No, the square root of 8 is not a rational number. According to the provided reference, the square root of 8 is an irrational number.
Rational vs. Irrational Numbers
To understand why √8 is irrational, let's briefly define rational and irrational numbers:
- Rational Numbers: Can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. They have terminating or repeating decimal representations.
- Irrational Numbers: Cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating.
Why √8 is Irrational
√8 can be simplified to 2√2. The number √2 itself is a well-known irrational number. Since multiplying an irrational number (√2) by a rational number (2) results in another irrational number, 2√2 (or √8) is therefore irrational.
Further Explanation
Consider trying to express √8 as a fraction:
- If √8 = p/q (where p and q are integers), then squaring both sides gives us 8 = p²/q².
- Rearranging, we get 8q² = p².
- This implies p² is an even number (because it's a multiple of 8). Therefore, p must also be an even number. Let p = 2k.
- Substituting p = 2k into 8q² = p² gives us 8q² = (2k)² = 4k².
- Dividing both sides by 4 gives us 2q² = k².
- This implies k² is even, so k must also be even.
We have shown that if √8 can be written as p/q, then both p and k must be even. This means we can keep simplifying the fraction infinitely, which contradicts the initial assumption that p/q is in simplest form. Therefore, √8 cannot be expressed as a fraction of two integers and is hence, irrational.
