Question

A 18-bit floating point binary number has 12 bits (including a sign bit) for the mantissa 6 bits (including a sign bit) for the exponent. The mantissa is a normalized fraction. Numbers in the mantissa and exponent are in signed-magnitude representation. What are the largest and smallest positive quantities that can be represented, excluding zero? Give your answers in decimal.

Answer 1

The largest positive quantity represented by this floating point system is approximately 1.99902 × 2^+31, and the smallest positive quantity is approximately 1.00000 × 2^-31.

Step-by-step explanation:

The question involves finding the largest and smallest positive quantities that can be represented by an 18-bit floating point binary number with a 12-bit mantissa (including a sign bit) and a 6-bit exponent (including a sign bit), with numbers in both mantissa and exponent being in signed-magnitude representation. To determine these quantities, we will consider the range of values the mantissa and exponent can represent in this system. The largest value the mantissa can represent is when all bits are 1 except for the sign bit, which would be 0 to indicate a positive number. The mantissa would then be 1.1111111111 in binary, which equals 1 + (1/2) + (1/4) +... + (1/1024) = 1.99902 in decimal, rounded to six significant figures. To get the largest possible number, we would then pair this mantissa with the largest exponent, which would be 011111 in binary, representing +31 in decimal (since the exponent is also in signed-magnitude format).

The smallest positive value is when the mantissa is just over the smallest normalized positive value, which would be 1.0000000000 in binary (or just 1 in decimal), and the smallest positive exponent, which would be 100000 in binary, represents -31 in decimal. Therefore, the largest quantity is approximately 1.99902 × 2+31, and the smallest positive quantity is approximately 1.00000 × 2-31. A 18-bit floating-point binary number with 12 bits for the mantissa (including a sign bit) and 6 bits for the exponent (including a sign bit) can represent a range of positive quantities. Since the mantissa is normalized, the largest positive quantity that can be represented is when all mantissa bits are set to 1 and the exponent is set to its maximum value. This results in a value of 1.111111111111 x 10^31 in decimal notation. The smallest positive quantity that can be represented is when the mantissa is set to 0 and the exponent is set to its minimum value. This results in a value of 1.0 x 10^(-32) in decimal notation.