Explain how integers are represented in binary.

Integers in binary are represented as a sequence of 0s and 1s, based on the binary number system.

In more detail, the binary number system, also known as base-2, is a method of representing numbers that counts by using two numerals: 0 and 1. This is the fundamental language of computers, as it directly corresponds to the on and off states of a digital system.

To represent an integer in binary, you start by finding the highest power of 2 that fits into the number, and write a 1 in that position. Then, you subtract that value from the number, and repeat the process with the remainder. If a power of 2 does not fit into the number, you write a 0 in that position. This process continues until you reach the power of 2 that is less than or equal to 1.

For example, to represent the number 13 in binary, you would start with the highest power of 2 that fits into 13, which is 2^3 (or 8). So, you write a 1 in the 2^3 position. Then, you subtract 8 from 13, leaving 5. The next highest power of 2 that fits into 5 is 2^2 (or 4), so you write a 1 in the 2^2 position. Subtracting 4 from 5 leaves 1, which is 2^0, so you write a 1 in the 2^0 position. The final binary representation of 13 is 1101.

Negative integers can also be represented in binary using a method called two's complement. This involves flipping all the bits (changing 1s to 0s and vice versa) of the positive version of the number, and then adding 1. This allows for easy arithmetic operations even with negative numbers.

Understanding binary representation is crucial in computer science, as it forms the basis of all digital computation. It's the way computers store and manipulate numerical data, and understanding it can give you a deeper insight into how computers work at a fundamental level.