What is the binary representation of the decimal number -3 in a 4-bit system?
Understand the Problem
The question is asking for the binary representation of a negative decimal number (-3) in a 4-bit binary system, which involves understanding how negative numbers are represented in binary form, likely through two's complement.
Answer
$1101_2$
Answer for screen readers
The binary representation of -3 in a 4-bit binary system is $1101_2$.
Steps to Solve
- Find the positive binary representation First, we need to convert the positive version of the number, which is 3, into binary.
The binary representation of 3 is: $$ 3 = 2^1 + 2^0 = 11_2 $$
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Pad to the desired bit width Since we need a 4-bit binary representation, we pad the binary number with zeros to the left: $$ 3 \to 0011_2 $$
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Invert the bits Next, we find the one's complement by flipping the bits of the 4-bit representation: $$ 0011_2 \to 1100_2 $$
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Add one to find two's complement Now, we add 1 to the one's complement: $$ 1100_2 + 0001_2 = 1101_2 $$
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Final result The two's complement representation of -3 in a 4-bit binary system is: $$ -3 \to 1101_2 $$
The binary representation of -3 in a 4-bit binary system is $1101_2$.
More Information
In binary systems, negative numbers are often represented using a method called two's complement. This allows for efficient arithmetic operations, as the same circuitry can be used for both addition and subtraction.
Tips
One common mistake is forgetting to pad the binary representation to the correct bit width, which can lead to incorrect values. Always ensure that the binary number is of the required length (in this case, 4 bits) before proceeding with further calculations.
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