Convert the following from binary to hex, then from hex to decimal. Show your work. Hex: 0x0f 0x100 0x0d 0xffa Decimal: Binary:
The Correct Answer and Explanation is:
Let’s go through each conversion step by step, starting with the binary to hex and then hex to decimal. We’ll break it down clearly so you can follow along:
1. Binary to Hex Conversion:
Binary 1: 0x0f
This is already in hexadecimal format, so no conversion from binary is needed for this part. Hexadecimal “0x0f” is just a representation in base-16. Now, to understand the decimal equivalent:
- Hexadecimal 0x0f = 0 × 16^1 + 15 × 16^0
- = 0 + 15
- Decimal: 15
Binary 2: 0x100
Again, this is in hex format. Let’s break it down into its decimal equivalent.
- Hexadecimal 0x100 = 1 × 16^2 + 0 × 16^1 + 0 × 16^0
- = 1 × 256 + 0 + 0
- Decimal: 256
Binary 3: 0x0d
- Hexadecimal 0x0d = 0 × 16^1 + 13 × 16^0
- = 0 + 13
- Decimal: 13
Binary 4: 0xffa
- Hexadecimal 0xffa = 15 × 16^2 + 15 × 16^1 + 10 × 16^0
- = 15 × 256 + 15 × 16 + 10
- = 3840 + 240 + 10
- Decimal: 4090
2. Hex to Decimal Conversion:
Here’s the summary of our decimal results after converting each hexadecimal number:
- 0x0f = 15
- 0x100 = 256
- 0x0d = 13
- 0xffa = 4090
Explanation:
The hexadecimal system is base-16, so each position represents a power of 16 (just as in decimal, each place represents a power of 10). In hexadecimal, the digits are 0-9 and A-F, where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.
For example, to convert 0x0f to decimal:
- The “0x” prefix indicates it’s in hexadecimal. The “f” represents the number 15 in decimal, so 0x0f = 15 in decimal.
To convert 0x100 to decimal:
- The first digit (1) is in the 16^2 place, so we multiply 1 by 256 (16^2). The other digits are 0, so 0x100 = 256.
For 0x0d:
- The “d” in hex represents 13, so 0x0d = 13.
And for 0xffa:
- The “f” represents 15, so 15 × 256 (16^2), 15 × 16 (16^1), and 10 for “a” (16^0), which gives 0xffa = 4090.
This process involves understanding the positional value of each hex digit and converting each into its corresponding decimal value.
