How do you represent an irrational number in a computer system?

In a computer system, an irrational number is typically represented as a floating-point number.

In more detail, an irrational number is a number that cannot be expressed as a simple fraction. It has non-repeating, non-terminating decimal parts. Examples include numbers like Pi (π), the square root of 2 (√2), and Euler's number (e). These numbers have decimal representations that go on forever without repeating. However, computers have finite memory, so they cannot store an infinite number of digits. Therefore, they need a way to represent these numbers in a finite way.

The most common method used to represent irrational numbers in computer systems is the floating-point representation. This is a system where a number is represented by two parts: a significand (or mantissa) and an exponent. The number is then calculated as the significand multiplied by the base (usually 2) raised to the power of the exponent. This allows for a wide range of values and precision levels, but it is not exact. The precision of a floating-point number is limited by the number of bits used to store it.

For example, in most systems, a double-precision floating-point number uses 64 bits: 1 bit for the sign, 11 bits for the exponent, and 52 bits for the significand. This gives about 15 to 17 decimal digits of precision. So, while the representation is not exact, it is usually precise enough for practical purposes.

It's important to note that not all irrational numbers can be represented exactly, even with floating-point numbers. For example, the number π is often approximated as 3.14159 in calculations. However, many computer systems and programming languages provide built-in constants for common irrational numbers like π and e, which are represented as floating-point numbers with the maximum possible precision for the system.

In conclusion, while it's impossible to represent irrational numbers exactly in a computer system due to their infinite nature, floating-point representation provides a practical and widely used solution.