Convert −9.125 to IEEE 754 Single and Double Precision Step by Step

 To convert the decimal number −9.125 into its IEEE 754 double-precision(64) and single-precision (32-bit) representation, we’ll follow these steps:

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✅ Step-by-Step Conversion of −9.125 to IEEE 754 (64-bit)

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a) Normalization of the Binary Number

1. Convert the absolute value of the number (9.125) to binary:

• 9 in binary = 1001

• 0.125 in binary:

  • 0.125 × 2 = 0.25 → 0

  • 0.25 × 2 = 0.5 → 0

  • 0.5 × 2 = 1.0 → 1
    → So, 0.125 = 0.001

✅ Therefore,
9.125 in binary = 1001.001

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2. Normalize it (move the binary point after the first 1):

1001.001 → 1.001001 × 2³

✅ So, the normalized binary form is:

1.001001 × 2³

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b) Determination of the Sign Bit, Exponent, and Mantissa

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1. Sign Bit (s):

• The number is negative → sign bit = 1

✅ s = 1

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2. Exponent (e):

IEEE 754 double-precision uses:

• 11 bits for exponent

• Bias = 1023

Since exponent = 3 (from 1.001001 × 2³):

e = 3 + 1023 = 1026

Convert 1026 to binary:

• 1026 = 10000000010 (11 bits)

✅ Exponent = 10000000010

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3. Mantissa (Fraction):

• From the normalized form 1.001001, the fraction part is everything after the decimal point: 001001

• Fill with zeros to make it 52 bits:

0010010000000000000000000000000000000000000000000000

✅ Mantissa = 001001 followed by 46 zeros

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✅ Final IEEE 754 Double-Precision Representation

Put all parts together:

Sign (1 bit)	Exponent (11 bits)	Mantissa (52 bits)
1	10000000010	0010010000000000000000000000000000000000000000000000

 here’s −9.125 in IEEE 754 single-precision (32-bit) too.

a) Normalization (same as before)

$9.125_{10} = 1001.001_2 = 1.001001_2 \times 2^3$

b) Sign, exponent, mantissa (single precision)

• Sign bit (1 bit): number is negative → 1

• Exponent (8 bits): bias = 127 → $3 + 127 = 130$ → 10000010

• Mantissa (23 bits): fraction after the leading 1 ⇒ 001001 then pad zeros to 23 bits
  → 00100100000000000000000

Final 32-bit pattern

1 10000010 00100100000000000000000