Number Systems and Conversions Quiz

Number Systems

• Number systems are used to represent quantities in different bases.
• Common Number Systems:
  • Decimal (Base 10): Uses symbols 0-9, used by humans.
  • Binary (Base 2): Uses symbols 0 and 1, used by computers.
  • Octal (Base 8): Uses symbols 0-7, used in some computing contexts.
  • Hexadecimal (Base 16): Uses symbols 0-9 and A-F, used extensively due to its compact representation.

Conversion Between Bases

• Various methods exist to convert between number systems.
  • Decimal to Binary: Repeated division by 2, retaining remainders to form the binary representation.
  • Binary to Decimal: Multiply each bit by 2 raised to its respective position from right (0 for rightmost). Sum all the products.
  • Decimal to Octal: Similar to Decimal to Binary, by dividing successively by 8.
  • Octal to Decimal: Similar to Binary to Decimal, but each octal digit is multiplied by 8 raised to its position.
  • Decimal to Hexadecimal: Divide by 16 and retain the remainders.
  • Hexadecimal to Decimal: Multiply each hex digit by 16 raised to its position from right (0 for rightmost) and sum the products.

Converting Fractions

• For fractional conversions, the standard method of using a base-10 representation must be translated for a different base, like base 2 or 8.
• Specific method depends if the number is before or after the decimal point.

Quantities and Counting

• Conversion tables for common decimal, binary, octal and hexadecimal values are presented in the slides.

Conversion Among Bases

• Illustrates possible conversions from decimal, binary, octal, to hexadecimal

Quick Example

• An example illustrating how these concepts are implemented in real use-cases is shown.

Decimal to Binary Conversion Technique

• Divide the decimal number by 2, and record the remainder.
• Repeat this process until the quotient becomes zero.
• Concatenate the remainders in reverse order to obtain the binary equivalent.

Example of Decimal to Binary Conversion

• An example of decimal to Binary conversion is displayed, showing the calculation steps and result.

Decimal to Fractional Binary Conversion Technique

• Multiply the fractional part of the decimal by 2.
• If the result is greater than or equal to 1, append a 1 and subtract 1 from the result.
• If the result is less than 1, append a 0 to the binary fractional part.
• Repeat until the result is exactly 0 or until you have reached a desired level of precision.
• Append the integer part to the binary fractional part, separated by a decimal point, to assemble the full binary result.

Binary Addition and Multiplication

• Binary addition is performed bit by bit and carries are propagated in the addition process
• Binary multiplication procedures for various values, similar to decimal arithmetic, but using 0s and 1s instead of base-10 digits.
• Charts that demonstrate two and n-bit value operations

Fractions (decimal and binary conversions)

• Decimal to decimal fraction examples.
• Binary to decimal fraction examples

Converting base 16 (Hexadecimal) to base 10 (Decimal)

• Multiply each hexadecimal digit by the appropriate power of 16 (starting from the rightmost digit).

Converting base 8 (Octal) to base 10 (Decimal)

• Multiply each octal digit by the appropriate power of 8 and sum the results.

Converting base 2 (Binary) to base 8 (Octal)

• Group the bits in sets of 3(From right to left). Add leading zeros if necessary.
• Convert each group of 3 binary digits into its respective octal equivalent.

Converting base 2 (Binary) to base 16 (Hexadecimal)

• Group the bits in sets of 4 (from right to left). Add leading zeros if necessary.
• Convert each group of 4 binary digits into its respective Hexadecimal equivalent.

Converting base 8 (Octal) to base 16 (Hexadecimal)

• Convert the octal number in the first step to binary.
• Then convert the binary number to hexadecimal.

Converting base 16 (Hexadecimal) to base 8 (Octal)

• Convert the hexadecimal number to binary.
• Then convert the binary number to octal.