- #1
micromass said:
DrClaude said:
ltkach said:
rbj said:
An irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, it cannot be written as a fraction in the form of a/b where a and b are integers.
Square Root 2 is considered an irrational number because it cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating, meaning it goes on infinitely without following a specific pattern.
The proof that Square Root 2 is irrational was first demonstrated by the ancient Greek mathematician Pythagoras. A more formal proof was later developed by the Greek mathematician Euclid in his book "Elements". It involves assuming that Square Root 2 can be expressed as a ratio of two integers, and then using logical deductions to show that this assumption leads to a contradiction.
No, not all square roots are irrational. Some square roots, such as the square root of 4, can be expressed as a ratio of two integers (in this case, 2). These are called rational square roots.
Irrational numbers are used in various fields such as physics, engineering, and computer science. For example, the value of pi (π), which is an irrational number, is used in calculations involving circles and spheres. In computer science, irrational numbers are used in algorithms for tasks such as generating random numbers or calculating the speed of an object.
