To convert the decimal number 125 to its 8-bit binary representation, we can follow these steps:
1. Understand the place values in an 8-bit binary number:
Each bit from left to right represents:
[tex]\[
(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0)
\][/tex]
or in decimal terms:
[tex]\[
(128, 64, 32, 16, 8, 4, 2, 1)
\][/tex]
2. Determine which of these values sum to 125:
- Start with the largest value, 128. Since 125 is less than 128, the first bit must be 0.
- Move to the next value, 64. 125 is greater than 64, so the second bit is 1: [tex]\( 125 - 64 = 61 \)[/tex].
- Next, 32. 61 is greater than 32, so the third bit is 1: [tex]\( 61 - 32 = 29 \)[/tex].
- For 16, 29 is greater than 16, so the fourth bit is 1: [tex]\( 29 - 16 = 13 \)[/tex].
- For 8, 13 is greater than 8, so the fifth bit is 1: [tex]\( 13 - 8 = 5 \)[/tex].
- For 4, 5 is greater than 4, so the sixth bit is 1: [tex]\( 5 - 4 = 1 \)[/tex].
- For 2, 1 is less than 2, so the seventh bit is 0.
- Finally, for 1, 1 is equal to 1, so the eighth bit is 1: [tex]\( 1 - 1 = 0 \)[/tex].
3. Construct the binary representation:
The binary digits, from left to right, are: [tex]\( 0, 1, 1, 1, 1, 1, 0, 1 \)[/tex].
Thus, the 8-bit binary representation of 125 is:
[tex]\[
01111101
\][/tex]
4. Map this binary representation to the LED lights representation:
- A bit value of 1 means the corresponding LED light is on and will be represented as 1.
- A bit value of 0 means the corresponding LED light is off and will be represented as Ω.
Mapping the binary representation [tex]\( 01111101 \)[/tex] to the LED lights:
[tex]\[
[Ω, 1, 1, 1, 1, 1, Ω, 1]
\][/tex]
So, the 8-bit binary representation of 125 and the LED lights representation are:
[tex]\[
01111101 \quad \text{and} \quad [Ω, 1, 1, 1, 1, 1, Ω, 1]
\][/tex]
