Number to Octal
A base (also known as radix) of a number system tells us about the unique or different symbols and notations it uses to represent a value. For example, the base 8 tells us that it uses 8 different numbers i.e. 0 to 7 to represent a value. The most commonly used base in the number system is the base 10 which uses 10 different digits i.e. 0 to 10 to represent a value. Bases can be 0, positive, negative, complex, and non-integral. Other frequently used bases are base 2, base 8, and base 16. They are used in computing and are known as binary, octal, and hexadecimal respectively.
We can convert one base into the other base by following the simple conversion steps. Here, we will discuss the steps on how to convert a number from decimal to octal, octal to decimal, octal to binary, binary to octal, octal to hexadecimal, hexadecimal to octal, and so on.
Let us begin with the octal to decimal conversion
Decimal to Octal Conversion
To convert decimal to octal, we use the following steps:
The steps given below show how to convert a number from the decimal system to the octal system.
Example
Convert 560₁₀ into the octal system
Step 1: Observe the base of the required number.
As we have to convert the given decimal number into the octal number system, the base of the required number is 8.
Step 2: Divide the given decimal number by the base of the required number and list down the quotient and remainder in the form of quotient-remainder.
Repeat the process (dividing the quotient value again by base 8) till we get the quotient value less than the base.
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Octal to Decimal Conversion
To convert a number from an octal system to a decimal system, multiply each digit of the given octal number by the reducing power of 8.
Let us discuss octal to decimal conversion through an example:
Example:
Convert 764₈ into the decimal system
764₈ = 7 x 8² + 6 x 8¹ + 4 x 8º
= 7 x 64 + 6 x 8 + 4 x 1
= 448 + 48 + 4
= 500
Binary to Octal Conversion
To convert a number from binary system to the octal system, we use the following binary to the octal table:
Example:
Convert 1010111100₂ to octal number system
To convert binary to octal, we will group the binary digits by three, starting from the far right to left. You can add trailing zeros to the left of the last digit to complete the last group of three if required. Then replace each group of three digits with the equivalent octal digit.
1010111100₂ in a group of 3 is represented by \[\overline{001}\] \[\overline{010}\] \[\overline{111}\] \[\overline{100}\]. Here we have added trailing zeros to the left of the last digit to complete the last group of three.
Let us now convert base₂ to base₈
Octal to Binary Conversion
To convert a number from the octal system to the decimal number system, we simply need to follow the 2 steps given below:
Convert the given octal number into the decimal number system.
Convert the decimal number that is obtained in Step 1 into the binary number.
Let us understand octal to binary conversion through an example:
Example: Convert 205₈ into the decimal number system.
Step 1: Convert a number from the octal system to the decimal system
205₈ = 2 x 8² + 0 x 8¹ + 5 x 8º
= 2 x 64 + 0 x 8 + 5 x 1
= 128 + 0 + 5
= 133₁₀
Step 2: Convert a decimal number that is obtained in the previous step into the binary system.
To get the equivalent binary number, we will list down all the remainder values from bottom to top.
Octal to Hexadecimal Conversion
To convert octal to hexadecimal or hexadecimal to octal, we need to memorize the table given below:
Octal to Hexadecimal Conversion Steps
To convert a number from the octal system to the decimal number system, we simply need to follow the 2 steps given below:
Convert the given number from the octal system into the binary number system.
Convert the binary number that is obtained in Step 1 into the hexadecimal number system.
Let us understand octal to hexadecimal conversion through an example.
Example: Convert 536₈ into the hexadecimal decimal number system
Step 1: Convert the given number from the octal system into the binary number system
To convert a number from the octal system to the binary number system, we simply need to follow the 2 steps given below:
Convert each octal digit into the equivalent three-digit binary number. Each of the digits must be treated as a decimal value.
Combine these groups of 3 binary values into a single binary number.
Convert each octal digit into the three-digit binary number as shown below:
\[\overline{101}\] \[\overline{011}\] \[\overline{110}\]
Combine these groups of 3 binary values (starting from the far right) into a single binary number.
536₈ = 101011110₂
Step 2: Convert the binary number that is obtained in the previous step into the hexadecimal number system.
101011110₂
To convert a number from binary system to the hexadecimal system, we will group the binary digits by four, starting from the far right and proceeding to the left as shown below. You can add trailing zeros to the left of the last digit to complete the last group of four if required.
101011110₂ in a group of 4 is represented by as \[\overline{0001}\] \[\overline{0101}\] \[\overline{1110}\]. Here we have added trailing zeros to the left of the last digit to complete the last group of four.
Now, let us convert the 1101011110₂ into the base₁₆.
Hexadecimal to Octal Conversion
To perform the hexadecimal to octal conversion, we simply need to follow the below-mentioned steps.
Write each of the given hexadecimal number digits into its equivalent binary number.
Combine and make the groups of binary bits, each group containing 3 bits from right to left. You can add trailing 0’s to the left of the last digit if there are less than 3 digits in the last group.
Find the equivalent hexadecimal number of each binary group.
Let us understand hexadecimal to octal conversion, through an example.
Example: Convert (1BC)₁₆ into the octal number system
Step 1: Convert each hexadecimal digit into the four-digit binary number as shown below:
Step 2: Combine and make the groups of binary bits, each group containing 3 bits from right to left.
000, 101 , 111, 100
Step 3: Now, write down the equivalent hexadecimal number of each binary group of 3 bits.
FAQs on Base Conversion
1. What exactly is base conversion in Maths?
Base conversion is the process of changing the representation of a number from one base to another without changing its actual value. A base is simply the number of unique digits used in a number system. For example, our everyday decimal system is base-10 because it uses ten digits (0-9), while computers use a base-2 or binary system.
2. What are the most common number systems used in base conversion?
The four most common number systems you'll encounter are:
- Decimal (Base-10): Uses the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the system we use daily.
- Binary (Base-2): Uses only two digits: 0 and 1. This is the fundamental language of all digital computers.
- Octal (Base-8): Uses eight digits from 0 to 7. It's sometimes used as a shorter way to represent binary numbers.
- Hexadecimal (Base-16): Uses sixteen characters: digits 0-9 and the letters A (10), B (11), C (12), D (13), E (14), and F (15).
3. How do you convert a number from any base to the decimal (base-10) system?
To convert any number to the decimal system, you use the positional value method. You multiply each digit by its base raised to the power of its position, starting from 0 on the right. For example, to convert the binary number 1101₂ to decimal, you calculate:
(1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 8 + 4 + 0 + 1 = 13. So, 1101₂ is equal to 13 in base-10.
4. What is the method for converting a decimal (base-10) number to another base?
To convert a decimal number to another base, you use the repeated division method. Continuously divide the decimal number by the new base until the quotient becomes 0. The remainders, read from the bottom up, form the new number in that base. For instance, to convert 13₁₀ to binary (base-2), you would divide by 2 repeatedly and note the remainders.
5. Why is learning base conversion so important, especially for subjects like computer science?
Base conversion is crucial because it bridges the gap between how humans and computers handle numbers. While we use the decimal system, all computer operations are performed in binary (base-2). Programmers often use hexadecimal (base-16) or octal (base-8) as a more compact and human-friendly way to write long binary code, making it essential for understanding computer architecture, data storage, and low-level programming.
6. Is there a quick trick to convert between binary, octal, and hexadecimal systems?
Yes, there's a simple shortcut that avoids converting to decimal first. Since 8 is 2³ and 16 is 2⁴, you can group binary digits.
- Binary to Octal: Group the binary digits into sets of three from right to left, and convert each group to its octal equivalent. For example, 110101₂ becomes (110)(101), which is 65₈.
- Binary to Hexadecimal: Group the binary digits into sets of four from right to left, and convert each group. For example, 11010101₂ becomes (1101)(0101), which is D5₁₆.
This method is much faster for these specific conversions.
7. What are some common mistakes to avoid when converting bases?
Students often make a few common errors. Be careful to avoid these:
- Incorrect Positional Values: Forgetting that the position starts from 0 on the right (e.g., b⁰, b¹, b², etc.).
- Reading Remainders Wrong: When converting from decimal using repeated division, the remainders must be read from the bottom to the top to get the correct answer.
- Hexadecimal Symbols: Forgetting to use letters A-F for the numbers 10-15 in hexadecimal. For instance, writing '10' instead of 'A'.
- Calculation Errors: Simple mistakes in multiplication or division can lead to a completely wrong result. It's always good to double-check your arithmetic.
