Squares and Square Roots

Square Numbers

• A square number is a natural number $m$ that can be expressed as $n^2$, where $n$ is also a natural number.
• Examples: 1, 4, 9, 16, 25, …

Properties of Square Numbers

• Square numbers end with 0, 1, 4, 5, 6, or 9 in the units place.
• Square numbers can only have an even number of zeros at the end.

Interesting Patterns

• Combining two consecutive triangular numbers results in a square number.
• Between $n^2$ and $(n + 1)^2$, there are $2n$ non-perfect square numbers.
• The sum of the first $n$ odd natural numbers is $n^2$.
• $(a + 1) × (a – 1) = a^2 – 1$.

Finding the Square of a Number

• $(a5)^2 = (10a + 5)^2 = a(a + 1) \, hundred + 25$

Pythagorean Triplets

• For any natural number m > 1, $(2m)^2 + (m^2 – 1)^2 = (m^2 + 1)^2$.
• $2m$, $m^2 – 1$, and $m^2 + 1$ form a Pythagorean triplet.

Square Roots

• Square root is the inverse operation of squaring.
• Positive square root of a number is denoted by the symbol $\sqrt{}$.

Finding Square Root Through Repeated Subtraction

• Every square number can be expressed as a sum of successive odd natural numbers starting from 1.

Finding Square Root Through Prime Factorisation

• Each prime factor in the prime factorisation of the square of a number occurs twice the number of times it occurs in the prime factorisation of the number itself.

Finding Square Root by Division Method

• If a perfect square is of $n$-digits, then its square root will have $\frac{n}{2}$ digits if $n$ is even or $\frac{(n + 1)}{2}$ if $n$ is odd.