Why Sum And Product Of Irrational Numbers Are Also Irrational – A Proof.

sum or product of two irrational numbers is

rational or irrational

an irrational number.

To prove this, let’s assume that there are two irrational numbers, x and y, such that their sum is a rational number.

So, x + y = a/b, where a and b are coprime integers, and b is not equal to zero.

We can rewrite this equation as y = a/b – x. Since a/b is a rational number, we know that its decimal expansion either terminates or repeats. This means that we can write a/b as a finite or infinite decimal.

Now, let’s assume that x is also represented as a decimal expansion that either terminates or repeats. In other words, x is a rational number. This means that y would also be a rational number because the difference of two rational numbers is always a rational number.

However, we know that x is an irrational number, and therefore, it cannot be represented as a decimal expansion that terminates or repeats. This contradicts our assumption that x is a rational number, which means that x must also be an irrational number.

Therefore, the sum of two irrational numbers is also an irrational number.

Now, let’s consider the product of two irrational numbers.

Again, let’s assume that x and y are irrational numbers such that their product is a rational number.

So, xy = a/b, where a and b are coprime integers, and b is not equal to zero.

As we have seen earlier, if a/b is a rational number, then its decimal expansion either terminates or repeats. This means that we can write a/b as a finite or infinite decimal.

Now, let’s assume that x is also represented as a decimal expansion that either terminates or repeats. In other words, x is a rational number.

If xy = a/b, then y = a/(bx). Since a/b is rational and x is rational, bx is also rational. This means that y would also be a rational number because the quotient of two rational numbers is always a rational number.

However, we know that y is an irrational number, and therefore, it cannot be represented as a decimal expansion that terminates or repeats. This contradicts our assumption that y is a rational number, which means that y must also be an irrational number.

Therefore, the product of two irrational numbers is also an irrational number.

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