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Chapter 0: Problem 9

Give an example of two irrational numbers whose sum is an irrational number.

Short Answer

Expert verified
An example of two irrational numbers whose sum is an irrational number is \(\sqrt{2}\) and \(\pi\).

Step by step solution

01

Choose the first irrational number

We can select a well-known irrational number, such as the square root of 2, denoted as \(\sqrt{2}\), for our first irrational number.
02

Choose the second irrational number

We can choose another well-known irrational number, such as pi. Pi, denoted as \(\pi\), is an irrational number representing the ratio of a circle's circumference to its diameter.
03

Sum the two irrational numbers

Now, we need to add the two irrational numbers together: \(\sqrt{2} + \pi\).
04

Verify that the sum is irrational

The sum of \(\sqrt{2} + \pi\) is highly likely to be irrational due to the properties of transcendental numbers like pi and the square root of a prime number, both of which cannot be expressed as the quotient of two integers. Since no known algebraic relationship exists between \(\sqrt{2}\) and \(\pi\), we can safely assume the sum is also an irrational number. In conclusion, an example of two irrational numbers whose sum is an irrational number is \(\sqrt{2}\) and \(\pi\).

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