Sum of Surds Irrationality - Proof by Contradiction

Problem Statement

Prove by contradiction that the real number $x = \sqrt{2} + \sqrt{3} + \sqrt{5}$ is irrational.

Hints

• The Setup: Begin with the standard assumption for contradiction: let $x \in \mathbb{Q}$.
• Strategic Isolation: Do not square the expression as it is. Squaring a trinomial creates a mess of cross-terms. Isolate one of the surds (e.g., move $\sqrt{5}$ to the left) before squaring.
• The Double Square: Squaring once will leave you with a $\sqrt{5}$ term on the left and a $\sqrt{6}$ term on the right. Isolate the $\sqrt{6}$ term and square a second time to eliminate it.
• The Contradiction Engine: Rearrange your final equation to make $\sqrt{5}$ the subject. Rely on the closure property of rational numbers (addition, subtraction, multiplication, and non-zero division of rationals always yield a rational) to force the contradiction.

Solutions

Proof by Contradiction

Step 1: Assume the negation

Assume, for the purpose of contradiction, that $x$ is a rational number ($x \in \mathbb{Q}$).

Step 2: Isolate one surd

Rearrange the equation to isolate one surd: $x - \sqrt{5} = \sqrt{2} + \sqrt{3}$

Step 3: Square both sides

Step 4: Square a second time

Step 5: Make $\sqrt{5}$ the subject

Since $x > 0$, division by $4x^3$ is valid.

Step 6: Derive the contradiction

Because the rational numbers are closed under addition, subtraction, multiplication, and non-zero division, the right-hand side evaluates to a rational number. However, the left-hand side, $\sqrt{5}$, is known to be irrational.

Conclusion

A rational number cannot equal an irrational number. This is a contradiction. Therefore, the initial assumption is false, and $x = \sqrt{2} + \sqrt{3} + \sqrt{5}$ must be irrational.

Takeaways

• Algebraic Foresight: Recognising that squaring $(a + b + c)^2$ yields too many irrational cross-terms is the key separator for top-band students. Isolate one surd before squaring.
• The Power of Closure: The entire proof hinges on knowing that manipulating rational $x$ through integer powers and coefficients guarantees a rational output on the right-hand side.
• Structured Argumentation: The proof demands a clear logical flow from the initial assumption to an explicit statement of why the final line constitutes a mathematical contradiction.
• Double Squaring Technique: This method extends the squaring strategy used in simpler surd proofs (such as comparing $\sqrt{3} + \sqrt{5}$ with $\sqrt{11}$) to irrationality arguments involving multiple surds.
• You are encouraged to prove that $\sqrt{2} + \sqrt{3} + \sqrt{6}$ is irrational using a different method, which can be smarter than the technique used here. Note that $6 = 2 \cdot 3$, which may be useful.