How are negative numbers represented using two's complement in binary?

Negative numbers are represented in binary using two's complement by inverting the bits of the positive number and adding one.

In more detail, two's complement is a mathematical operation on binary numbers. It is used in computing as a method of signed number representation. To understand two's complement, it's important to first understand how positive numbers are represented in binary. Binary numbers are base 2 numbers where each digit is either a 0 or a 1. For example, the number 5 is represented as 101 in binary.

To represent a negative number using two's complement, you start by writing down the binary representation of the positive version of that number. Then, you invert the bits, which means changing all the 1s to 0s and all the 0s to 1s. This is known as the one's complement. Finally, you add one to the result.

For example, to represent -5 in binary using two's complement, you would start with the binary representation of 5, which is 101. Inverting the bits gives you 010, and adding one gives you 011. So, -5 is represented as 011 in binary using two's complement.

It's important to note that the leftmost bit in a two's complement number is known as the sign bit. If the sign bit is 0, the number is positive. If the sign bit is 1, the number is negative. This is why when you invert the bits of a positive number, the result is a negative number.

Two's complement is widely used in computing because it allows for simple addition and subtraction operations. When you add or subtract two's complement numbers, the arithmetic works out the same way as it does with positive numbers, which simplifies the design of computer hardware.