Calculating Bits Needed for 4K Memory Addressing

Understanding memory addressing is crucial in computer architecture. The number of bits required to address a memory system depends directly on the total number of unique memory locations it contains. Each memory location needs a unique address, and the number of bits determines how many unique addresses can be represented.

Understanding Memory Size Notation

In computing, particularly when discussing memory capacity, the prefix 'K' (Kilo) often represents $2^{10}$, which is 1024, rather than $10^3$ (1000). Therefore, a 4K memory refers to memory with 4 times 1024 locations.

Calculating Total Memory Locations

To find the total number of addressable locations in a 4K memory, we perform the following calculation:

• Memory Size = 4K locations
• Since K = $2^{10}$, we have:
• Total Locations = $4 \times 2^{10}$
• We can express 4 as $2^2$:
• Total Locations = $2^2 \times 2^{10}$
• Using the rule of exponents ($a^m \times a^n = a^{m+n}$), we get:
• Total Locations = $2^{(2+10)} = 2^{12}$

So, the 4K memory has $2^{12}$ unique locations.

Determining the Number of Address Bits

The number of bits required to represent N unique addresses is given by the formula: Number of bits = $\log_2(N)$.

In this case, the total number of locations (N) is $2^{12}$.

• Number of bits = $\log_2(\text{Total Locations})$
• Number of bits = $\log_2(2^{12})$
• Using the logarithm property $\log_b(b^x) = x$, we get:
• Number of bits = 12

Therefore, 12 bits are needed to address 4K memory locations.

Conclusion

The calculation shows that 12 bits are required to provide unique addresses for all $2^{12}$ locations in a 4K memory system. This matches one of the options provided.