Decimal to binary and Hex Conversion

Decimal to binary and Hex Conversion: a. Convert the decimal number 115 into binary. b. Convert the decimal number 898 into hexadecimal.

The Correct Answer and Explanation is:

a. Convert Decimal 115 to Binary

To convert a decimal number to binary, we repeatedly divide the number by 2, recording the remainder at each step, and then read the remainders from bottom to top.

Let’s convert 115 into binary:

1. 115 ÷ 2 = 57, remainder 1
2. 57 ÷ 2 = 28, remainder 1
3. 28 ÷ 2 = 14, remainder 0
4. 14 ÷ 2 = 7, remainder 0
5. 7 ÷ 2 = 3, remainder 1
6. 3 ÷ 2 = 1, remainder 1
7. 1 ÷ 2 = 0, remainder 1

Now, reading the remainders from bottom to top: 115 in binary is 1110011.

b. Convert Decimal 898 to Hexadecimal

For hexadecimal conversion, we divide the decimal number by 16, noting the remainder each time. The remainders correspond to hexadecimal digits: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15).

Let’s convert 898 into hexadecimal:

1. 898 ÷ 16 = 56, remainder 2 (2 in hex)
2. 56 ÷ 16 = 3, remainder 8 (8 in hex)
3. 3 ÷ 16 = 0, remainder 3 (3 in hex)

Now, reading the remainders from bottom to top: 898 in hexadecimal is 382.

Explanation

1. Binary Conversion (115): Each division by 2 breaks down the number by halving it and tracking the remainder, which gives us the individual binary digits (bits). Since binary is a base-2 system, each remainder is either 0 or 1, and the remainders form the binary representation when read in reverse order.
2. Hexadecimal Conversion (898): Hexadecimal is a base-16 number system, which is a more compact way to represent binary numbers. By dividing by 16, we group the binary digits in 4-bit chunks, as one hexadecimal digit can represent four binary digits. The remainder after each division tells us the value of the hexadecimal digit.

In both cases, the processes are similar but differ in the base used (2 for binary, 16 for hexadecimal). The goal is to break the decimal number into smaller, easier-to-read chunks in their respective bases.