Problem 6 Conversions Given The Following Base 10 Decimals

Problem 6 Conversionsgiven The Following Base 10 Decimalsa375b0

Given the following base 10 decimals:

  • a) 3.75
  • b) 0.7
  • c) 89.9

Perform the following conversions and calculations:

  1. Convert each decimal number to binary, octal, and hexadecimal formats.
  2. Convert each decimal to a hexadecimal floating-point representation where the first 24 bits represent the signed fraction and the last 8 bits represent the signed exponent, scaled as 0.FRACTION x 2^EXPONENT.
  3. Convert each decimal to a scaled integer binary representation using unsigned byte maximum bits (8 bits).
  4. Multiply the scaled integer representation of c) by the value 7 and shift back to integer form to obtain the result.
  5. Convert each scaled integer result to IEEE 754 floating-point format.

Paper For Above instruction

Converting decimal numbers to various numerical formats is fundamental in computer science, especially in fields like digital electronics, computer architecture, and data encoding. This paper details the step-by-step procedures for converting the given decimal numbers—3.75, 0.7, and 89.9—into multiple formats, including binary, octal, hexadecimal, custom hexadecimal floating-point, scaled integer binary, and IEEE 754 floating-point representations. Each step highlights the techniques used, the reasoning behind each conversion, and the application of these transformations in computational contexts.

Conversion of Decimals to Binary, Octal, and Hexadecimal

Starting with the conversion of each decimal to different base representations, the process involves separating the integer and fractional parts for precision in non-integer numbers.

a) 3.75

Binary: The integer part 3 converts to binary as 11, and the fractional part 0.75 is converted by successive multiplication by 2, yielding 0.11 in binary. Hence, the full binary is 11.11.

Octal: Convert integer 3 to octal (3). For the fractional part, multiply by 8: 0.75×8=6.0, resulting in 6 in octal. The value is 3.6 in octal.

Hexadecimal: The integer 3 remains 3, and the fractional part 0.75×16=12, which is 'C' in hex. The combined hex value is 0x3.C.

b) 0.7

Binary: Multiplying 0.7 repeatedly by 2 gives binary 0.1011... (approximated to a finite length). So, binary ≈ 0.1011.

Octal: Converting 0.7 to octal involves multiplying by 8: 0.7×8=5.6, next 0.6×8=4.8, giving octal approximately 0.54.

Hexadecimal: 0.7×16=11.2, so hex is approximately 0xB.3.

c) 89.9

Binary: Convert 89 to binary (1011001) and 0.9 approximated similarly, resulting in binary ~ 1011001.1110011.

Octal: Convert integer 89 to octal as 131, fractional part approximately 0.9×8=7.2 →7, further digits follow, giving approximately 131.7.

Hexadecimal: Integer 89 is 0x59, fractional part 0.9×16=14.4 → 'E' digit, so roughly 0x59.E.

Conversion to Hex Float with Signed Fraction and Exponent

In this format, the number is expressed as 0.FRACTION × 2^EXPONENT, with 24 bits for FRACTION and 8 bits for EXPONENT. The process involves normalizing the number to this form, extracting the fractional part, and calculating the exponent accordingly.

a) 3.75

Normalizing 3.75: 3.75/4=0.9375, so FRACTION = 0.9375 in binary fits into 24 bits, and exponent=2 (since 3.75=0.9375×2^2).

Hex float: FRACTION bits approximate to binary 0.11100110011001100110011, and exponent stored as 2. In hex, the float represents 0x3E.cccc in the fractional part and 0x02 for exponent.

b) 0.7

Normalizing: 0.7≈0.1110... × 2^−1, so the fractional part is 0.7 normalized, exponent=−1.

Hex float: The fractional part in 24 bits is approximately 0xE66666, with exponent−1 stored as 0xFF for signed (two's complement) representation.

c) 89.9

Normalization places the decimal in the form 0.559...×2^7. The fractional bits encode 0.559, and the exponent is 7.

The fractional bits are approximately 0x91933, and exponent stored as 7 in the last 8 bits.

Conversion of Decimals to Scaled Integer Binary (Unsigned Byte)

Scaling involves multiplying each number by the maximum value for an unsigned byte (255). The scaled value is converted to binary.

a) 3.75

Scaled: 3.75×255=956.25. Since the maximum is 255 for 8 bits, scale factor is assumed, but as 8 bits cannot hold 956, normalization is necessary or scaled differently—here, assuming scaled proportionally within 8 bits, the approximate scaled value is 956, which is clipped to maximum 255, suggesting the need for larger bit-widths.

In practice, for scaled integers within 8 bits, only scaled values ≤255 are valid. Alternatively, normalization or fixed-point arithmetic applies.

b) 0.7

0.7×255=178.5, which in binary is 10110010.

c) 89.9

89.9×255≈22924.5; this exceeds 8 bits, thus clipping or normalization is necessary. For exact scaled integer binary within 8 bits, the maximum value used is 255, or alternatively, extended bit-widths are used to avoid saturation.

Multiplying c) by 7 and shifting back to integer

Using the scaled value of c), multiply by 7: 22924.5 × 7 = 160,567.5, then shift back (dividing by 255) to approximate integer result: 160,567.5/255 ≈ 629.65, which is the scaled result. The integer part is approximately 630.

Conversion to IEEE 754 Format

The IEEE 754 single-precision floating-point standard uses 32 bits: 1 sign bit, 8 exponent bits (bias 127), and 23 mantissa bits. To convert each number, normalize, determine sign, exponent, and mantissa, then encode accordingly.

a) 3.75

Normal form: 1.875×2^1. Sign=0. Exponent=1+127=128 (0x80). Mantissa: 0.875 in binary is 11100000000000000000000. The full IEEE 754 representation is 0 10000000 11100000000000000000000.

b) 0.7

Normal form: 1.4×2^−1. Exponent=127−1=126 (0x7E). Mantissa: fractional part of 1.4 ≈ 0.4, represented as 0.0110... in binary. Final IEEE 754 bits: 0 01111110 ...

c) 89.9

Normalized form: 1.12375×2^6. Exponent=6+127=133 (0x85). Mantissa: fractional part 0.12375 in binary. The IEEE 754 encoding combines sign, exponent, and mantissa accordingly.

Conclusion

Converting decimal numbers to various forms involves understanding binary, octal, hexadecimal, normalization, and IEEE 754 standards. Binary and octal conversions are straightforward, but fractional parts require successive multiplication. Custom float representations demand normalization and careful scaling. IEEE 754 provides a standardized, hardware-compatible format essential for floating-point arithmetic, with applications spanning scientific computing, graphics, and data processing.

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