Why The Sum And Product Of Two Irrational Numbers Are Always Irrational? – A Deeper Look Into The Properties Of Irrational Numbers.

sum or product of two irrational numbers is

rational or irrational

an irrational number.

To understand why, let’s first define what is an irrational number. An irrational number is a number that cannot be expressed as a ratio of two integers. Some examples of irrational numbers include the square root of 2, pi, and the golden ratio.

Now, let’s consider the sum of two irrational numbers. Let’s say we have two irrational numbers, x and y. We know that x and y cannot be expressed as a ratio of two integers. If we add these two numbers together, we get:

x + y

This sum cannot be expressed as a ratio of two integers either. To see why, let’s assume the sum can be expressed as a ratio of two integers, a/b:

x + y = a/b

Now, if we rearrange this equation, we get:

x = a/b – y

However, we know that x and y cannot be expressed as a ratio of two integers. Therefore, a/b must also be an irrational number. This means that the sum of two irrational numbers is also an irrational number.

Similarly, if we consider the product of two irrational numbers, x and y, we get:

xy

Again, let’s assume that this product can be expressed as a ratio of two integers, a/b:

xy = a/b

We can rearrange this equation to get:

x = a/(by)

However, we know that x and y cannot be expressed as a ratio of two integers. Therefore, a/b must also be an irrational number. This means that the product of two irrational numbers is also an irrational number.

Therefore, we can conclude that the sum or product of two irrational numbers is always an irrational number.

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