Converting numbers from base 10 to other bases is a fundamental concept in mathematics and computer science. It allows us to represent numbers using different symbol sets and perform computations in different number systems. In this article, we will explore the process of converting numbers from base 10 to other bases, such as binary, octal, and hexadecimal, and discuss some of their practical applications.
To convert a number from base 10 to another base, we can use the process of repeated division by the new base. The remainder of each division will give us the digits of the new number, starting from the rightmost digit. Let’s explore this process in detail with some examples.
CONVERSION FROM BASE 10 TO BINARY
To convert a number from base 10 to binary (base 2), we can repeatedly divide the number by 2 and record the remainder of each division. We continue dividing until the quotient is 0 and then write the remainder in reverse order to get the binary representation of the number. Here’s an example:
Convert the decimal number 27 to binary.
Step 1: Divide 27 by 2, giving a quotient of 13 with a remainder of 1.
Step 2: Divide 13 by 2, giving a quotient of 6 with a remainder of 1.
Step 3: Divide 6 by 2, giving a quotient of 3 with a remainder of 0.
Step 4: Divide 3 by 2, giving a quotient of 1 with a remainder of 1.
Step 5: Divide 1 by 2, giving a quotient of 0 with a remainder of 1.
| 2 | 27 | R |
| 2 | 13 | 1 |
| 2 | 6 | 1 |
| 2 | 3 | 0 |
| 2 | 1 | 1 |
| 0 | 1 |
The remainders in reverse order are 11011, so the binary representation of the decimal number 27 is 11011_2.
CONVERSION FROM BASE 10 TO OCTAL
To convert a number from base 10 to octal (base 8), we can use a similar process of repeated division by 8. We continue dividing until the quotient is 0 and then write the remainder in reverse order to get the octal representation of the number. Here’s an example:
Convert the decimal number 75 to octal.
Step 1: Divide 75 by 8, giving a quotient of 9 with a remainder of 3.
Step 2: Divide 9 by 8, giving a quotient of 1 with a remainder of 1.
Step 3: Divide 1 by 8, giving a quotient of 0 with a remainder of 1.
| 8 | 75 | R |
| 8 | 9 | 3 |
| 8 | 1 | 1 |
| 0 | 1 |
The remainders in reverse order are 113, so the octal representation of the decimal number 75 is 113_8.
See also: CONVERSION FROM OTHER BASES TO BASE 10 WITH VIDEO – NUMBER BASES
CONVERSION FROM BASE 10 TO HEXADECIMAL
To convert a number from base 10 to hexadecimal (base 16), we can use a similar process of repeated division by 16. However, since hexadecimal uses 16 symbols (0-9 and A-F), we need to be able to represent remainders greater than 9 using letters. We can use the following table to represent remainders 10-15 in hexadecimal:
| Remainder | Hexadecimal Symbol |
|---|---|
| 10 | A |
| 11 | B |
| 12 | C |
| 13 | D |
| 14 | E |
| 15 | F |
Let’s look at an example of converting a decimal number to hexadecimal:
Convert the decimal number 137 to hexadecimal.
Step 1: Divide 137 by 16, giving a quotient of 8 with a remainder of 9.
Step 2: Divide 8 by 16, giving a quotient of 0 with a remainder of 8.
| 16 | 137 | R |
| 16 | 8 | 9 |
| 0 | 8 |
The remainders in reverse order are 98, so the hexadecimal representation of the decimal number 137 is 98_16.
In some cases, it may be necessary to pad the hexadecimal number with leading zeros to ensure that it has the correct number of digits. For example, the decimal number 10 is represented as A in hexadecimal, but we can also write it as 0A to indicate that it has two digits.
If it’s still unclear, watch the detailed video on conversion from base 10 to other bases below
APPLICATIONS
Converting numbers from base 10 to other bases has several practical applications in computer science, including:
- The binary representation of data: Computers use the binary number system to represent data, and converting decimal numbers to binary is an important step in programming and data analysis.
- Network addressing: IP addresses in computer networks are often represented in dotted-decimal notation, where each of the four numbers represents an octet in the address. Converting decimal numbers to octal or hexadecimal can be useful for understanding and working with IP addresses.
- Cryptography: Many encryption algorithms rely on the manipulation of numbers in different bases, including binary, octal, and hexadecimal.
CONCLUSION
Converting numbers from base 10 to other bases is a fundamental concept in mathematics and computer science. By using the process of repeated division, we can convert decimal numbers to binary, octal, or hexadecimal representation. This process has several practical applications in computer science, including data representation, network addressing, and cryptography. Understanding how to convert between different number bases is an important skill for anyone working in the fields of mathematics or computer science.
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