Problem 1. (a) Suppose the binary (base two) representation of a number x is 1.1011×two110 (where the exponent is also written in base two). What is the base ten representation of the number? (b) How is fifty-seven fourths represented in base four?

Let's solve both parts of this problem step by step.

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### Part (a): Base Two to Base Ten Conversion

We are given a binary number \( x = 1.1011 \times 2^{\text{two} 110} \). The exponent \( 110 \) is in binary, so let's break it down.

#### Step 1: Convert the exponent from binary to decimal.

The exponent \( 110_2 \) in binary equals \( 6_{10} \) in decimal. This means that the number is \( 1.1011_2 \times 2^6 \).

#### Step 2: Convert the binary fraction \( 1.1011_2 \) to decimal.

The number \( 1.1011_2 \) can be broken into its components:

\[
1.1011_2 = 1 \times 2^0 + 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} + 1 \times 2^{-4}
\]

This simplifies to:

\[
1.1011_2 = 1 + \frac{1}{2} + \frac{0}{4} + \frac{1}{8} + \frac{1}{16} = 1 + 0.5 + 0 + 0.125 + 0.0625 = 1.6875
\]

So, \( 1.1011_2 = 1.6875_{10} \) in decimal.

#### Step 3: Multiply by \( 2^6 \).

Now, multiply \( 1.6875_{10} \) by \( 2^6 \):

\[
1.6875 \times 64 = 108
\]

Thus, the base ten representation of \( x \) is \( \boxed{108} \).

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### Part (b): Representing \( \frac{57}{4} \) in Base Four

We need to convert the fraction \( \frac{57}{4} \) to base four.

#### Step 1: Simplify \( \frac{57}{4} \).

First, simplify the fraction:

\[
\frac{57}{4} = 14.25
\]

#### Step 2: Convert the integer part (14) to base four.

To convert \( 14_{10} \) to base four, divide 14 by 4:

\[
14 \div 4 = 3 \text{ remainder } 2
\]
\[
3 \div 4 = 0 \text{ remainder } 3
\]

So, \( 14_{10} = 32_4 \).

#### Step 3: Convert the decimal part (0.25) to base four.

Now, convert the decimal part \( 0.25_{10} \) to base four. Multiply \( 0.25 \) by 4:

\[
0.25 \times 4 = 1
\]

Thus, \( 0.25_{10} = 1_4 \).

#### Step 4: Combine the results.

The base four representation of \( 14.25_{10} \) is \( 32.1_4 \).

Thus, \( \frac{57}{4} \) is represented in base four as \( \boxed{32.1_4} \).

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### Summary:
- (a) The base ten representation of \( 1.1011 \times 2^{110_2} \) is \( 108 \).
- (b) The base four representation of \( \frac{57}{4} \) is \( 32.1_4 \).

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Would you like more details on any part of the solution or further explanation? 

Here are five questions to explore this topic further:

1. How do you convert a binary number with a fractional part to base ten?
2. Why is \( 110_2 \) equal to \( 6_{10} \) in decimal?
3. What are the steps to convert a decimal fraction to a different base?
4. How does the base-four system differ from the base-ten system in representing fractions?
5. Can you generalize the method for converting binary numbers with floating-point notation to other bases?

**Tip:** When converting fractions to other bases, multiplying the fractional part repeatedly by the base helps extract the digits after the point!

Step-by-step solution to converting a binary number (1.1011 × 2^110) to base ten and representing 57/4 in base four. Learn essential base conversion techniques, including binary fractions and positional notation.

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