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Encapsulate
math

SqUArES ANd SqUArE rooTS

Square Numbers Perfect Squares Pythagorean Triplets

A number multiplied by A natural number n is said to A set of three numbers a, b, and c is called
2
2
2
2
itself. E.g., a × a = a . Read be a perfect square if there Pythagorean triplet if a + b = c . For any
2
as a squared and is always exists some natural number natural number m > 1, 2m, (m – 1) and
2
2
positive. m such that n = m × m = m . (m + 1) form a Pythagorean triplet.
Properties of Perfect Squares Square root of Numbers
• A number having 2, 3, 7 or 8 at unit The square root of a number n is that number which when
place is never be a perfect square multiplied by itself gives the number n as the product. is
number. the symbol for square root. The square root of n is denoted
• The number of zeros at the end of a by n .
perfect square number is always even.
• Squares of even numbers are even and • a × b = ab or ab = a × b
×
squares of odd numbers are odd.
• For every natural number n, the • a = a or a = a
difference of squares of two consecutive b b b b
+
−
natural numbers is equal to their sum. • a + b ≠ ab • a − b ≠ ab
• The number of non-perfect square
numbers between the squares of two Patterns of Square Numbers
consecutive numbers n and (n + 1) is
2n, where n is the natural number. • By adding two triangular numbers we get a square number.
• For any natural number n, the square of • The square of any odd number, other than 1 can be
n, i.e., n is equal to the sum of the first expressed as the sum of two consecutive natural numbers.
2
n odd natural numbers. • 11 = 121
2
• The square of a natural number (other 111 = 12321
2
than 1) is either a multiple of 3 or M M
exceeds a multiple of 3 by 1. 111111 = 12345654321
2
2
• The difference between the squares of • 7 = 49
two consecutive natural numbers is 67 = 4489
2
equal to their sum, i.e., for every natural 667 = 444889
2
number n, (n + 1)² – n² = {(n + 1) + n}. 6667 = 44448889
2

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