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Decimal and octal are two numbering systems used to represent numerical values. The decimal system is widely used in everyday life and is a base 10 numbering system, while the octal system is less common and is a base 8 numbering system.
In this article, we will explore the basics of decimal and octal numbering systems, their relationship, and provide examples of how to convert between the two. I have placed the tool at the end of the article.
Decimal Number System:
The decimal number system is a base 10 numbering system, meaning it uses ten digits (0-9) to represent numbers. The rightmost digit represents the ones place, the next digit to the left represents the tens place, and so on, with each digit representing a power of ten.
For example, the number 123 can be broken down as follows:
1 x 10^2 + 2 x 10^1 + 3 x 10^0 = 100 + 20 + 3 = 123
The decimal system is commonly used in everyday life for counting and arithmetic operations, such as addition, subtraction, multiplication, and division.
Octal Number System:
The octal number system is a base 8 numbering system, meaning it uses eight digits (0-7) to represent numbers. The rightmost digit represents the ones place, the next digit to the left represents the eights place, and so on, with each digit representing a power of eight.
For example, the number 173 can be broken down as follows:
1 x 8^2 + 7 x 8^1 + 3 x 8^0 = 64 + 56 + 3 = 123
The octal system is less commonly used in everyday life, but it is often used in computer programming and other technical fields.
Relationship between Decimal and Octal Number Systems:
The relationship between the decimal and octal number systems is based on the fact that 8 is a power of 2, and 2 is a factor of 10. This means that any octal number can be converted to a decimal number by multiplying each digit by the appropriate power of 8 and then adding the results.
For example, the octal number 173 can be converted to decimal as follows:
1 x 8^2 + 7 x 8^1 + 3 x 8^0 = 64 + 56 + 3 = 123
Conversely, any decimal number can be converted to octal by repeatedly dividing the number by 8 and recording the remainders.
For example, the decimal number 123 can be converted to octal as follows:
123 ÷ 8 = 15 remainder 3
15 ÷ 8 = 1 remainder 7
1 ÷ 8 = 0 remainder 1
Therefore, the octal representation of 123 is 173.
Converting between Decimal and Octal:
To convert a decimal number to octal, you can use the repeated division method described above. Alternatively, you can use the following algorithm:
Divide the decimal number by 8.
Record the remainder.
Divide the quotient by 8 and repeat step 2 until the quotient is 0.
The octal representation of the decimal number is the remainders in reverse order.
For example, to convert the decimal number 275 to octal, we can use the following algorithm:
275 ÷ 8 = 34 remainder 3
34 ÷ 8 = 4 remainder 2
4 ÷ 8 = 0 remainder 4
Therefore, the octal representation of 275 is 423.
To convert an octal number to decimal, you can use the following algorithm:
Write the octal number as a sum of powers of 8, starting with the rightmost digit as the first power.
For example, to convert the octal number 423 to decimal, we can use the following algorithm:
4 x 8^2 + 2 x 8^1 + 3 x 8^0 = 256 + 16 + 3 = 275
Therefore, the decimal representation of 423 is 275.
Advantages and Disadvantages of Decimal and Octal Number Systems:
The decimal system is easy to understand and widely used in everyday life. It is also easy to perform arithmetic operations such as addition, subtraction, multiplication, and division in the decimal system.
The octal system is less commonly used in everyday life but is often used in computer programming and other technical fields. The octal system has the advantage of being easy to convert to binary, another important numbering system used in computing. This is because each octal digit can be represented by three binary digits, making it easy to convert between the two.
However, the octal system can be more difficult to use for arithmetic operations than the decimal system, and it is less intuitive for most people.
Table of Octal and decimal:
Here's a table showing the correspondence of octal 0 to 50 in hexadecimal, decimal, binary, and text:
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