How do you represent a negative decimal number using 16-bit binary?

A negative decimal number can be represented in 16-bit binary using the two's complement method.

In computer science, negative numbers are typically represented using a method known as two's complement. This method is widely used because it allows for simple arithmetic operations and efficient use of binary number systems. To represent a negative decimal number in 16-bit binary, you first convert the absolute value of the number to binary, then invert the bits (change 1s to 0s and 0s to 1s), and finally add 1 to the result.

Let's take -18 as an example. The binary representation of 18 (in 16 bits) is 0000000000010010. To get the two's complement, you first invert the bits, which gives you 1111111111101101. Then, you add 1 to the result, which gives you 1111111111101110. So, -18 is represented as 1111111111101110 in 16-bit binary.

The beauty of the two's complement method is that it allows for straightforward arithmetic operations. When you add a positive number and its negative counterpart together in two's complement, you always get zero. This is because the carry from the highest bit (the sign bit) is discarded in fixed-width binary numbers, which results in a phenomenon known as 'wrap around'.

It's also worth noting that the leftmost bit in a two's complement binary number is used as a sign bit. If the sign bit is 0, the number is positive; if it's 1, the number is negative. This makes it easy to determine the sign of a number at a glance.

A-Level Computer Science Tutor Summary: To represent a negative decimal number in 16-bit binary, we use the two's complement method. First, change the number to binary, then invert the bits, and finally add 1. This method, with the sign bit indicating the number's sign (0 for positive, 1 for negative), simplifies arithmetic operations, making it easy to work with negative numbers in binary.