Sqares & Square rootsShyam Prasad Sahoo

Class VIII sec eRoll No 31 School No 5363

DAV Cspur ,BBSR

Properties of Square Numbers

We have learnt about various types of numbers such as

Natural Numbers, Whole Numbers, Integers and Rational Numbers.

Properties of Square NumbersDefinition :Numbers that can be expressed as the

square of a number are called square numbers or perfect squares.

If a and b are two natural numbers such that a = b2,  then a is called a square number or a perfect square of b.

Every natural number has a square, but every natural number is not a perfect square itself.

Numbers that cannot be expressed as the square of another number are not perfect squares.

Since there are infinite natural numbers, there are infinite numbers of perfect squares.

Properties of Square NumbersAll perfect squares:Have 0, 1, 4, 5, 6 or 9 in their units place. Never have 2, 3, 7 or 8 in their units place. The numbers that have 0, 1, 4, 5, 6 or 9 in

their units place maybe perfect squares whereas the numbers that have 2, 3, 7 or 8 in their units place are never perfect squares. 

Properties of Square NumbersAll numbers ending with 1 or 9 have 1 in

the units place of their squares. All squares having 1 in their units place

are squares of the numbers ending with either 1 or 9.

All numbers ending with 2 or 8 have 4 in the units place of their squares

Properties of Square NumbersAll squares having 4 in their units place

are squares of the numbers ending with either 2 or 8.

All numbers ending with 3 or 7 have 9 in the units place of their squares.

All squares having 9 in their units place are squares of the numbers ending with either 3 or 7.

Properties of Square NumbersAll numbers ending with 4 or 6 have 6 in the

units place of their squares. All squares having 6 in their units place are

squares of the numbers ending with either 4 or 6. All numbers ending with 5 have 5 in the units place of their squares.

All squares having 5 in their units place are squares of the numbers ending with 5.

All numbers ending with 0 have 0 in the units place of their squares.

All squares having 0 in their units place are squares of the numbers ending with 0.

Properties of Square NumbersSquare of a number ending with zero(s)

contains double the number of zeroes than the number.

All square numbers contain an even number of zeroes. Odd square numbers are squares of numbers ending with 1, 3, 5, 7 or 9.

Even square numbers are squares of numbers ending with 0, 2, 4, 6 or 8.

Properties of Square NumbersThe numbers whose dot patterns can be

arranged as triangles are called triangular numbers. The sum of any two consecutive triangular numbers is a square number.

Properties of Square NumbersFor two consecutive numbers n and n+1,

there are 2n non square numbers between n2 and (n+1)2.

Sum of first n odd natural numbers is n2. The square of any odd number can be

written as the sum of two consecutive positive integers.

Properties of Square Numbers

Finding Square and Square roots

We know that a number m is called a square number if it is expressed as n2. Here n is called the square root of m. Square root is the inverse operation of squares. Every square number is a sum of the first n odd natural numbers.

Finding Square rootsSquare root of a given number is a number

whose square is equal to the given number. Positive square root of a number is denoted

by the symbol √.  Square root of a number can be found using the following three methods:

Repeated Subtraction MethodPrime Factorisation Method Long Division Method

Finding Squares

Pythagorean Triplet

Finding Square roots

Finding Square roots

Finding Square and Square roots

Cubes and Cube Roots

Shyam Prasad Sahoo Class VIII sec E

Roll No 31, School no 5363DAV Cspur

BBSR

Cubes and Cube Roots

If a and b are two natural numbers such that a3 = b,  then b is called the cube of a.

If the units digit of a3 is b, then the cubes of all numbers ending with a will have their units digit as b.

Properties of Cubes and Cube Roots

The cubes of all numbers that end in 2 have 8 as the units digit. The cubes of all numbers that end in 3 have 7as the units digit.

Properties of Cubes and Cube Roots

Definition:The cube root of a given number is a number, which, when

multiplied with itself three times, gives the number.

The first odd natural number is the cube of 1. The sum of the next two odd natural numbers is the cube of 2. The sum of the next three odd natural numbers is the cube of 3, and so on.

If a given number is a perfect cube, then its prime factors will always occur in groups of three.

The cube root of a number can be found using the prime factorisation method or estimation method.

Properties of Cubes and Cube Roots

Properties of Cubes and Cube Roots

Properties of Cube Roots