Answer :
To add two numbers in scientific notation, there are a few step: 1. Make sure they have the same exponent, 2. Add their mantissas, 3. Normalize the result if necessary and 4. Convert to hexadecimal. Based on the calculation we've done, none of the choices a) 0x2.5, b) 0x6.25, c) 0x4.1, and d) 0x7.3125 match 0x1.54.
To add two numbers in scientific notation, there are a few steps:
1. Make sure they have the same exponent.
2. Add their mantissas.
3. Normalize the result if necessary.
4. Convert to hexadecimal.
Let's walk through these steps for the numbers 1.5625 × 10⁻¹ and 3.75 × 10⁻¹.
Both numbers already have the same exponent of -1, so the first step is satisfied.
Now, let's convert them to binary, since hexadecimal is more easily related to binary:
1.5625 in decimal is 1.1001 in binary since:
1.5625 = 1 + 1/2 + 0/4 + 0/8 + 1/16.
3.75 in decimal is 11.11 in binary since:
3.75 = 3 + 3/4.
Now we should adjust these binary numbers so they both have the same exponent when represented in binary scientific notation:
1.5625 (1.1001 binary) can be written as:
1.1001 × 2⁰
Since we wantto add it to 3.75, we can represent it with the same exponent:
1.1001 (binary) = 0.11001 × 2¹
3.75 (11.11 binary) can be written as:
1.111 × 2¹.
Now we add these numbers:
0.11001
+ 1.11100
----------
10.10101 which is the result in binary with an exponent of 2¹ (since we carried over the leading 1).
Now we normalize this binary number:
2.10101 × 2¹ turns into:
1.010101 × 2²
This is our result in binary scientific notation.
However, 16-bit half precision has a specific format:
1 bit for sign (we assume both numbers are positive so this is 0),
5 bits for exponent (with a bias of 15),
10 bits for the fraction (mantissa).
First, we convert our exponent to 5-bit binary with a bias of 15.
Our exponent is 2, so we add the bias:
2 + 15 = 17.
17 in binary is:
10001.
Next, we take the mantissa from the normalized number, which is the part after the binary point:
010101 (we only need the first 10 digits but we have fewer, so we fill with zeros).
Now we create the binary representation, starting with the sign bit:
0 (since the number is positive), followed by the exponent, and then the mantissa:
0 10001 0101010000
This binary number corresponds to the floating-point number we've calculated. To convert to hexadecimal, we divide into groups of four binary digits (starting from the binary point):
1. 0101 (5 in hexadecimal)
2. 0100 (4 in hexadecimal)
So the hexadecimal representation is:
0x1.54 (we truncate the mantissa to fit the available bits).
However, none of the options you provided exactly matches this representation.
Based on the calculation we've done, none of the choices a) 0x2.5, b) 0x6.25, c) 0x4.1, and d) 0x7.3125 match 0x1.54.
