least weight is the Least Significant Bit (LSB).
• It is used in computers and digital circuits to represent data in bits, which are binary digits.
• Logical operations like AND, OR, NOT, and XOR use binary numbers.
• 1101.011 can be expressed as (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) + (0 × 2-1) + (1 × 2-2) + (1 × 2-3).

Octal Number System

• Digital systems operate using binary, however, octal and hexadecimal shorthand is used for large binary numbers.
• The octal system has a base/radix of 8 using digits 0 to 7.
• The part of the number to the left of the octal point is the integer part.
• The part to the right is the fractional part.
• Weights to the left of the octal point are 80, 81, 82, 83, and so on.
• Weights to the right are 8-1, 8-2, 8-3, and so on.
• Each position has power of base 8.
• Zero to 7 are represented by corresponding binary numbers, for example 0 = 000, 1 = 001, 7 = 111.
• 1457.236 = (1 × 83) + (4 × 82) + (5 × 81) + (7 × 80) + (2 × 8-1) + (3 × 8-2) + (6 × 8-3). are the mathematical representation for the octal system.
• Each digit in an octal number represents a power of 8.
• The octal number 237 converts to 159 in decimal using (2 x 8²) + (3 × 8¹) + (7 × 8°).
• Octal numbers are used in older computing systems.
• They are used as shorthand for binary numbers; one octal digit represents three binary digits.

Hexadecimal Number System

• The hexadecimal system uses base/radix 16.
• The digits 0-9 and letters A-F represent numbers.
• The digits A-F are equivalent to decimals 10-15, respectively.
• The part to the left of the hexadecimal point is known as the integer part.
• The part to the right is the fractional part.
• Positions to the left have weights of 16°, 161, 162, 163, and so on.
• Positions to the right have weights of 16-¹, 16-2, 16-3, and so on.
• Each position has specific weight, which is power of base 16.
• The system uses 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F; where A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
• Examples include: 3F7A = 0011 1111 0111 1010 and 53CE = 0101 0011 11001110.
• Hexadecimal is used in computing and digital systems.
• Memory addresses and color codes in web design use hexadecimal notation.
• This notation is used because one hexadecimal digit represents four binary digits.
• Hexadecimal number 1A05.2C4 can be written in decimal form as (1 × 163) + (10 × 162) + (0 × 161) + (5 × 160) + (2 × 16-1) + (12 × 16-2) + (4 × 16-3).

Conversion Between Number Systems

• To convert a decimal number with both integer and fractional parts, each part is converted separately to the new base.
• The steps include dividing the integer part by the new base 'r', noting remainders until the quotient is zero then reading the remainders in reverse order to get the new integer part.
• Multiplie the fractional part by 'r' successively, noting the carries until the result is zero (or desired digits are obtained), then combine the integer part and the normal sequence of carry to get the fractional part of the equivalent number of based 'r'.

Decimal to Binary Conversion

• To convert from decimal to binary, divide the number by 2.
• Continue the division process with each quotient until it becomes zero.
• Write down each remainder, and then read the binary number from the bottom (last remainder) to the top (first remainder).
• To convert decimal 13 to binary:
• 13 ÷ 2 = 6, Remainder = 1
• 6 ÷ 2 = 3, Remainder = 0
• 3 ÷ 2 = 1, Remainder = 1
• 1 ÷ 2 = 0, Remainder = 1
• 13 in decimal is therefore 1101 in binary.
• When converting a decimal number containing an integer and fractional part into a binary number, two operations take place.
• The integer is divided and the fractional is multiplied.

Conversion Examples

• The decimal number 58.25 converted to binary using division and multiplication.
• Integer 58 converts using division:
• 58 ÷ 2 = 29, Remainder 0(LSB)
• 29 ÷ 2 = 14, Remainder 1
• 14 ÷ 2 = 7, Remainder 0
• 7 ÷ 2 = 3, Remainder 1
• 3 ÷ 2 = 1, Remainder 1
• 1 ÷ 2 = 0, Remainder 1(MSB)
• Giving an equivalent binary form of 111010.
• Fractional 0.25 converts using multiplication given as, .2510=.012.
  • 0.25 x 2 = 0.5, Carry 0
  • 0.5 x 2 = 1.0, Carry 1
• Therefore, 58.2510 = 111010.012 is an equivalent in binary form.
• The decimal number 58.25 converted to octal has integer 58 and fractional parts as 0.25.
• Integer converts: 5810 = 728
• 58 ÷ 8 = 7, Remainder 2
• 7 ÷ 8= 0, Remainder 7
• Fractional converts .2510=.28
• 0.25 x 8 = 2.00 Carry 2
• The final results give us 58.2510 = 72.28.
• The decimal number 58.25 expressed to hexa-decimal.
• 58 ÷ 16 = 3, Remainder 10=A
• 3 ÷ 16=0, Remainder 3
• The result of 5810 is 3A16 expressed as a hexagon.
• Lastly the fractional number converts to .2510=.416 where 0.25 x 16 = 4.0 .

Binary Number to Other Bases Conversion

• Converting from binary to decimal differs from converting to octal and Hexa-decimal.
• Converting a binary number into its decimal equivalent by multiplying binary bits with their respective positional weights and summing all products.
• For example
• 1101.112 = (1 x 23) + (1 x 22) + (0 x 21) + (1 x 20) + (1 × 2-1) + (1 × 2-2)
• 1101.112 = 8+4+0+1+0.5 +0.25 giving you 13.75. which is its decimal equal.
  • 1101.112 =13.7510.

Binary to Octal and Hexadecimal conversion

• Binary to octal conversion uses the bases 2 and 8, respectively, where three bits of binary equals one octal due to 23 = 8.
• Group the bits into 3 from the binary point on both sides and pad lacking zeros on the extremes.
• Write the octal digits corresponding to each group of 3 bits.
• For example, 101110.011010 converts to groups of 101 110 . 011 010 which is the equivalent of 56.328.
• Binary to Hexa conversion uses bases 2 and 16
• Four bits of a binary number is equivalent to one Hexa-decimal digit because 24=16.
• Pad zeros on either side if there are not enough bits.
• For example for number 101110.01101, you make 10 1110.0110 1 groups of four bits so the solution can be found by 0010 1110.0110 1000= 2E.6816.

Binary Arithmetic

• It is essential for all digital computers and related components.
• The following rules are of binary addition are required:
  • 0+0 = 0, w/ a carry of 0
  • 0 + 1 =, w/ a carry of 0
  • 1+0= 1 w/ a carry of 0
  • 1 + 1= 0, w/ a carry of 1
• Subtraction & Borrow are used for binary subtraction
• The four rules of binary arithmetic are:
  • 0 -0 = 0, w/ a borrow of 0
  • 1 -0 = 1, w/ a borrow of 0
  • 1 - 1 = 0, w/ a borrow of 0
  • 0 -1 = 0, w/ a borrow of 1
• Binary multiplication is similar, but Simpler because it involves Os and 1s so,
• 0 X 0= 0
• 0 X 1 =0
• 1 X 0 =0
• 1 X 1 =1
• Steps for Binary Division
• Align the dividend and divisor like in long division
• Ensure divisor is fits into the current section the dividend.
• If YES write a 1 in quotient, subtract
• if no Write 0 in the quotient then bring down the next but.
• Ensure entire dividend has processed with any remainder left.

Binary System Complements

• The binary system has base r = 2, thus 2's complement and 1's complement exist
• A 1's complement can be by changing all 1's to 0's and all 0's to 1's. It often called the complement
• A 2's complement can be found by, adding a 1 to the Least Significant Bit (LSB) of the 1's complement of a number