How do you convert octal to decimal and vice versa?
2 Answers
### Converting Octal to Decimal
**Octal** is a base-8 numbering system, meaning it uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. **Decimal** is a base-10 system, using ten digits: 0 through 9. To convert an octal number to its decimal equivalent, follow these steps:
1. **Identify the octal number** and its digits.
2. **Multiply each digit** of the octal number by \(8\) raised to the power of its positional index (starting from 0 for the rightmost digit).
3. **Sum** all the products to get the decimal equivalent.
#### Example 1: Convert Octal 157 to Decimal
1. Write the octal number: \(157\).
2. Break it down by each digit:
- \(1 \times 8^2\)
- \(5 \times 8^1\)
- \(7 \times 8^0\)
3. Calculate each term:
- \(1 \times 8^2 = 1 \times 64 = 64\)
- \(5 \times 8^1 = 5 \times 8 = 40\)
- \(7 \times 8^0 = 7 \times 1 = 7\)
4. Sum all the products:
- \(64 + 40 + 7 = 111\)
So, the decimal equivalent of the octal number \(157\) is **111**.
#### Formula Recap
Given an octal number \(O\), its decimal equivalent \(D\) is:
\[
D = \sum_{i=0}^{n} O_i \times 8^i
\]
where \(O_i\) is the \(i\)-th digit of the octal number starting from the right (with \(i = 0\)).
### Converting Decimal to Octal
To convert a decimal number to an octal number, use the following steps:
1. **Divide** the decimal number by 8.
2. **Record the remainder**.
3. **Divide the quotient** by 8 again and record the next remainder.
4. **Repeat** this process until the quotient becomes 0.
5. The **octal number** is the sequence of remainders read in reverse (from last to first).
#### Example 2: Convert Decimal 111 to Octal
1. Divide 111 by 8:
- Quotient: \(111 \div 8 = 13\)
- Remainder: \(111 \mod 8 = 7\)
2. Divide the quotient 13 by 8:
- Quotient: \(13 \div 8 = 1\)
- Remainder: \(13 \mod 8 = 5\)
3. Divide the quotient 1 by 8:
- Quotient: \(1 \div 8 = 0\)
- Remainder: \(1 \mod 8 = 1\)
4. Since the quotient is now 0, stop the division.
5. Read the remainders in reverse order: **157**.
So, the octal equivalent of the decimal number \(111\) is **157**.
### Summary
- **Octal to Decimal:** Multiply each digit by \(8^n\) where \(n\) is the position of the digit, starting from 0 on the right.
- **Decimal to Octal:** Divide the decimal number by 8 repeatedly, record the remainders, and read them in reverse order.
**Octal** is a base-8 numbering system, meaning it uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. **Decimal** is a base-10 system, using ten digits: 0 through 9. To convert an octal number to its decimal equivalent, follow these steps:
1. **Identify the octal number** and its digits.
2. **Multiply each digit** of the octal number by \(8\) raised to the power of its positional index (starting from 0 for the rightmost digit).
3. **Sum** all the products to get the decimal equivalent.
#### Example 1: Convert Octal 157 to Decimal
1. Write the octal number: \(157\).
2. Break it down by each digit:
- \(1 \times 8^2\)
- \(5 \times 8^1\)
- \(7 \times 8^0\)
3. Calculate each term:
- \(1 \times 8^2 = 1 \times 64 = 64\)
- \(5 \times 8^1 = 5 \times 8 = 40\)
- \(7 \times 8^0 = 7 \times 1 = 7\)
4. Sum all the products:
- \(64 + 40 + 7 = 111\)
So, the decimal equivalent of the octal number \(157\) is **111**.
#### Formula Recap
Given an octal number \(O\), its decimal equivalent \(D\) is:
\[
D = \sum_{i=0}^{n} O_i \times 8^i
\]
where \(O_i\) is the \(i\)-th digit of the octal number starting from the right (with \(i = 0\)).
### Converting Decimal to Octal
To convert a decimal number to an octal number, use the following steps:
1. **Divide** the decimal number by 8.
2. **Record the remainder**.
3. **Divide the quotient** by 8 again and record the next remainder.
4. **Repeat** this process until the quotient becomes 0.
5. The **octal number** is the sequence of remainders read in reverse (from last to first).
#### Example 2: Convert Decimal 111 to Octal
1. Divide 111 by 8:
- Quotient: \(111 \div 8 = 13\)
- Remainder: \(111 \mod 8 = 7\)
2. Divide the quotient 13 by 8:
- Quotient: \(13 \div 8 = 1\)
- Remainder: \(13 \mod 8 = 5\)
3. Divide the quotient 1 by 8:
- Quotient: \(1 \div 8 = 0\)
- Remainder: \(1 \mod 8 = 1\)
4. Since the quotient is now 0, stop the division.
5. Read the remainders in reverse order: **157**.
So, the octal equivalent of the decimal number \(111\) is **157**.
### Summary
- **Octal to Decimal:** Multiply each digit by \(8^n\) where \(n\) is the position of the digit, starting from 0 on the right.
- **Decimal to Octal:** Divide the decimal number by 8 repeatedly, record the remainders, and read them in reverse order.
Sure, let's dive into converting between octal (base-8) and decimal (base-10) number systems.
### **Converting Octal to Decimal**
To convert an octal number to decimal, follow these steps:
1. **Write Down the Octal Number**: Take the octal number and write it out. For example, let's use \( 257_8 \) (where the subscript 8 indicates that it's an octal number).
2. **Identify the Place Values**: Each digit in the octal number has a place value. The rightmost digit has the place value \( 8^0 \), the next digit to the left has the place value \( 8^1 \), and so on.
3. **Multiply Each Digit by Its Place Value**: Multiply each digit of the octal number by the place value of its position. For example:
\[
257_8 = 2 \cdot 8^2 + 5 \cdot 8^1 + 7 \cdot 8^0
\]
4. **Calculate the Powers of 8**: Compute the power of 8 for each position:
\[
8^2 = 64
\]
\[
8^1 = 8
\]
\[
8^0 = 1
\]
5. **Multiply and Sum Up**: Perform the multiplication and add up the results:
\[
2 \cdot 64 = 128
\]
\[
5 \cdot 8 = 40
\]
\[
7 \cdot 1 = 7
\]
\[
128 + 40 + 7 = 175
\]
So, \( 257_8 \) in decimal is \( 175_{10} \).
### **Converting Decimal to Octal**
To convert a decimal number to octal, follow these steps:
1. **Divide the Decimal Number by 8**: Start with the decimal number and divide it by 8. For example, let's convert \( 175_{10} \) to octal.
2. **Record the Remainder**: The remainder of this division is the least significant digit (rightmost digit) of the octal number.
3. **Update the Quotient**: Replace the decimal number with the quotient (result of the division).
4. **Repeat**: Repeat the division and remainder steps with the new quotient until the quotient is zero.
5. **Write Down the Remainders**: The octal number is formed by writing down all remainders in reverse order (from last remainder to first).
Here’s how this works for \( 175_{10} \):
- Divide 175 by 8:
\[
175 \div 8 = 21 \text{ with a remainder of } 7
\]
- Divide the quotient 21 by 8:
\[
21 \div 8 = 2 \text{ with a remainder of } 5
\]
- Divide the quotient 2 by 8:
\[
2 \div 8 = 0 \text{ with a remainder of } 2
\]
- Writing the remainders from last to first gives us \( 257_8 \).
So, \( 175_{10} \) in octal is \( 257_8 \).
### Summary
- **Octal to Decimal**: Multiply each digit by \( 8 \) raised to the power of its position index, then sum the results.
- **Decimal to Octal**: Divide the decimal number by \( 8 \), record the remainders, and read them in reverse order.
These methods will help you convert between octal and decimal systems easily.
### **Converting Octal to Decimal**
To convert an octal number to decimal, follow these steps:
1. **Write Down the Octal Number**: Take the octal number and write it out. For example, let's use \( 257_8 \) (where the subscript 8 indicates that it's an octal number).
2. **Identify the Place Values**: Each digit in the octal number has a place value. The rightmost digit has the place value \( 8^0 \), the next digit to the left has the place value \( 8^1 \), and so on.
3. **Multiply Each Digit by Its Place Value**: Multiply each digit of the octal number by the place value of its position. For example:
\[
257_8 = 2 \cdot 8^2 + 5 \cdot 8^1 + 7 \cdot 8^0
\]
4. **Calculate the Powers of 8**: Compute the power of 8 for each position:
\[
8^2 = 64
\]
\[
8^1 = 8
\]
\[
8^0 = 1
\]
5. **Multiply and Sum Up**: Perform the multiplication and add up the results:
\[
2 \cdot 64 = 128
\]
\[
5 \cdot 8 = 40
\]
\[
7 \cdot 1 = 7
\]
\[
128 + 40 + 7 = 175
\]
So, \( 257_8 \) in decimal is \( 175_{10} \).
### **Converting Decimal to Octal**
To convert a decimal number to octal, follow these steps:
1. **Divide the Decimal Number by 8**: Start with the decimal number and divide it by 8. For example, let's convert \( 175_{10} \) to octal.
2. **Record the Remainder**: The remainder of this division is the least significant digit (rightmost digit) of the octal number.
3. **Update the Quotient**: Replace the decimal number with the quotient (result of the division).
4. **Repeat**: Repeat the division and remainder steps with the new quotient until the quotient is zero.
5. **Write Down the Remainders**: The octal number is formed by writing down all remainders in reverse order (from last remainder to first).
Here’s how this works for \( 175_{10} \):
- Divide 175 by 8:
\[
175 \div 8 = 21 \text{ with a remainder of } 7
\]
- Divide the quotient 21 by 8:
\[
21 \div 8 = 2 \text{ with a remainder of } 5
\]
- Divide the quotient 2 by 8:
\[
2 \div 8 = 0 \text{ with a remainder of } 2
\]
- Writing the remainders from last to first gives us \( 257_8 \).
So, \( 175_{10} \) in octal is \( 257_8 \).
### Summary
- **Octal to Decimal**: Multiply each digit by \( 8 \) raised to the power of its position index, then sum the results.
- **Decimal to Octal**: Divide the decimal number by \( 8 \), record the remainders, and read them in reverse order.
These methods will help you convert between octal and decimal systems easily.