how would you represent a binary decimal  101.1101 represent. How would you convert a base 10 decimal into a binary decimal

To understand how to represent and convert between binary and decimal numbers, let's walk through the concepts step by step.

### 1. **Binary to Decimal Conversion (for `101.1101₂`)**

A binary number is a number expressed in base 2, where each digit is either 0 or 1. To convert a binary number into a decimal number, we sum up the powers of 2 for each digit. 

The given binary number is:  
\[ 101.1101₂ \]

#### Step 1: Split into whole and fractional parts
- **Whole part:** `101₂`
- **Fractional part:** `1101₂`

#### Step 2: Convert the whole part
The binary number `101₂` is evaluated as:

\[
1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 1 \times 4 + 0 \times 2 + 1 \times 1 = 4 + 1 = 5_{10}
\]

#### Step 3: Convert the fractional part
For the fractional part `1101₂`, we use negative powers of 2:

\[
1 \times 2^{-1} + 1 \times 2^{-2} + 0 \times 2^{-3} + 1 \times 2^{-4} = \frac{1}{2} + \frac{1}{4} + 0 + \frac{1}{16} 
\]

\[
= 0.5 + 0.25 + 0 + 0.0625 = 0.8125_{10}
\]

#### Step 4: Combine the whole and fractional parts
Now, combine the whole part (`5`) and the fractional part (`0.8125`):

\[
101.1101₂ = 5.8125_{10}
\]

### 2. **Decimal to Binary Conversion (Base 10 to Binary)**

Now, let’s convert a decimal number (base 10) into a binary number.

#### Example: Convert `13.375₁₀` to binary.

#### Step 1: Convert the whole part (integer) to binary
Start with the whole part (`13₁₀`), using successive division by 2:

\[
13 \div 2 = 6 \text{ remainder } 1
\]
\[
6 \div 2 = 3 \text{ remainder } 0
\]
\[
3 \div 2 = 1 \text{ remainder } 1
\]
\[
1 \div 2 = 0 \text{ remainder } 1
\]

So, `13₁₀ = 1101₂`.

#### Step 2: Convert the fractional part to binary
Now, convert the fractional part (`0.375₁₀`) using successive multiplication by 2:

\[
0.375 \times 2 = 0.75 \text{ (integer part is 0)}
\]
\[
0.75 \times 2 = 1.5 \text{ (integer part is 1)}
\]
\[
0.5 \times 2 = 1.0 \text{ (integer part is 1)}
\]

Reading the integer parts from top to bottom, `0.375₁₀ = 0.011₂`.

#### Step 3: Combine the whole and fractional parts
Now, combine the whole part (`1101₂`) and the fractional part (`0.011₂`):

\[
13.375_{10} = 1101.011_{2}
\]

### Summary of Steps for Decimal to Binary Conversion:
1. **Convert the integer part** by successive division by 2.
2. **Convert the fractional part** by successive multiplication by 2.
3. **Combine the results** for the binary equivalent.

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### Do you have any questions on these steps, or would you like more details?

Here are 5 related questions to deepen your understanding:
1. How would you convert a negative decimal number into binary?
2. What is the binary representation of 0.1 in base 10?
3. How do you convert a binary number to hexadecimal?
4. Can binary numbers represent irrational numbers exactly?
5. What are the advantages of using binary representation in computers?

**Tip:** When converting fractional decimals to binary, keep in mind that some fractions in decimal may result in repeating binary fractions!

how would you represent a binary decimal 101.1101? How would you convert a base 10 decimal into a binary decimal?

Learn how to convert the binary decimal 101.1101 into its decimal form and understand the process of converting a base 10 decimal into a binary decimal. This guide covers key concepts like binary to decimal conversion using powers of 2 and decimal to binary conversion through successive division and multiplication.

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