What is the largest 16-bit binary value that can be represented by

What is the largest 16-bit binary value that can be represented by:

(a) Unsigned number
(b) Signed magnitude
(c) Signed two’s complement

The Correct Answer and Explanation is:

Here are the correct answers for the largest 16-bit binary values under each representation:

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(a) Unsigned Number:

Answer: 1111111111111111 (binary) = 65,535 (decimal)

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(b) Signed Magnitude:

Answer: 0111111111111111 (binary) = 32,767 (decimal)

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(c) Signed Two’s Complement:

Answer: 0111111111111111 (binary) = 32,767 (decimal)

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✅ Explanation (300+ words):

The number of bits (in this case, 16 bits) determines how many binary values can be represented. Each of the three number formats interprets the same 16-bit binary sequence differently:

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(a) Unsigned Number:

In an unsigned representation, all 16 bits are used to represent the magnitude of the number. There is no sign bit (i.e., no bit is used to indicate positive or negative).

• The smallest unsigned value is 0000000000000000 = 0.
• The largest unsigned value is when all bits are 1s: 1111111111111111.
• This binary number is equal to: $$
  2^{16} – 1 = 65,535
  $$

So, the largest 16-bit unsigned binary number is 65,535.

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(b) Signed Magnitude:

In signed magnitude representation, the first bit is the sign bit:

• 0 indicates a positive number.
• 1 indicates a negative number.
• The remaining 15 bits represent the magnitude.

The largest positive number in signed magnitude is:

• 0 followed by 15 1s: 0111111111111111
• This represents: $$
  2^{15} – 1 = 32,767
  $$

So, the largest 16-bit signed magnitude number is 32,767.

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(c) Signed Two’s Complement:

In two’s complement representation:

• The first bit is still the sign bit, but the negative numbers are represented differently (using inversion and addition of 1).
• The range for a 16-bit two’s complement number is: $$
  -2^{15} \text{ to } 2^{15} – 1
  $$ That is: $$
  -32,768 \text{ to } 32,767
  $$
• The largest number is when the sign bit is 0 and the remaining bits are all 1s: 0111111111111111 = 32,767

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📌 Summary Table:

Representation	Largest 16-bit Value	Decimal Equivalent
Unsigned	1111111111111111	65,535
Signed Magnitude	0111111111111111	32,767
Signed Two’s Complement	0111111111111111	32,767

Each format has a different range because of how it uses the bits, particularly the most significant bit (MSB).