Perfect Square Definition

A perfect square is an integer that can be expressed as the product of an integer with itself. In simpler terms, it's the square of an integer. For example, $9$ is a perfect square because $9 = 3 \times 3 = 3^2$. Similarly, $100$ is a perfect square because $100 = 10 \times 10 = 10^2$. We need to find the perfect square number that is nearest to $2359$.

Finding Perfect Squares Near 2359

To find the perfect square nearest to $2359$, we first need to identify the perfect squares that are close to this number. A good way to start is by estimating the square root of $2359$.

• We know that $40^2 = 40 \times 40 = 1600$.
• We also know that $50^2 = 50 \times 50 = 2500$.

Since $2359$ is between $1600$ and $2500$, its square root must be between $40$ and $50$. As $2359$ is closer to $2500$ than $1600$, the square root will be closer to $50$. Let's try squaring integers around the middle point, or closer to $50$:

• Let's try $48^2$:

  $\qquad 48^2 = 48 \times 48 = 2304$

• Let's try $49^2$:

  $\qquad 49^2 = 49 \times 49 = 2401$

So, we have identified two perfect squares near $2359$: $2304$ and $2401$. Now, we need to determine which one is closer to $2359$.

Calculating Proximity to 2359

To find out which perfect square is nearest to $2359$, we will calculate the absolute difference between $2359$ and each of the perfect squares we found.

Comparing the two differences:

• The difference between $2359$ and $2304$ is $55$.
• The difference between $2359$ and $2401$ is $42$.

Since $42$ is less than $55$, the perfect square $2401$ is closer to $2359$ than $2304$ is.

Conclusion

Based on our calculations, the nearest perfect square number to $2359$ is $2401$. This corresponds to $49^2$.