# Irrational Numbers

#### Understanding Irrational Numbers

**Irrational numbers**are numbers that cannot be expressed as a ratio of two integers.- Their decimal representation is neither terminating nor repeating. In other words, the decimal goes on forever without repeating a pattern.
- Quite a few well known numbers are irrational, including
**Pi (π)**,**Euler’s number (e)**, and the**square root of two (√2)**. - Any number that can’t be placed in a simple fraction or whole number is an irrational number.

#### Representing Irrational Numbers in Computers

- Since irrational numbers are infinite and non-repeating, they can’t be represented exactly in a computer, which has finite memory.
- We most often use
**approximations**of irrational numbers in computing. - For example, Pi is often approximated as 3.14159 in programming and computer calculations.

#### Encoding and Decoding Irrational Numbers

**Encoding**an irrational number in a computer involves choosing an appropriate approximation and then representing that approximation in binary.- Like rational numbers, these approximations are typically represented as
**floating-point numbers**. **Decoding**an irrational number involves reversing the process – interpreting the binary as a floating-point number and then using the resulting approximation.

#### Importance of Irrational Numbers in Computing

- Despite their inability to be represented exactly, irrational numbers are immensely important in computer science and mathematics.
- They occur in many of the most important formulas in physics, engineering and theoretical mathematics.
- Understanding how they work is crucial, because the use of approximations can lead to
**round-off errors**. These errors can accumulate and become significant in computations, especially in iterative processes. Understanding this helps to prevent or mitigate such errors. - An awareness of the features of these numbers also helps in developing algorithms and solving problems that need
**high precision calculations**.