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Probably the ancient Egyptians who discovered that the diagonal of a unit square was not a rational number. And then discovered other such numbers.

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Q: Who invented the irrational number set?

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Yes it will be. The set of real numbers can be divided into two distinct sets: rational and irrational. So if it is not rational, then it is irrational.

Integers are rational. In the set of real numbers, every number is either rational or irrational; a number can't be both or neither.

No. You can well multiply two irrational numbers and get a result that is not an irrational number.

The set of irrational numbers is NOT denoted by Q.Q denotes the set of rational numbers. The set of irrational numbers is not denoted by any particular letter but by R - Q where R is the set of real numbers.

The set of real numbers is defined as the union of all rational and irrational numbers. Thus, the irrational numbers are a subset of the real numbers. Therefore, BY DEFINITION, every irrational number is a real number.

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No, a number is either rational or irrational

Its a null set.

Yes it will be. The set of real numbers can be divided into two distinct sets: rational and irrational. So if it is not rational, then it is irrational.

Integers are rational. In the set of real numbers, every number is either rational or irrational; a number can't be both or neither.

No; here's a counterexample to show that the set of irrational numbers is NOT closed under subtraction: pi - pi = 0. pi is an irrational number. If you subtract it from itself, you get zero, which is a rational number. Closure would require that the difference(answer) be an irrational number as well, which it isn't. Therefore the set of irrational numbers is NOT closed under subtraction.

No. You can well multiply two irrational numbers and get a result that is not an irrational number.

No. Irrational and rational numbers can be non-negative.

The set of irrational numbers is NOT denoted by Q.Q denotes the set of rational numbers. The set of irrational numbers is not denoted by any particular letter but by R - Q where R is the set of real numbers.

The set of real numbers is defined as the union of all rational and irrational numbers. Thus, the irrational numbers are a subset of the real numbers. Therefore, BY DEFINITION, every irrational number is a real number.

The intersection between rational and irrational numbers is the empty set (Ø) since no rational number (x∈ℚ) is also an irrational number (x∉ℚ)

Irrational.

An irrational number is any real number that cannot be expressed as a ratio of two integers.So yes, an irrational number IS a real number.There is also a set of numbers called transcendental numbers, which includes both real and complex/imaginary numbers. Of this set, all the real numbers are irrational numbers.