https://math.stackexchange.com › questions › 157245

2 Sum of a rational and an irrational is always irrational. Proof : Let the rational number be of the form $\dfrac{p}{q}$, where $p \in \mathbb{Z}$ and $q \in \mathbb{Z} \backslash \{0\}$ while the irrational number be $r$.

https://plainmath.org › discrete-math › 1861-proof-contradiction-prove-irrational-number...

To prove that the sum of an irrational number and a rational number is irrational using proof by contradiction, we assume the opposite, i.e., we assume that the sum of an irrational number a and a rational number b is rational. Let's denote the sum as c: c = a + b.

https://mathnovice.com › numbers › sum-of-rational-irrational-number-is-irrational

Learn how to add rational and irrational numbers and why the sum is always irrational. See proof, examples, and known irrational numbers.

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Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:irra...

https://openstax.org › books › contemporary-mathematics › pages › 3-5-irrational-numbers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point. So irrational numbers must be those whose decimal representations do not terminate or become a ...

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What about a rational number and an irrational number? The sum of a rational number and an irrational number is irrational. To explain why requires a slightly different argument: Let \(R\) be a rational number and \(I\) an irrational number. We want to show that \(R+I\) is irrational.

https://www.math.net › irrational-numbers

Learn what irrational numbers are, how to recognize them, and how they differ from rational numbers. Find out the properties of irrational numbers under addition, subtraction, multiplication, and division, and see examples of each operation.

https://math.stackexchange.com › questions › 1415735

When a statement like that is given to prove or disprove, " sum of two irrationals is irrational" , it is proved if it is found to be always true and disproved if at least one counter example can be given.