Irrational numbers are numbers that cannot be expressed as a ratio of two integers. This means they cannot be written as a fraction with a finite number of digits in the numerator and denominator. They are non-repeating and non-terminating decimals.
This statement is significant because it proves that there are numbers that cannot be expressed as a ratio of two integers. It also shows that there are numbers that cannot be represented by a decimal with a finite number of digits. This has many implications in mathematics and has been studied extensively by mathematicians.
The proof for this statement was first given by the ancient Greek mathematician, Pythagoras. He showed that the square root of 2 cannot be expressed as a ratio of two integers by using a geometric proof. This proof has been refined and expanded upon over the years by other mathematicians.
Yes, there are infinitely many irrational numbers. Some examples include pi, e, and the square root of any non-perfect square. In fact, the majority of numbers are irrational, and it is only a small subset that can be expressed as a ratio of two integers.
Irrational numbers are used in many areas of mathematics and science, including geometry, physics, and engineering. In real life, they are used to represent values that are not whole numbers, such as the circumference of a circle or the growth rate of a population. They are also important in computer science and cryptography.