Key Points

A natural number $m$ is called a square number or a perfect square if it can be expressed as $n^2$, where $n$ is also a natural number. For example, $25$ is a perfect square because $25 = 5^2$.

A perfect square number can only end with the digits $0, 1, 4, 5, 6,$ or $9$. Numbers ending in $2, 3, 7,$ or $8$ are never perfect squares.

A number that is a perfect square can only have an even number of zeros at the end. For example, $100$ ($10^2$) is a perfect square, but $1000$ is not.

The square of an even number is always an even number, for instance $6^2 = 36$. The square of an odd number is always an odd number, for instance $7^2 = 49$.

Finding the square root is the inverse operation of squaring. The positive square root of a number $m$ is denoted by the symbol $\sqrt{m}$. For example, since $9^2 = 81$, the square root of $81$ is $\sqrt{81} = 9$.

The sum of the first $n$ odd natural numbers is $n^2$. For example, $1 + 3 + 5 = 9 = 3^2$. This property can be used to identify perfect squares.

There are $2n$ non-perfect square numbers between the squares of two consecutive numbers, $n$ and $(n+1)$. For instance, between $3^2=9$ and $4^2=16$, there are $2 \times 3 = 6$ numbers: $10, 11, 12, 13, 14, 15$.

To find the square root of a perfect square, first find its prime factors. A number is a perfect square if all its prime factors exist in pairs. The square root is the product of one prime factor from each pair.

To find the smallest number to multiply or divide a given number by to make it a perfect square, find its prime factorization. The product of the unpaired factors is the required smallest number.

A collection of three positive integers $a, b, c$ is a Pythagorean triplet if $a^2 + b^2 = c^2$. For any natural number $m > 1$, the numbers $(2m, m^2-1, m^2+1)$ form a Pythagorean triplet.

A perfect square can be reduced to zero by successively subtracting odd numbers starting from $1$. The number of subtractions performed gives the square root of the number. For example, for $25$: $25-1=24, 24-3=21, 21-5=16, 16-7=9, 9-9=0$. It took 5 steps, so $\sqrt{25}=5$.

The long division method is used for finding square roots of large numbers. It involves placing bars over pairs of digits from the right, and then performing a division-like process to find the square root digit by digit.

If a perfect square has $n$ digits, its square root will have $\frac{n}{2}$ digits if $n$ is even, or $\frac{(n+1)}{2}$ digits if $n$ is odd. This can be quickly determined by placing bars over pairs of digits.

To find the square root of a decimal, use the long division method. Place bars on the integral part from right to left, and on the decimal part from left to right. Add zeros to the decimal part if needed to complete pairs.