How do you convert binary to decimal and vice versa?
2 Answers
Converting between binary and decimal involves understanding the different bases used in each numbering system. Here’s a detailed explanation of both conversions:
### Converting Binary to Decimal
Binary is a base-2 system that uses only two digits: 0 and 1. Each digit's position represents a power of 2, starting from the rightmost position, which is \(2^0\).
**Steps:**
1. **Write down the binary number.**
For example, consider the binary number \(1101_2\).
2. **Assign powers of 2 to each digit, starting from the right.**
\[
\begin{align*}
\text{Position:} & \quad 3 \quad 2 \quad 1 \quad 0 \\
\text{Binary:} & \quad 1 \quad 1 \quad 0 \quad 1 \\
\text{Powers of 2:} & \quad 2^3 \quad 2^2 \quad 2^1 \quad 2^0 \\
\end{align*}
\]
3. **Multiply each binary digit by its corresponding power of 2.**
\[
1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1
\]
4. **Add the results.**
\[
8 + 4 + 0 + 1 = 13
\]
So, \(1101_2\) equals \(13_{10}\) in decimal.
### Converting Decimal to Binary
To convert a decimal number to binary, you can use the method of repeated division by 2.
**Steps:**
1. **Start with the decimal number.**
For example, convert \(13_{10}\) to binary.
2. **Divide the number by 2 and record the quotient and the remainder.**
- \(13 \div 2 = 6\) remainder \(1\)
- \(6 \div 2 = 3\) remainder \(0\)
- \(3 \div 2 = 1\) remainder \(1\)
- \(1 \div 2 = 0\) remainder \(1\)
3. **Record the remainders in reverse order.**
The remainders from bottom to top give you the binary representation:
- So, \(1101_2\).
Thus, \(13_{10}\) converts to \(1101_2\) in binary.
### Summary
- **Binary to Decimal:** Multiply each digit by \(2^n\) (where \(n\) is the position from the right) and sum them up.
- **Decimal to Binary:** Divide the decimal number by 2 repeatedly, recording the remainders, and read them in reverse order.
With practice, these conversions become intuitive!
### Converting Binary to Decimal
Binary is a base-2 system that uses only two digits: 0 and 1. Each digit's position represents a power of 2, starting from the rightmost position, which is \(2^0\).
**Steps:**
1. **Write down the binary number.**
For example, consider the binary number \(1101_2\).
2. **Assign powers of 2 to each digit, starting from the right.**
\[
\begin{align*}
\text{Position:} & \quad 3 \quad 2 \quad 1 \quad 0 \\
\text{Binary:} & \quad 1 \quad 1 \quad 0 \quad 1 \\
\text{Powers of 2:} & \quad 2^3 \quad 2^2 \quad 2^1 \quad 2^0 \\
\end{align*}
\]
3. **Multiply each binary digit by its corresponding power of 2.**
\[
1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1
\]
4. **Add the results.**
\[
8 + 4 + 0 + 1 = 13
\]
So, \(1101_2\) equals \(13_{10}\) in decimal.
### Converting Decimal to Binary
To convert a decimal number to binary, you can use the method of repeated division by 2.
**Steps:**
1. **Start with the decimal number.**
For example, convert \(13_{10}\) to binary.
2. **Divide the number by 2 and record the quotient and the remainder.**
- \(13 \div 2 = 6\) remainder \(1\)
- \(6 \div 2 = 3\) remainder \(0\)
- \(3 \div 2 = 1\) remainder \(1\)
- \(1 \div 2 = 0\) remainder \(1\)
3. **Record the remainders in reverse order.**
The remainders from bottom to top give you the binary representation:
- So, \(1101_2\).
Thus, \(13_{10}\) converts to \(1101_2\) in binary.
### Summary
- **Binary to Decimal:** Multiply each digit by \(2^n\) (where \(n\) is the position from the right) and sum them up.
- **Decimal to Binary:** Divide the decimal number by 2 repeatedly, recording the remainders, and read them in reverse order.
With practice, these conversions become intuitive!
Converting between binary and decimal systems is a fundamental concept in computer science and digital electronics. Here’s a detailed explanation of how to perform these conversions:
### Converting Binary to Decimal
**Binary** is a base-2 number system that uses only two digits: 0 and 1. **Decimal** is a base-10 number system that uses ten digits: 0 through 9. To convert a binary number to decimal, follow these steps:
1. **Write down the binary number**. For example, let's convert `1101` to decimal.
2. **Identify the position of each bit**. In binary, the rightmost bit is the least significant bit (LSB) and its position is 0. Each position to the left increases by 1. For `1101`, the positions are 3, 2, 1, and 0, respectively.
3. **Multiply each bit by 2 raised to the power of its position**.
- For `1101`:
- Bit at position 3 (from the right): `1 × 2^3 = 1 × 8 = 8`
- Bit at position 2: `1 × 2^2 = 1 × 4 = 4`
- Bit at position 1: `0 × 2^1 = 0 × 2 = 0`
- Bit at position 0: `1 × 2^0 = 1 × 1 = 1`
4. **Sum the results**:
- `8 + 4 + 0 + 1 = 13`
So, the binary number `1101` converts to the decimal number `13`.
### Converting Decimal to Binary
To convert a decimal number to binary, follow these steps:
1. **Start with the decimal number**. For example, let's convert `13` to binary.
2. **Divide the number by 2** and keep track of the quotient and the remainder. Continue dividing the quotient by 2 until you get a quotient of 0. Record the remainders.
- `13 ÷ 2 = 6` with a remainder of `1`
- `6 ÷ 2 = 3` with a remainder of `0`
- `3 ÷ 2 = 1` with a remainder of `1`
- `1 ÷ 2 = 0` with a remainder of `1`
3. **Write the remainders in reverse order** to get the binary number.
- The remainders in reverse order are `1101`.
So, the decimal number `13` converts to the binary number `1101`.
### Summary
- **Binary to Decimal**: Multiply each bit by `2` raised to the power of its position and sum the results.
- **Decimal to Binary**: Divide the decimal number by `2`, keep track of remainders, and write the remainders in reverse order.
By following these steps, you can convert between binary and decimal systems efficiently.
### Converting Binary to Decimal
**Binary** is a base-2 number system that uses only two digits: 0 and 1. **Decimal** is a base-10 number system that uses ten digits: 0 through 9. To convert a binary number to decimal, follow these steps:
1. **Write down the binary number**. For example, let's convert `1101` to decimal.
2. **Identify the position of each bit**. In binary, the rightmost bit is the least significant bit (LSB) and its position is 0. Each position to the left increases by 1. For `1101`, the positions are 3, 2, 1, and 0, respectively.
3. **Multiply each bit by 2 raised to the power of its position**.
- For `1101`:
- Bit at position 3 (from the right): `1 × 2^3 = 1 × 8 = 8`
- Bit at position 2: `1 × 2^2 = 1 × 4 = 4`
- Bit at position 1: `0 × 2^1 = 0 × 2 = 0`
- Bit at position 0: `1 × 2^0 = 1 × 1 = 1`
4. **Sum the results**:
- `8 + 4 + 0 + 1 = 13`
So, the binary number `1101` converts to the decimal number `13`.
### Converting Decimal to Binary
To convert a decimal number to binary, follow these steps:
1. **Start with the decimal number**. For example, let's convert `13` to binary.
2. **Divide the number by 2** and keep track of the quotient and the remainder. Continue dividing the quotient by 2 until you get a quotient of 0. Record the remainders.
- `13 ÷ 2 = 6` with a remainder of `1`
- `6 ÷ 2 = 3` with a remainder of `0`
- `3 ÷ 2 = 1` with a remainder of `1`
- `1 ÷ 2 = 0` with a remainder of `1`
3. **Write the remainders in reverse order** to get the binary number.
- The remainders in reverse order are `1101`.
So, the decimal number `13` converts to the binary number `1101`.
### Summary
- **Binary to Decimal**: Multiply each bit by `2` raised to the power of its position and sum the results.
- **Decimal to Binary**: Divide the decimal number by `2`, keep track of remainders, and write the remainders in reverse order.
By following these steps, you can convert between binary and decimal systems efficiently.