The square root of 3 is not a rational number. A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. The square root of 3, however, cannot be expressed in this form.
To understand why, let’s assume that the square root of 3 is rational. This means there exist integers p and q (with q ≠ 0) such that:
√3 = p/q
Squaring both sides, we get:
3 = p²/q²
Multiplying both sides by q², we obtain:
3q² = p²
This implies that p² is divisible by 3, and therefore p must also be divisible by 3. Let p = 3k for some integer k. Substituting back into the equation:
3q² = (3k)²
3q² = 9k²
Dividing both sides by 3:
q² = 3k²
This shows that q² is also divisible by 3, and hence q must be divisible by 3. However, this leads to a contradiction because we initially assumed that p and q have no common factors other than 1 (i.e., they are in their simplest form). Since both p and q are divisible by 3, this assumption is false.
Therefore, the square root of 3 cannot be expressed as a fraction of two integers, and it is an irrational number.