Irrational numbers are numbers which cannot be expressed in the form of a fraction, and which are also impossible to record as a complete decimal. People have been working with irrational numbers since Greek and Roman times, and a number have been identified by mathematicians through the ages. There are a number of interesting applications and uses for irrational numbers, ranging from frustrating math students to completing complex equations.

So-called rational numbers can all be written in decimal form or the form of a fraction. ¾, for example, is a rational number, which can also be expressed as .75. When a number is irrational, it cannot be written out as a fraction with integers and the number will be impossible to record in decimal form. Pi is a famous example of an irrational number; while it is often simplified to 3.14 for the purpose of rough calculations, pi cannot actually be fully written out in decimal form because the decimal is endless.

Some other examples include the square root of two, Euler's number, and the golden ratio. For the purpose of simplicity, some of these numbers are written out as symbols, as in the case of “e” for Euler's number, and sometimes they will be represented in partial decimal form. When an irrational number is presented in decimal form, ellipses are usually used after the last number in the decimal to indicate that it continues, as in 3.14... for pi.

People often start working with these numbers at a young age, although they may not be introduced specifically to the concepts of rational and irrational numbers until later. Pi is one of the earliest irrational numbers which many people learn, because it is used in equations to find the area and circumference of a circle, and these equations often make an excellent introduction to more advanced mathematics for young children. People are also introduced to the concept in many of the sciences as they start to learn about equations which are commonly used.

These unusual numbers can be difficult to work with on a basic calculator, because of the limitations of the calculator. It is usually necessary to have an advanced scientific or graphing calculators which has been programmed with these numbers and their values.

Some mathematicians make the study of these numbers their life's work. These numbers often have a number of intriguing properties which are fun to explore for people who love mathematics, and a mathematician may also be able to come up with a new application for an irrational number.