Decimal Number System

• In early days, people used fingers, stones, pebbles, and sticks for counting due to the lack of other means.
• Over time, various number systems were introduced to overcome the limitations of early counting methods, including decimal, binary, octal, and hexadecimal systems.
• A number system includes a set of values used to represent different quantities.

Arhat

• Arhat was India's greatest mathematician and astronomer.
• He introduced the concept of zero.
• Modern computer technology would be non-existent without the concept of zero.

Decimal Number System

• The need for counting led to the introduction of the decimal number system.
• The decimal number system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
• Most arithmetic operations are performed using decimal numbers.
• The decimal number system consists of 10 digits (0 to 9) and has a base of 10.
• The base of a number system is the number of digits it uses.
• Numbers can be used individually or grouped to form a numeric value, such as 82, 256, or 52.87.
• The value of each digit depends on its face value, the base of the number system, and its position in the number.
• Each position represents a specific power of base 10.
• The rightmost digit is the least significant digit, and the leftmost digit is the most significant digit.
• The positional value of each digit increases tenfold as you move from right to left.
• Place value depends on the position of the digit in the number.

Binary Number System

• The binary number system consists of only two digits: 0 and 1.
• It has a base of 2.
• All digital computers use the binary number system
• Computers convert input data from decimal to binary format.
• Gottfried Leibniz, a German mathematician, is credited with the invention of the modern binary number system.
• Computers do not understand human language but understand binary code.
• Data entered into a computer is converted into its binary equivalent.
• The computer converts the binary results back into their decimal equivalents for output

Decimal to Binary Conversion

• Divide the decimal number by 2.
• Write down the remainder.
• Divide the quotient again by 2.
• Repeat until the quotient is 0.
• Write the remainders in reverse order to form the binary equivalent.
• Example: 25 (base 10) = 11001 (base 2)
• Example: 321 (base 10) = 101000001 (base 2)

Binary to Decimal Conversion

• Multiply each binary number with its positional value in powers of 2, starting from the rightmost digit.
• Sum up all the products to get the decimal number.
• Example: 1010 (base 2) = 0*(2^0) + 1*(2^1) + 0*(2^2) + 1*(2^3) = 0 + 2 + 0 + 8 = 10 (base 10)
• Example: 1001 (base 2) = 1*(2^0) + 0*(2^1) + 0*(2^2) + 1*(2^3) = 1 + 0 + 0 + 8 = 9 (base 10)
• Example: 110001001 (base 2) = 393 (base 10)

Bytes

• Most computer operations use the byte or multiples of bytes (16 bits, 24, 32, 64, etc.).
• Hexadecimal makes it easier to write these large binary numbers.

Octal Number System

• The octal number system consists of eight digits: 0 to 7.
• The octal number system has base 8.
• The concept of the octal number system came from Native Americans, who counted using the spaces between their fingers.
• Octal to decimal conversion is similar to binary to decimal conversion, but with a base of 8.
• To convert an octal number to decimal, multiply each digit by increasing powers of 8 from right to left, starting from zero, and sum the products.
• Example: 345 (base 8) = 3*(8^2) + 4*(8^1) + 5*(8^0) = 192 + 32 + 5 = 229 (base 10)
• Example: 317 (base 8) = 3*(8^2) + 1*(8^1) + 7*(8^0) = 192 + 8 + 7 = 207 (base 10)

Hexadecimal Number System

• Consists of 16 digits: 0 to 9 and letters A to F, where A to F represent digits 10 to 15.
• It has a base of 16.
• Hexadecimal is also known as hex (hex = 6, decimal = 10).
• Converting hexadecimal to decimal is similar to the methods used for binary and octal, but with a base of 16.
• To convert hexadecimal to decimal, multiply each digit by base 16.
• Example: 3B (base 16) = 3*(16^1) + 11*(16^0) = 48 + 11 = 59 (base 10)
• Example: 4D2 (base 16) = 4*(16^2) + 13*(16^1) + 2*(16^0) = 1024 + 208 + 2 = 1234 (base 10)

Computer Arithmetic

• Computers understand only binary code, so input data is converted to binary for processing.
• Processing includes arithmetic operations like addition, subtraction, multiplication, and division on binary numbers.
• 4 bits = 1 nibble
• 2 nibbles = 8 bits
• 8 bits = 1 byte

Binary Addition Rules

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (0 with a carry-over of 1 to the next column)
• Example: 1000 (base 2) + 0111 (base 2) = 1111 (base 2)
• Example: 1111 (base 2) + 1011 (base 2) = 11010 (base 2)

Binary Subtraction Rules

• 0 - 0 = 0
• 1 - 0 = 1
• 1 - 1 = 0
• 0 - 1 = 1 (with a borrow of 1 from the next column)
• 10 - 1 = 1
• Example: 1111 (base 2) - 1010 (base 2) = 0101 (base 2)
• Example: 1100 (base 2) - 0011 (base 2) = 1001 (base 2)

Binary Multiplication Rules

• 0 * 0 = 0
• 0 * 1 = 0
• 1 * 0 = 0
• 1 * 1 = 1
• Follow the same method as decimal multiplication.
• Example: 101 (base 2) * 11 (base 2) = 1111 (base 2)
• Example: 1111 (base 2) * 11 (base 2) = 101101 (base 2)

Binary Division

• The method for dividing binary numbers is the same as for decimal numbers.

Summary of Key Points

• The decimal number system is commonly used, with base 10.
• The rightmost digit of a number is the least significant digit; the leftmost digit is the most significant digit.
• The binary number system uses digits 0 and 1, with base 2.
• The octal number system uses eight digits and has base 8.
• The hexadecimal number system uses 16 digits and has base 16.