Binary, Octal, and Hexadecimal Conversions

Number System Conversions

• To convert a decimal number to hexadecimal, divide the decimal number by 16 and keep track of the remainders; the remainders will be the hexadecimal digits.
• The decimal number 422 is converted to hexadecimal by dividing it by 16: 422 ÷ 16 = 26 with a remainder of 10, and 26 ÷ 16 = 1 with a remainder of 10, so the hexadecimal representation is A6.
• To convert the decimal number 139 to octal, divide it by 8: 139 ÷ 8 = 17 with a remainder of 3, and 17 ÷ 8 = 2 with a remainder of 1, so the octal representation is 213.

Fraction Representation

• In a radix r number system, fractions are represented by dividing the fractional part by r and keeping track of the remainders; the remainders will be the digits of the fractional part.
• The significance of the remainder when converting from decimal to hexadecimal is that it determines the next hexadecimal digit.

Binary, Octal, and Hexadecimal Conversions

• When converting fractions from binary to hexadecimal, group the binary digits into sets of 4 and convert each set to a hexadecimal digit.
• 4 binary digits are grouped to form a hexadecimal digit.
• The simplified method for converting fractions between binary, octal, and hexadecimal is to convert the fraction to one system and then convert the result to the desired system.

Binary Number Representation

• In binary, 2 should be represented as (10)2 because the binary system can only represent two values: 0 and 1.
• The widely used representation for signed numbers is 2's complement.
• Signed numbers are represented in 2's complement by inverting the bits and adding 1.
• The standard way to represent signed integers in computers is using 2's complement.

2's Complement Operations

• The 2's complement of a binary number is calculated by inverting the bits and adding 1.
• The range of represented values in 2's complement form for n bits is -2^(n-1) to 2^(n-1) - 1.
• The property of the 2's complement in relation to the original number is that the sum of the original number and its 2's complement is 2^n.
• Subtraction can be converted into addition using 2's complement by adding the 2's complement of the subtrahend to the minuend.
• In the case of overflow when adding signed integers, the result will be incorrect.
• Sign extension is performed in binary numbers by copying the sign bit (most significant bit) to the left to extend the number.
• The final carry in 2's complement addition indicates an overflow.
• The key advantage of using 2's complement for subtraction is that it allows for efficient and simple implementation.
• Overflow can be detected in binary addition by checking the carry bit.