Convert each decimal fraction to binary using the sum-of-weights method

Convert each decimal fraction to binary using the sum-of-weights method:

(a) 0.26

(b) 0.762

(c) 0.0975

The Correct Answer and Explanation is :

To convert each decimal fraction to binary using the sum-of-weights method, we follow these steps:

Steps for Conversion:

1. Multiply the decimal fraction by 2.
2. Note the integer part (0 or 1); this is the next binary digit.
3. Retain the fractional part and repeat step 1 until the fraction reaches 0 or a repeating cycle is noticed.

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(a) 0.26 in Binary

1. 0.26×2=0.520.26 \times 2 = 0.52 → Integer: 0
2. 0.52×2=1.040.52 \times 2 = 1.04 → Integer: 1
3. 0.04×2=0.080.04 \times 2 = 0.08 → Integer: 0
4. 0.08×2=0.160.08 \times 2 = 0.16 → Integer: 0
5. 0.16×2=0.320.16 \times 2 = 0.32 → Integer: 0
6. 0.32×2=0.640.32 \times 2 = 0.64 → Integer: 0
7. 0.64×2=1.280.64 \times 2 = 1.28 → Integer: 1
8. 0.28×2=0.560.28 \times 2 = 0.56 → Integer: 0
9. 0.56×2=1.120.56 \times 2 = 1.12 → Integer: 1

Binary Representation: 0.0100001101… (approximated)

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(b) 0.762 in Binary

1. 0.762×2=1.5240.762 \times 2 = 1.524 → Integer: 1
2. 0.524×2=1.0480.524 \times 2 = 1.048 → Integer: 1
3. 0.048×2=0.0960.048 \times 2 = 0.096 → Integer: 0
4. 0.096×2=0.1920.096 \times 2 = 0.192 → Integer: 0
5. 0.192×2=0.3840.192 \times 2 = 0.384 → Integer: 0
6. 0.384×2=0.7680.384 \times 2 = 0.768 → Integer: 0
7. 0.768×2=1.5360.768 \times 2 = 1.536 → Integer: 1
8. 0.536×2=1.0720.536 \times 2 = 1.072 → Integer: 1

Binary Representation: 0.11000011… (approximated)

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(c) 0.0975 in Binary

1. 0.0975×2=0.1950.0975 \times 2 = 0.195 → Integer: 0
2. 0.195×2=0.390.195 \times 2 = 0.39 → Integer: 0
3. 0.39×2=0.780.39 \times 2 = 0.78 → Integer: 0
4. 0.78×2=1.560.78 \times 2 = 1.56 → Integer: 1
5. 0.56×2=1.120.56 \times 2 = 1.12 → Integer: 1
6. 0.12×2=0.240.12 \times 2 = 0.24 → Integer: 0
7. 0.24×2=0.480.24 \times 2 = 0.48 → Integer: 0
8. 0.48×2=0.960.48 \times 2 = 0.96 → Integer: 0
9. 0.96×2=1.920.96 \times 2 = 1.92 → Integer: 1

Binary Representation: 0.0001100110… (approximated)

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Explanation:

The sum-of-weights method represents decimal fractions as sums of negative powers of two. Each binary digit corresponds to a fraction of 1/2,1/4,1/8,…1/2, 1/4, 1/8, …. This method efficiently represents fractions in binary, but some decimal numbers have repeating patterns in binary, similar to how 1/31/3 repeats in decimal (0.333…).

Now, I’ll generate an image illustrating these conversions.

Here is the mathematical illustration showing the step-by-step conversion of decimal fractions to binary using the sum-of-weights method. Let me know if you need any modifications or explanations!