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Proof: No Two Integers Equidistant from Square Root of 2
Published on August 23, 2024
Algebra
Discover why no two integers can be the same distance from the square root of 2. This math problem explores the concept of irrational numbers and distance, proving that integers and the square root of 2 cannot share equidistant points.
Question
How to proof it
Answer
Let’s prove this statement by contradiction.
1. Assume the opposite:
Suppose there are two integers, let’s call them ‘a’ and ‘b’ (where a > b), that are equidistant from the square root of 2.
2. Express the assumption mathematically:
This means the distance between ‘a’ and √2 is equal to the distance between ‘b’ and √2. We can write this as:
- a - √2 = √2 - b
3. Solve for √2:
- Add √2 to both sides: a = 2√2 - b
- Add ‘b’ to both sides: a + b = 2√2
- Divide both sides by 2: (a + b) / 2 = √2
4. The contradiction:
We’ve arrived at a contradiction! Here’s why:
- ‘a’ and ‘b’ are integers, so their sum (a + b) is also an integer.
- Dividing an integer by 2 results in either an integer or a number with a decimal part of 0.5.
- However, √2 is irrational. This means it cannot be expressed as a simple fraction of two integers, and its decimal representation neither terminates nor repeats.
5. Conclusion:
Our initial assumption that two integers are equidistant from √2 led to a contradiction. Therefore, the original statement is true: No two integers are equidistant from the square root of 2.