Irrational

The existence of irrational numbers challenged the ancient Greek belief in the perfection of mathematics.

The number e is an irrational number that is often used in calculus.

The set of irrational numbers is uncountable, meaning it cannot be put into a one-to-one correspondence with the set of natural numbers.

The existence of irrational numbers was discovered by the ancient Greeks.

The discovery of irrational numbers was a major breakthrough in the history of mathematics.

Irrational numbers are an essential concept in the study of calculus.

The square root of 2 is also an irrational number.

An irrational number can be approximated by a rational number, but not expressed exactly.

Irrational numbers have decimal expansions that never terminate or repeat.

An irrational number cannot be expressed as a fraction.

Irrational numbers are sometimes called "non-repeating" or "non-terminating" decimals.

The value of the golden ratio, approximately 1.618, is an irrational number.

The discovery of irrational numbers was a major breakthrough in mathematics and helped lay the groundwork for modern calculus.

The set of irrational numbers is infinite and uncountable, meaning that there are more irrational numbers than there are natural numbers.

Irrational numbers are an important part of number theory, a branch of mathematics that deals with the properties of numbers.

It is impossible to calculate the exact value of some square roots, such as the square root of 3, because they are irrational numbers.

The square root of two is another example of an irrational number.

The mathematical constant pi, which is used to calculate the circumference and area of circles, is an example of an irrational number.

Irrational numbers are often represented by decimal points that go on forever without repeating.

Irrational numbers are essential to many fields of science and technology, including cryptography and computer science.

The famous mathematician Pythagoras discovered that the square root of two is irrational.

The existence of irrational numbers challenges the traditional view of numbers as being either rational or irrational, and raises questions about the nature of mathematical knowledge itself.

The decimal expansion of an irrational number never repeats and never ends.

The number pi is an irrational number that cannot be expressed as a fraction.

Irrational numbers are a fascinating topic in mathematics that have intrigued scholars for centuries.

It is impossible to write the number e as a fraction because it is an irrational number.

The weather forecast seemed irrational with conflicting predictions.

The value of the golden ratio, approximately equal to 1.618, is an irrational number.

Her boss was an irrational person who made decisions based on emotion rather than logic.

The irrational refused to take advice from anyone and insisted on doing things their own way.

The criminal's irrational actions led to his capture by the police.