Decimal to binary and Hex Conversion:
Convert the decimal number 115 into binary.
Convert the decimal number 898 into hexadecimal.
The correct answer and explanation is :
Let’s break down the process of converting the decimal numbers to binary and hexadecimal:
1. Converting the decimal number 115 to binary:
To convert a decimal number to binary, we repeatedly divide the number by 2, recording the remainders at each step. The binary number is the sequence of remainders, read from bottom to top.
Let’s convert 115 into binary:
- 115 ÷ 2 = 57, remainder = 1
- 57 ÷ 2 = 28, remainder = 1
- 28 ÷ 2 = 14, remainder = 0
- 14 ÷ 2 = 7, remainder = 0
- 7 ÷ 2 = 3, remainder = 1
- 3 ÷ 2 = 1, remainder = 1
- 1 ÷ 2 = 0, remainder = 1
Now, reading the remainders from bottom to top, we get: 1110011.
So, the binary equivalent of 115 is 1110011.
2. Converting the decimal number 898 to hexadecimal:
To convert a decimal number to hexadecimal, we repeatedly divide the number by 16, recording the remainders. The hexadecimal system uses digits 0-9 and letters A-F to represent values 0-15.
Let’s convert 898 into hexadecimal:
- 898 ÷ 16 = 56, remainder = 2
- 56 ÷ 16 = 3, remainder = 8
- 3 ÷ 16 = 0, remainder = 3
Now, reading the remainders from bottom to top, we get 382.
So, the hexadecimal equivalent of 898 is 382.
Explanation:
- Binary system is a base-2 system, which means it only uses two digits: 0 and 1. Each binary digit (bit) represents a power of 2. For example, the binary number 1110011 is calculated as: $$
1 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 115
$$ The process of dividing by 2 repeatedly helps to break down the number into its binary representation. - Hexadecimal system is a base-16 system, which means it uses digits 0-9 and letters A-F to represent values 0-15. It is commonly used in computing as a more compact way to represent binary numbers. In the example of 898, we divide by 16 because 16 is the base of the system. Each remainder corresponds to a hexadecimal digit, either a number or a letter.
In conclusion, understanding the processes of division and remainders allows us to convert decimal numbers to binary and hexadecimal systems effectively, which are commonly used in computing and digital electronics.