Squares and Square Roots

Square Numbers

  • A square number is a natural number mm that can be expressed as n2n^2, where nn 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 n2n^2 and (n+1)2(n + 1)^2, there are 2n2n non-perfect square numbers.
  • The sum of the first nn odd natural numbers is n2n^2.
  • (a+1)×(a–1)=a2–1(a + 1) × (a – 1) = a^2 – 1.

Finding the Square of a Number

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

Pythagorean Triplets

  • For any natural number m > 1, (2m)2+(m2–1)2=(m2+1)2(2m)^2 + (m^2 – 1)^2 = (m^2 + 1)^2.
  • 2m2m, m2–1m^2 – 1, and m2+1m^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 nn-digits, then its square root will have n2\frac{n}{2} digits if nn is even or (n+1)2\frac{(n + 1)}{2} if nn is odd.