Number Systems Conversion

To convert decimal to octal, we have to learn about both the number systems first. A number with base 8 is the octal number and a number with base 10 is the decimal number. Here we will convert a decimal number to an equivalent octal number. It is the same as converting any decimal number to binary or decimal to hexadecimal.

In decimal to binary, we divide the number by 2, in decimal to hexadecimal we divide the number by 16. In case of decimal to octal, we divide the number by 8 and write the remainders in the reverse order to get the equivalent octal number.

Decimal Number: All the numbers to the base ten are called decimal numbers. These are the commonly used numbers, which are 0-9. It has both integer part and the decimal part. It is separated by a decimal point (.). Numbers on the left of the decimal are integers and numbers on the right of the decimal is the decimal part. Example: (236.89)₁₀, (54.2)₁₀, etc.

Octal number: These are the numbers with base 8. If x is a number then the octal number is denoted as x₈. It contains digits from 0 to 7. Example: (212)₈, (121)₈, etc.

Octal

Octal number is a number expressed in the base 8 numeral system. Octal number's digits have 8 symbols: 0,1,2,3,4,5,6,7. Each digit of an octal number counts a power of 16.

Octal number example: 627₈ = 6×8² + 2×8¹ + 7×8⁰ = 1580₁₀

Decimal

Decimal number is a number expressed in the base 10 numeral system. Decimal number's digits have 10 symbols: 0,1,2,3,4,5,6,7,8,9. Each digit of a decimal number counts a power of 10.

Decimal number example: 653₁₀ = 6×10² + 5×10¹ + 3×10⁰

How to convert from octal to decimal

A regular decimal number is the sum of the digits multiplied with 10ⁿ.

Example #1: 137 in base 10 is equal to each digit multiplied with its corresponding 10ⁿ:

137₁₀ = 1×10² + 3×10¹ + 7×10⁰ = 100 + 30 + 7

Octal numbers are read the same way, but each digit counts 8ⁿ instead of 10ⁿ.

Example #2: 37 in base 8 is equal to each digit multiplied with its corresponding 8ⁿ:

37₈ = 3×8¹ + 7×8⁰ = 24 + 7 = 31

Example #3: 7014 in base 8 is equal to each digit multiplied with its corresponding power of 8:

7014₈ = 7×8³ + 0×8² + 1×8¹ + 4×8⁰ = 3584 + 0 + 8 + 4 = 3596

Example

Understanding Octal-to-Decimal Conversion

Octal-to-decimal conversion involves converting numbers in octal (base 8) into their equivalent decimal (base 10) values. Octal uses digits 0-7 to represent values, while decimal uses digits 0-9.

The general approach to converting octal numbers to decimal includes:

Write down the octal number.
Assign powers of 8 to each digit based on its position (rightmost digit is \(8^0\), next is \(8^1\), and so on).
Multiply each digit by \(8^n\), where \(n\) is the position of the digit, starting from 0.
Add the results to get the decimal equivalent.

Steps for Octal-to-Decimal Conversion

Step 1: Write down the octal number.

Step 2: Assign powers of 8 to each digit starting from the rightmost digit.

Step 3: Multiply each digit by \(8^n\), where \(n\) is the positional power.

Step 4: Add all the results to get the decimal equivalent.

Example: Converting Octal to Decimal

Convert \( 345 \) (octal) to decimal:

Step 1: Write the octal number \( 345 \).
Step 2: Assign powers of 8:
- \( 5 = 5 \) at \(8^0\)
- \( 4 = 4 \) at \(8^1\)
- \( 3 = 3 \) at \(8^2\)
Step 3: Multiply each digit by \(8^n\):
- \( 5 \times 8^0 = 5 \times 1 = 5 \)
- \( 4 \times 8^1 = 4 \times 8 = 32 \)
- \( 3 \times 8^2 = 3 \times 64 = 192 \)
Step 4: Add the results: \( 192 + 32 + 5 = 229 \).
Final decimal result: \( 229 \).

Conversion Table for Quick Reference

Here is a quick reference for octal digits and their decimal equivalents:

0 → 0
1 → 1
2 → 2
3 → 3
4 → 4
5 → 5
6 → 6
7 → 7

Applications of Octal-to-Decimal Conversion

Octal-to-decimal conversion is commonly used in:

Understanding file permissions in Unix/Linux systems.
Interpreting memory addresses in computing systems.
Working with digital systems that use base-8 numbering.

Practice Problem

Convert \( 127 \) (octal) to decimal:

Solution:
- \( 7 = 7 \) at \(8^0\)
- \( 2 = 2 \) at \(8^1\)
- \( 1 = 1 \) at \(8^2\)
Multiply and add:
- \( 7 \times 8^0 = 7 \times 1 = 7 \)
- \( 2 \times 8^1 = 2 \times 8 = 16 \)
- \( 1 \times 8^2 = 1 \times 64 = 64 \)
- \( 64 + 16 + 7 = 87 \).
Final decimal result: \( 87 \).

Octal-to-Binary Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a single octal digit to its binary equivalent.	Write down the octal digit. Convert the octal digit to its 3-bit binary equivalent.	For `5`: ‘5’ in binary: `101`. Binary: `101`.
Handling Larger Numbers	Converting a multi-digit octal number to binary.	Convert each octal digit to its 3-bit binary equivalent. Combine the binary equivalents into one binary string.	For `275`: ‘2’ in binary: `010`. ‘7’ in binary: `111`. ‘5’ in binary: `101`. Combine: `010 111 101`. Binary: `010111101`.
Verifying Conversion	Checking the accuracy of octal-to-binary conversion.	Group the binary string into 3-bit groups from the right. Convert each 3-bit group back to its octal equivalent. Ensure the resulting octal number matches the original input.	For `010111101` (binary for `275`): Group into 3 bits: `010 111 101`. ‘010’ in octal: `2`. ‘111’ in octal: `7`. ‘101’ in octal: `5`. Octal: `275`, matches the original input.
Handling Special Cases	Converting octal numbers with leading zeroes or specific patterns to binary.	Include all leading zeroes in the binary equivalent for precision. Convert each digit individually to ensure no loss of information.	For `007`: ‘0’ in binary: `000`. ‘0’ in binary: `000`. ‘7’ in binary: `111`. Combine: `000 000 111`. Binary: `000000111`.
Applications	Understanding where octal-to-binary conversion is used in computer science and digital systems.	Octal and binary are widely used in computing systems for compact representation of data. Commonly applied in permissions (e.g., file permissions in UNIX), low-level programming, and memory addressing.	Example: Converting octal permissions `754` to binary: ‘7’ in binary: `111`. ‘5’ in binary: `101`. ‘4’ in binary: `100`. Combine: `111 101 100`. Binary: `111101100`.