Squares and Square Roots - Properties of Square Numbers

A square number, also called a perfect square, is obtained by multiplying a number by itself, denoted as $n^2 = n \times n$. Visually, this can be represented as an $n \times n$ grid of dots where the total number of dots equals the square number.

The ending digit (unit's place) of a perfect square follows a specific pattern: it must end in $0, 1, 4, 5, 6,$ or $9$. Numbers ending in $2, 3, 7,$ or $8$ are never perfect squares. Furthermore, if a number ends in $1$ or $9$, its square ends in $1$; if it ends in $4$ or $6$, its square ends in $6$.

Regarding zeros at the end of a number, if a number ends in $k$ zeros, its square will end in $2k$ zeros. This implies that a perfect square always contains an even number of zeros at the end (e.g., $10^2 = 100$, $100^2 = 10000$).

Between the squares of two consecutive natural numbers $n^2$ and $(n+1)^2$, there are exactly $2n$ non-perfect square numbers. For example, between $2^2=4$ and $3^2=9$, there are $2 \times 2 = 4$ non-square numbers ($5, 6, 7, 8$).

The sum of the first $n$ odd natural numbers is equal to $n^2$. This can be visualized as 'adding layers' to a square: $1$ (a $1 \times 1$ square), $1+3=4$ (a $2 \times 2$ square), $1+3+5=9$ (a $3 \times 3$ square).

For any natural number $m > 1$, we can generate a Pythagorean Triplet using the values $2m, m^2-1,$ and $m^2+1$. These three numbers satisfy the property that the sum of the squares of the first two equals the square of the third, representing the sides of a right-angled triangle.

A special property for numbers ending in $5$: the square of a number $(a5)$ can be calculated by multiplying $a$ by its successor $(a+1)$ and suffixing $25$ to the result. Visually, this simplifies calculating areas of squares with side lengths like $15, 25, 35, \dots$.