Question

As you know computer system stores all types of data as stream of binary digits (0 and 1). This also includes the numbers having fractional values, where placement of radix point is also incorporated along with the binary representation of the value. There are different approaches available in the literature to store the numbers having fractional part. One such method, called Floating-point notation is discussed in your week 03 lessons. The floating point representation need to incorporate three things:

• Sign
• Mantissa
• Exponent

A. Encode the (negative) decimal fraction -9/2 to binary using the 8-bit floating-
point notation.
B. Determine the smallest (lowest) negative value which can be
incorporated/represented using the 8-bit floating point notation.
C. Determine the largest (highest) positive value which can be
incorporated/represented using the 8- bit floating point notation.

Answer 1

A. Encode the (negative) decimal fraction -9/2 to binary using the 8-bit floating-point notation.

First, let's convert -9/2 to a decimal number: -9/2 = -4.5

Now, let's encode -4.5 using the 8-bit floating-point notation. We'll use the following format for 8-bit floating-point representation:

1 bit for the sign (S), 3 bits for the exponent (E), and 4 bits for the mantissa (M): SEEE MMMM

Sign bit: Since the number is negative, the sign bit is 1: 1

Mantissa and exponent: Convert -4.5 into binary and normalize it:

-4.5 in binary is -100.1. Normalize it to get the mantissa and exponent: -1.001 * 2^2

Mantissa (M): 001 (ignoring the leading 1 and taking the next 4 bits)

Exponent (E): To store the exponent (2) in 3 bits with a bias of 3, add the bias to the exponent: 2 + 3 = 5. Now, convert 5 to binary: 101

Now, put the sign, exponent, and mantissa together: 1101 0010

So, the 8-bit floating-point representation of -9/2 (-4.5) is 1101 0010.

B. Determine the smallest (lowest) negative value which can be incorporated/represented using the 8-bit floating-point notation.

To get the smallest negative value, we'll set the sign bit to 1 (negative), use the smallest possible exponent (excluding subnormal numbers), and the smallest mantissa:

Sign bit: 1

Exponent: Smallest exponent is 001 (biased by 3, so the actual exponent is -2)

Mantissa: Smallest mantissa is 0000

The 8-bit representation is 1001 0000. Converting this to decimal:

-1 * 2^{-2} * 1.0000 which is -0.25.

The smallest (lowest) negative value that can be represented using the 8-bit floating-point notation is -0.25.

C. Determine the largest (highest) positive value which can be incorporated/represented using the 8-bit floating-point notation.

To get the largest positive value, we'll set the sign bit to 0 (positive), use the largest possible exponent (excluding infinity), and the largest mantissa:

Sign bit: 0

Exponent: Largest exponent is 110 (biased by 3, so the actual exponent is 3)

Mantissa: Largest mantissa is 1111

The 8-bit representation is 0110 1111. Converting this to decimal:

1 * 2^3 * 1.1111 which is approximately 1 * 8 * 1.9375 = 15.5.

The largest (highest) positive value that can be represented using the 8-bit floating-point notation is 15.5.

Step-by-step explanation: