Binary to Octal Conversion - Definition, Methods and Examples

Binary to octal conversion is a method of changing a number from binary form (made of 0s and 1s) into octal form (which uses numbers from 0 to 7). Learning how to convert binary to octal is important because computers use binary numbers, and octal makes them easy to read and work with.

Binary to octal conversion can be done directly without first changing the number into a decimal. There is a simple trick: just group the binary digits into sets of three from the right side and then write their octal values. This binary to octal conversion method is quick and easy to learn for students.

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Binary Numbers Definition

Binary number system is a method of writing numbers using only two digits that is 0 and 1. It is also called the base 2 number system. This system is mostly used in computers because they understand only two signals: ON and OFF, which are shown as 1 and 0.

Each digit in a binary number is called a "Bit". When we put many bits together, we can make bigger numbers. For example, (01101)₂, (01000010)₂, and (11000101)₂ are all binary numbers.

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Octal Numbers Definition

Octal numbers are numbers that use base 8. This means they only use digits from 0 to 7. Numbers like 8 and 9 are not used in this system. Octal numbers are important and used mostly in computer programming and electronics.

We can write each octal number using just the digits 0, 1, 2, 3, 4, 5, 6, and 7. We represent a number in octal form by writing a small 8 next to it. Binary to Octal Conversion Examples are 2145₈, 7165₈, 46₈, etc.

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Binary to Octal Conversion

In number system conversions, the binary to octal procedure is a crucial concept. One must first learn how to convert binary to octal using the binary to octal formula to understand binary to octal conversion. This approach often involves grouping binary digits into groups of three before writing them as a single octal digit. With binary to octal conversion examples, students can simply practice and learn the procedures necessary to convert binary to octal. More practice with binary to octal conversion questions will improve comprehension and clarify the concept.

How to Convert Binary to Octal?

Using two methods, we can convert binary to octal numbers. Students can choose any method they find more comfortable. Let’s understand both one by one.

Method 1: Binary to Decimal, Then Decimal to Octal

This method of binary to octal conversion has two steps. First, we convert the binary number into a decimal number. Then we convert that decimal number into octal. Here is how students can understand how to convert binary to octal using this method:

• First, write down the binary number.

• Multiply each digit by powers of 2. Start from the right, where the power is 0, then 1, 2, 3, and so on.

• Add all the values together. This gives the decimal number.

• Now, divide that decimal number by 8.

• Write down the remainder.

• Keep dividing the number by 8 until the result is 0. Write all the remainders.

• Finally, reverse the remainders. That is the octal number.

While this method is long, it's useful if students are already learning decimal numbers.

Method 2: Grouping the Binary Digits

• Start with the binary number. Make sure it only has 0s and 1s.

• Group the digits in sets of three, starting from the right.

• If the last group has less than 3 digits, add extra 0s to the left side to make a full group.

• Now look at each group and find its octal value using a binary to octal table.

• Write those values together. That is the final octal number.

This binary to octal conversion method is simple and can be used for quick conversions.

Solved Binary to Octal Conversion Examples

Example: Convert (1111110)₂ = (?)₈

Step 1: Group the binary digits in sets of three from the right side: 001 111 110

Step 2: Find octal values of each group: 001 = 1, 111 = 7, 110 = 6

Final Answer: (1111110)₂ = 176₈

Example: Convert (11101010.010)₂ = (?)₈

Step 1: Group whole number part: 11101010
→ 011 101 010
Group decimal part: 010
→ 010

Step 2: Convert to octal: 011 = 3, 101 = 5, 010 = 2
Decimal part: 010 = 2

Final Answer: (11101010.010)₂ = 352.2₈

Example: Convert (110.00101)₂ = (?)₈

Step 1: Group the whole number part: 110
→ 110
Group decimal part: 00101 → add one 0 to make it three digits each.
→ 001 010

Step 2: Convert to octal: 110 = 6, 001 = 1, 010 = 2

Final Answer: (110.00101)₂ = 6.12₈

Example: Convert (010.11101)₂ = (?)₈

Step 1: Group whole number part: 01
→ 010
Group decimal part: 11101 → Add one 0 → 111 010

Step 2: Convert to octal: 010 = 2, 111 = 7, 010 = 2

Final Answer: (010.11101)₂ = 2.72₈

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Why Convert Binary to Octal?

Binary to octal conversion is helpful because it makes long binary numbers easy to read and solve. Binary numbers are made up of just 0s and 1s, which can become very long and confusing. But when we convert them to octal form, they become shorter and simpler.

This method is also useful in programming and electronics. It helps in checking and fixing problems in computer codes. That’s why students often find questions on binary to octal conversion in exams. It saves time and makes understanding easier. It is also an important part of learning number systems.

Mistakes to Avoid in Binary to Octal Conversion

When students try to convert binary to octal, they sometimes make small mistakes that can change the answer.

• One common mistake is grouping the binary digits from the left side. But the correct way is to start grouping from the right side.

• Another mistake is not adding extra zeros when the last group has less than three digits. Every group must have exactly three binary digits, so we add zeros at the start if needed.

• Some students also get confused between decimal, binary, and octal numbers. It is important to write the correct number system while giving the answer.

• Also, it is okay to miss leading zeros in the final octal answer, but keeping them during the process helps in checking if the conversion is done correctly.

Avoiding these common mistakes during examinations is important for students to prevent losing marks.

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Properties of Binary to Octal Conversion

Binary to octal conversion has some special properties, as outlined below, that make it useful and easy to understand.

1. Base Relationship

Binary numbers are written in base 2 and use only 0 and 1. Octal numbers are written in base 8 and use digits from 0 to 7. Since 8 is equal to 2 × 2 × 2 (or 2³), each octal digit can be made from three binary digits.

2. Grouping of Digits

To change a binary number into octal, we group the binary digits in sets of three. We start grouping from the right side. If the last group has less than three digits, we add extra zeros to the left to complete it.

3. Compact Form

Octal numbers are shorter than binary numbers. A big binary number becomes much easier to read when it is written in octal. This is helpful when working with large numbers, especially in computer systems.

4. Exact Conversion

When we convert binary to octal, we get the exact answer. No digits are lost or rounded off. This makes the method safe and accurate for all types of questions and real-life uses.

5. Quick and Easy

Since one octal digit is equal to three binary digits, the conversion becomes fast. We do not have to do any long calculations. Just group the digits and write the matching octal number.

6. Useful in Computers

Octal numbers are often used in computer programming. For example, in Linux or Unix systems, file permissions are shown using octal numbers. So learning this method is helpful even outside of exams.

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