Irrational Numbers: Explained Through The Product Of A Rational And An Irrational Number.

sum or product of two irrational numbers is

rational or irrational

generally irrational.

Proof for the sum of two irrational numbers being generally irrational:

Let’s suppose we have two irrational numbers a and b, and let’s assume that their sum is a rational number c:

a + b = c

Rearranging this equation, we can get:

a = c – b

Since c is a rational number, and b is an irrational number, then the difference c – b is also an irrational number. Therefore, a is the difference of two irrational numbers, which means that it must be irrational as well. This is a contradiction because we assumed that a is an irrational number. Therefore, because our assumption leads to a contradiction, we can conclude that the sum of two irrational numbers is generally irrational.

Proof for the product of two irrational numbers being generally irrational:

Let’s suppose we have two irrational numbers a and b, and let’s assume that their product is a rational number c:

ab = c

We can rewrite this equation as:

a = c/b

Since b is an irrational number, and c is a rational number, then the quotient c/b is also an irrational number. Therefore, a is the quotient of two irrational numbers, which means that it must be irrational as well. This is a contradiction because we assumed that a is an irrational number. Therefore, because our assumption leads to a contradiction, we can conclude that the product of two irrational numbers is generally irrational.

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