Squares and Square Roots - Properties of Square Numbers

Definition of a Perfect Square: A number $n$ is called a perfect square if there exists an integer $m$ such that $n = m^2$. Visually, a perfect square can be represented as a square grid of dots where the number of rows equals the number of columns, such as $3 \times 3 = 9$.

Ending Digits (Units Place): Perfect squares always end with the digits $0, 1, 4, 5, 6,$ or $9$ at the unit's place. A number ending in $2, 3, 7,$ or $8$ is never a perfect square. For example, $121$ (ends in $1$) is a square, while $127$ is not.

Property of Zeroes: A perfect square always ends in an even number of zeroes. For instance, $100$ ($2$ zeroes) and $10,000$ ($4$ zeroes) are perfect squares, but $1,000$ ($3$ zeroes) is not. Visually, this corresponds to pairs of powers of $10$.

Sum of Odd Numbers: The sum of the first $n$ consecutive odd natural numbers is exactly $n^2$. This can be visualized by adding 'L-shaped' borders of dots (gnomons) to a single dot: $1$ (which is $1^2$), $1+3=4$ (which is $2^2$), $1+3+5=9$ (which is $3^2$).

Numbers between Successive Squares: Between any two consecutive square numbers $n^2$ and $(n+1)^2$, there are $2n$ non-perfect square numbers. For example, between $2^2=4$ and $3^2=9$, there are $2(2) = 4$ non-square numbers: $5, 6, 7,$ and $8$.

Parity of Squares: The square of an even number is always even, and the square of an odd number is always odd. For example, $12^2 = 144$ (even) and $15^2 = 225$ (odd).

Pythagorean Triplets: For any natural number $m > 1$, the numbers $2m, m^2-1,$ and $m^2+1$ form a Pythagorean Triplet. This means $(2m)^2 + (m^2-1)^2 = (m^2+1)^2$, which geometrically represents the sides of a right-angled triangle.

Square of a Number ending in 5: If a number has $5$ in its units place, its square ends in $25$. The digits before $25$ can be calculated by multiplying the tens digit $a$ by $(a+1)$. For example, $25^2 = (2 \times 3)$ followed by $25$, which is $625$.