Question

How do you convert decimals to binary and vice versa?

Answer 1

Converting between decimal and binary is a fundamental skill in computer science and electrical engineering. Here’s a detailed guide on how to do both conversions:

### Converting Decimal to Binary

To convert a decimal number to binary, follow these steps:

1. **Divide the Decimal Number by 2**: Write down the integer quotient and the remainder. The remainder will be 0 or 1.

2. **Repeat the Division**: Continue dividing the quotient by 2 until the quotient is 0. Write down each remainder.

3. **Write the Binary Number**: The binary representation is obtained by reading the remainders in reverse order (from last to first).

#### Example

Convert the decimal number 13 to binary:

1. **13 ÷ 2** = 6 remainder **1**
2. **6 ÷ 2** = 3 remainder **0**
3. **3 ÷ 2** = 1 remainder **1**
4. **1 ÷ 2** = 0 remainder **1**

Reading the remainders from bottom to top, the binary representation of 13 is **1011**.

### Converting Binary to Decimal

To convert a binary number to decimal, use the following method:

1. **Write Down the Binary Number**: List each bit (0 or 1) and their corresponding powers of 2.

2. **Multiply Each Bit by Its Power of 2**: The rightmost bit is \(2^0\), the next is \(2^1\), then \(2^2\), and so on.

3. **Sum the Results**: Add up all the products to get the decimal number.

#### Example

Convert the binary number 1011 to decimal:

1. Write down the binary number with powers of 2:

   ```
   1   0   1   1
   2^3 2^2 2^1 2^0
   ```

2. Multiply each bit by its power of 2:

   ```
   (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0)
   ```

   Which simplifies to:

   ```
   (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 8 + 0 + 2 + 1 = 11
   ```

So, the decimal representation of 1011 is 11.

### Summary

- **Decimal to Binary**: Divide the number by 2, record the remainders, and read them in reverse order.
- **Binary to Decimal**: Multiply each bit by the corresponding power of 2 and sum the results.

Feel free to ask if you need further examples or clarification on either conversion process!

Answer 2

Converting between decimal and binary is a fundamental concept in computer science and digital electronics. Here’s a detailed explanation of how to do both conversions:

### **1. Converting Decimal to Binary**

To convert a decimal number to binary, you repeatedly divide the number by 2 and record the remainder. The binary number is formed by the remainders read in reverse order.

**Steps:**

- **Step 1:** Divide the decimal number by 2.
- **Step 2:** Write down the remainder (either 0 or 1).
- **Step 3:** Update the decimal number by taking the quotient obtained from the division.
- **Step 4:** Repeat steps 1 to 3 until the quotient is 0.
- **Step 5:** The binary equivalent is the sequence of remainders read in reverse order.

**Example:** Convert 13 (decimal) to binary.

1. 13 ÷ 2 = 6, remainder = 1
2. 6 ÷ 2 = 3, remainder = 0
3. 3 ÷ 2 = 1, remainder = 1
4. 1 ÷ 2 = 0, remainder = 1

Now, reading the remainders in reverse order, you get **1101**.

So, \( 13_{10} = 1101_2 \).

### **2. Converting Binary to Decimal**

To convert a binary number to decimal, you multiply each bit by 2 raised to the power of its position (counting from right, starting with 0) and sum them up.

**Steps:**

- **Step 1:** Write down the binary number.
- **Step 2:** Starting from the rightmost digit (least significant bit), multiply each bit by 2 raised to the power of its position.
- **Step 3:** Sum all the values obtained in step 2.

**Example:** Convert 1101 (binary) to decimal.

- \( 1 \times 2^3 = 8 \)
- \( 1 \times 2^2 = 4 \)
- \( 0 \times 2^1 = 0 \)
- \( 1 \times 2^0 = 1 \)

Sum: \( 8 + 4 + 0 + 1 = 13 \).

So, \( 1101_2 = 13_{10} \).

### **Summary**

- **Decimal to Binary:** Repeated division by 2 and recording remainders.
- **Binary to Decimal:** Summing products of each bit and its corresponding power of 2.

Understanding these conversions is crucial for various applications in programming, networking, and digital circuit design.