Proof: No Two Integers Equidistant from Square Root of 2
Published on August 23, 2024
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.