Question

4.Convert the following representation to base 2.

(a) Convert ( A F116) 16 to base 2.
(b) Convert (23) 10 to base 2.
(c) Convert (245) 8 to decimal integer

Answer 1

The conversions are as follows: (AF16)16 to binary is 1010111111010110, (23)10 to binary is 10111, and the octal (245)8 to decimal is 165.

Step-by-step explanation:

The student's question involves converting numbers between different bases and notations, which is a common topic in mathematics. They are converting from hexadecimal to binary, from decimal to binary, and from octal to decimal.

Converting Hexadecimal to Binary:

To convert (AF16)16 to base 2, each hexadecimal digit is replaced with its 4-bit binary equivalent.

• A in hexadecimal is 1010 in binary.
• F in hexadecimal is 1111 in binary.
• 1 in hexadecimal is 0001 in binary (often written simply as 1).
• 6 in hexadecimal is 0110 in binary.

So, (AF16)16 becomes 1010111111010110 in binary.

Converting Decimal to Binary:

To convert (23)10 to binary, we divide the number by 2 and record the remainders in reverse order as the binary number.

1. 23 ÷ 2 = 11 with a remainder of 1.
2. 11 ÷ 2 = 5 with a remainder of 1.
3. 5 ÷ 2 = 2 with a remainder of 1.
4. 2 ÷ 2 = 1 with a remainder of 0.
5. 1 ÷ 2 = 0 with a remainder of 1.

The binary number becomes 10111.

Converting Octal to Decimal:

For (245)8, each digit is multiplied by 8 raised to the power of its position, starting from 0 at the rightmost digit.

• 5 * 80 = 5
• 4 * 81 = 32
• 2 * 82 = 128

The sum of these is the decimal number: 128+32+5=165.