Converting Between Number Systems
How do we convert binary to decimal?
Convert the 8-binary binary expansion (0101 1010 ) to a decimal expansion.
How do we convert decimal to 8-bit binary?
Convert the following decimal expansion (106)10 to an 8-bit binary expansion.
How do we convert hexadecimal to octal?
Convert the following hexadecimal expansion (CAB)16 to an octal expansion.
How do we convert binary to hexadecimal?
Convert the following binary expansion (0110 0101 1001 1010 )z to a hexadecimal expansion.
Conversion Methods for Different Number Systems
Converting between various number systems necessitates a grasp of their fundamental bases for accurate digit translation.
Binary to Decimal
To convert the binary expansion (0101 1010) to decimal, each binary digit is multiplied by the corresponding power of 2 and summed up. Starting from the right, the calculation is: 0*(2^0) + 1*(2^1) + 0*(2^2) + 1*(2^3) + 0*(2^4) + 1*(2^5) + 0*(2^6) + 1*(2^7), resulting in a decimal value of 90.
Decimal to 8-bit Binary
Converting decimal 106 to an 8-bit binary entails representing it using 8 binary digits. In binary, 106 is 1101010, which fits into 8 bits as 01101010.
Hexadecimal to Octal
To convert the hexadecimal CAB to octal, first, convert it to binary: CAB is 1100 1010 1011 in binary. Then, group the binary digits into sets of three (starting from the right), resulting in 3 groups: 011 001 010 101. Convert each group to octal: 3 1 2 5. Thus, CAB (hexadecimal) translates to 3125 (octal).
Binary to Hexadecimal
For the binary expansion (0110 0101 1001 1010), split it into groups of four digits each: 0110 0101 1001 1010. Convert each group to its hexadecimal equivalent: 6 5 9 A. Therefore, the binary expansion translates to 659A in hexadecimal.
Understanding the conversion methods between different number systems is crucial for various applications in mathematics and computer science. Whether converting binary to decimal, decimal to binary, hexadecimal to octal, or binary to hexadecimal, each process follows specific rules based on the numbering system's base.
Binary to decimal conversion involves multiplying each binary digit by increasing powers of 2 and summing them to obtain the decimal equivalent. Similarly, converting decimal to 8-bit binary requires representing the decimal number using 8 binary digits, padding with zeros if necessary.
When converting hexadecimal to octal, first, convert the hexadecimal number to binary and then group the binary digits into sets of three to translate them into octal form. For binary to hexadecimal conversion, splitting the binary number into groups of four digits each and converting each group to its hexadecimal equivalent yields the final result.
By mastering these conversion techniques, individuals can efficiently work with different number systems and perform accurate translations between them, which is essential in fields such as computer programming, digital electronics, and cryptography.