Irrational Numbers Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

easy
Is 49+3\sqrt{49} + \sqrt{3} rational or irrational? Explain.

Solution

  1. 1
    49=7\sqrt{49} = 7 (rational). 3\sqrt{3} is irrational (3 is not a perfect square).
  2. 2
    The sum of a rational and an irrational number is always irrational.

Answer

Irrational\text{Irrational}
Adding a rational number to an irrational number always produces an irrational result, because if the sum were rational, subtracting the rational part would make the irrational part rational—a contradiction.

About Irrational Numbers

An irrational number is a real number that cannot be expressed as a ratio of two integers pq\frac{p}{q}; its decimal expansion goes on forever without repeating any fixed block of digits.

Learn more about Irrational Numbers →

More Irrational Numbers Examples

Example 1 easy

Classify each number as rational or irrational: [formula], [formula], [formula], [formula].

Example 2 medium

Show that [formula] lies between [formula] and [formula], then estimate it to one decimal place.

Example 3 medium

Prove that [formula] is irrational by contradiction.

Example 5 easy

Is [formula] rational or irrational?

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