To determine the largest decimal number that can be represented with eight bits, let's understand how binary numbers work.
1. Understand the concept of bits: Each bit in binary represents an exponent of 2, starting from [tex]\(2^0\)[/tex] for the rightmost bit and increasing by powers of 2 as you move to the left. An 8-bit number uses eight binary digits, from [tex]\(b_7 b_6 b_5 b_4 b_3 b_2 b_1 b_0\)[/tex].
2. All bits set to 1: To find the maximum value, all eight bits should be set to 1. This would look like:
[tex]\[
11111111_2
\][/tex]
This binary number needs to be converted to its decimal equivalent.
3. Calculate the decimal equivalent: Begin by expanding the binary number in terms of powers of 2:
[tex]\[
1 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0
\][/tex]
Calculating each term:
[tex]\[
1 \times 128 + 1 \times 64 + 1 \times 32 + 1 \times 16 + 1 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1
\][/tex]
Which simplifies to:
[tex]\[
128 + 64 + 32 + 16 + 8 + 4 + 2 + 1
\][/tex]
4. Sum the values:
[tex]\[
128 + 64 = 192
\][/tex]
[tex]\[
192 + 32 = 224
\][/tex]
[tex]\[
224 + 16 = 240
\][/tex]
[tex]\[
240 + 8 = 248
\][/tex]
[tex]\[
248 + 4 = 252
\][/tex]
[tex]\[
252 + 2 = 254
\][/tex]
[tex]\[
254 + 1 = 255
\][/tex]
Therefore, the largest decimal number that can be represented with eight bits is [tex]\( \boxed{255} \)[/tex].
Given the multiple-choice options:
a. 512
b. 255
c. 128
d. 256
The correct answer is [tex]\( \boxed{255} \)[/tex].
