1. **Problem Statement:** Convert the decimal number 25.375 to binary, octal, hexadecimal, single precision IEEE 754, and double precision IEEE 754 formats. 2. **Convert 25.375 to Binary:** - Separate integer and fractional parts: 25 and 0.375 - Integer 25 to binary: $25_{10} = 11001_2$ - Fraction 0.375 to binary: multiply by 2 repeatedly $$0.375 \times 2 = 0.75 \rightarrow 0$$ $$0.75 \times 2 = 1.5 \rightarrow 1$$ $$0.5 \times 2 = 1.0 \rightarrow 1$$ - Fractional binary: $0.011$ - Combine: $25.375_{10} = 11001.011_2$ 3. **Convert 25.375 to Octal:** - Group binary digits in 3s from decimal point: Integer: $11001_2 = 011 001$ Fraction: $011$ - Integer groups: $011_2=3$, $001_2=1$ - Fraction group: $011_2=3$ - Octal: $31.3_8$ 4. **Convert 25.375 to Hexadecimal:** - Group binary digits in 4s: Integer: $11001_2 = 0001 1001$ Fraction: $0110$ (add zero to complete 4 bits) - Integer groups: $0001_2=1$, $1001_2=9$ - Fraction group: $0110_2=6$ - Hexadecimal: $19.6_{16}$ 5. **Convert 25.375 to Single Precision IEEE 754:** - Binary: $11001.011_2$ - Normalize: $1.1001011 \times 2^4$ - Sign bit: 0 (positive) - Exponent: $4 + 127 = 131 = 10000011_2$ - Mantissa: bits after leading 1: $10010110000000000000000$ - IEEE 754: $0\ 10000011\ 10010110000000000000000$ 6. **Convert 25.375 to Double Precision IEEE 754:** - Exponent bias: 1023 - Exponent: $4 + 1023 = 1027 = 10000000011_2$ - Mantissa: $1001011$ followed by zeros to fill 52 bits - IEEE 754: $0\ 10000000011\ 1001011000000000000000000000000000000000000000000000$ **Final answers:** - Binary: $11001.011_2$ - Octal: $31.3_8$ - Hexadecimal: $19.6_{16}$ - Single Precision IEEE 754: $0\ 10000011\ 10010110000000000000000$ - Double Precision IEEE 754: $0\ 10000000011\ 1001011000000000000000000000000000000000000000000000$