What is the significand? \( \begin{array}{l}\text { Represent }(2748)_{10} \text { as floating point number in the computer with } \\ 16 \text {-bit } \\ 12\end{array} \)

To represent the decimal number \( (2748)_{10} \) as a floating-point number in a 16-bit format, we need to follow the structure of a typical floating-point representation. A common format is the IEEE 754 standard, which divides the bits into three parts: the sign bit, the exponent, and the significand (also known as the mantissa). ### Step 1: Determine the Sign Bit Since \( 2748 \) is positive, the sign bit is: - **Sign bit**: \( 0 \) ### Step 2: Convert to Binary Next, we convert \( 2748 \) to binary. 1. Divide \( 2748 \) by \( 2 \) and record the remainders: - \( 2748 \div 2 = 1374 \) remainder \( 0 \) - \( 1374 \div 2 = 687 \) remainder \( 0 \) - \( 687 \div 2 = 343 \) remainder \( 1 \) - \( 343 \div 2 = 171 \) remainder \( 1 \) - \( 171 \div 2 = 85 \) remainder \( 1 \) - \( 85 \div 2 = 42 \) remainder \( 1 \) - \( 42 \div 2 = 21 \) remainder \( 0 \) - \( 21 \div 2 = 10 \) remainder \( 1 \) - \( 10 \div 2 = 5 \) remainder \( 0 \) - \( 5 \div 2 = 2 \) remainder \( 1 \) - \( 2 \div 2 = 1 \) remainder \( 0 \) - \( 1 \div 2 = 0 \) remainder \( 1 \) Reading the remainders from bottom to top, we get: \[ (2748)_{10} = (101010101100)_{2} \] ### Step 3: Normalize the Binary Number To normalize \( (101010101100)_{2} \), we express it in the form \( 1.x \times 2^n \): \[ (101010101100)_{2} = 1.01010101100 \times 2^{11} \] Here, \( n = 11 \) because we moved the binary point 11 places to the left. ### Step 4: Determine the Exponent In a 16-bit floating-point representation, we typically have: - 1 bit for the sign - 5 bits for the exponent - 10 bits for the significand The exponent is stored in a biased form. For a 5-bit exponent, the bias is \( 2^{(5-1)} - 1 = 15 \). Therefore, we calculate the biased exponent: \[ \text{Biased exponent} = n + \text{bias} = 11 + 15 = 26 \] Now, we convert \( 26 \) to binary: \[ (26)_{10} = (11010)_{2} \] ### Step 5: Determine the Significand The significand (or mantissa) is taken from the normalized binary number, excluding the leading 1 (which is implicit in normalized form). We take the next 10 bits after the binary point: \[ \text{Significand} = 0101010110 \] ### Step 6: Assemble the Floating-Point Representation Now we can assemble the 16-bit floating-point representation: - Sign bit: \( 0 \) - Exponent: \( 11010 \) - Significand: \( 0101010110 \) Putting it all together, we have: \[ \text{Floating-point representation} = 0 \, 11010 \, 0101010110 \] ### Final Answer Thus, the significand of \( (2748)_{10} \) represented as a 16-bit floating-point number is: \[ \text{Significand} = 0101010110 \]