Convert $$ 101.011_{ ext {two }} $$ to base ten

Question

Convert $$ 101.011_{	ext {two }} $$ to base ten

UpStudy AI · Accepted Answer

**Step 1: Separate the integer and fractional parts**

The binary number is $$101.011_{\text{two}}$$. Its integer part is $$101_{\text{two}}$$ and the fractional part is $$0.011_{\text{two}}$$.

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**Step 2: Convert the integer part**

For $$101_{\text{two}}$$:

$$
101_{\text{two}} = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0
$$

Calculate each term:

- $$1 \times 2^2 = 4$$
- $$0 \times 2^1 = 0$$
- $$1 \times 2^0 = 1$$

Sum the values:

$$
4 + 0 + 1 = 5
$$

Thus, the integer part is $$5$$ in base ten.

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**Step 3: Convert the fractional part**

For $$0.011_{\text{two}}$$, the digits after the decimal point correspond to the fractions with denominators $$2^1$$, $$2^2$$, $$2^3$$, respectively.

$$
0.011_{\text{two}} = 0 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3}
$$

Calculate each fraction:

- $$0 \times 2^{-1} = 0 \times \frac{1}{2} = 0$$
- $$1 \times 2^{-2} = 1 \times \frac{1}{4} = \frac{1}{4}$$
- $$1 \times 2^{-3} = 1 \times \frac{1}{8} = \frac{1}{8}$$

Sum the fractions:

$$
0 + \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}
$$

Thus, the fractional part is $$\frac{3}{8}$$ in base ten.

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**Step 4: Combine the integer and fractional parts**

The total decimal value is:

$$
5 + \frac{3}{8} = \frac{40}{8} + \frac{3}{8} = \frac{43}{8}
$$

In decimal form, this is:

$$
\frac{43}{8} = 5.375
$$

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**Final Answer:** $$ 101.011_{\text{two}} $$ equals $$ \frac{43}{8} $$ or $$ 5.375 $$ in base ten.
$$101.011_{\text{two}}$$ is equal to $$5.375$$ in base ten.

Convert \( 101.011_{\text {two }} \) to base ten

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