Computer Number Systems Quiz

Computer Number Systems

• Computers use binary (base-2) number systems internally. This is because electronic circuits are easily represented as "on" (1) or "off" (0).
• Other number systems include decimal (base-10), hexadecimal (base-16), and octal (base-8). These are used for human readability and sometimes for efficiency in representing binary numbers.
• Binary numbers use only two digits: 0 and 1.
• Each position in a binary number represents a power of 2, increasing from right to left.

Decimal to Binary Conversion

• To convert a decimal number to binary, repeatedly divide the decimal number by 2 and record the remainder. The remainders, read from bottom to top, form the binary equivalent.
• Example: Convert 13 to binary.
  • 13 / 2 = 6 with a remainder of 1
  • 6 / 2 = 3 with a remainder of 0
  • 3 / 2 = 1 with a remainder of 1
  • 1 / 2 = 0 with a remainder of 1
  • The binary equivalent is 1101.

Binary to Decimal Conversion

• To convert a binary number to decimal, multiply each digit by the corresponding power of 2 and sum the results.
• Example: Convert 1101 to decimal.
  • (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 4 + 0 + 1 = 13

Hexadecimal (Base-16)

• Hexadecimal uses 16 digits: 0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, F=15.
• Hexadecimal is a convenient shorthand for representing groups of four binary digits. Each hexadecimal digit corresponds to four binary digits.
• Example: The binary number 1010 1101 is equivalent to the hexadecimal number AA. (1010 is A and 1101 is D)

Octal (Base-8)

• Octal uses 8 digits: 0 to 7.
• Octal is another shorthand representation of binary, using groups of three binary digits for each octal digit.
  • Example: The binary number 100 101 111 is 457 in octal.

Computer Representation of Numbers

• Integers and floating-point numbers are both represented in the computer's memory using a specific number of bits.
• The representation depends on the data type and the number of bits used.
• The representation can be unsigned or signed, with different ranges and limitations.
• Positive and negative numbers can be stored using signed magnitude, one's complement, or two's complement representations. Two's complement is the most common for signed integers.

Conversion between different Number Systems

• It's important to recognize the pattern of powers of the base during conversions. Practice is crucial for mastering these conversions (between binary, decimal, octal, hexadecimal).

Error Handling and Overflow/Underflow

• Computers have limited storage capacity for numbers.
• Overflow occurs when the result of an arithmetic operation is greater than or lesser than the maximum/minimum value that can be represented.
• Underflow occurs when the result of an arithmetic operation is smaller than or less than the minimum value that can be represented.

Signed Number Representation

• Different representations (sign-magnitude, one's complement, two's complement) are used for negative numbers.
  • Two's complement is commonly used because it simplifies arithmetic operations. It is the standard for representing signed integers in most computer architectures. It makes adding signed numbers efficient.
• The number of bits used impacts the range of representable values.

Floating-Point Representation

• Used to represent real numbers (numbers with decimal points).
• It leverages a format using sign, mantissa (fractional part), and exponent to accommodate a wide range of numbers.
• This representation has limitations concerning precision. Round-off errors can occur due to the loss of precision during calculations.