1.2. Squares and square roots:
We have
studied in previous classes that the area of a rectangle is =
length * breadth When
the length and breadth are equal we have a Square figure like the adjacent
figure (PQ=QR) Its
area is PQ*QR= length*length = (length)^{2} 

Have you observed some
thing unique about the numbers 1, 4, 9, 16 among 1,2,3,4,5…
Let us look at the
multiplication table:
Multiplication table for 2 
Multiplication table for 3 
Multiplication table for 4 
Multiplication table for 5 
2*1 =
2 
3*1 =
3 
4*1 = 4 
5*1 = 5 
2*2 = 4 
3*2
=6 
4*2 =8 
5*2 =10 
2*3 = 6 
3*3 = 9 
4*3 =12 
5*3 =15 

3*4 =
12 
4*4 =16 
5*4 =20 


4*5 =
20 
5*5 =25 
What do you observe? The
numbers 4,9,16 and 25 are squares of 2, 3, 4 and 5 respectively.
Definition
: A perfect square
is a number which can be expressed as a
product of two same numbers
Also notice:
1^{2} = 1 = (1)^{2}
2^{2} = 4 = (2)^{2}
3^{2} = 9 = (3)^{2}
Square numbers are integers
raised to the power of 2. They are of the form n^{2}
Properties
of Square numbers:
1. The
digit in the units place of a perfect square is always 0,1,4,5,6,9(
1,4,9,16,25,36,49,64,81,100,121…)
2. Numbers
ending with 2, 3, 7 or 8 cannot be perfect squares (43, 62, 57, 98 are not
perfect squares)
3. The
square of an even number is always even (4, 16, 36, 64, 100 …)
4. The
square of an odd number is always odd (1, 9, 25, 49, 81, 121…)
5. The
square numbers do not have (2, 3, 7, and 8) as digits in their unit place.
6. A
perfect square cannot have a reminder of 2 when divided by 3(400÷3 gives a
reminder of 1. 324÷3 gives a reminder of 0. Since 455÷3 gives a reminder of 2,
it cannot be a perfect square).
7. Number
of zeros at the end of a perfect square is always even (Ex. 100, 2500, etc).
8. If
n is a perfect square and p is a prime number then p*n cannot be a perfect
square (81 is a prefect square but 2*81, 3*81 are not perfect squares).
9. Square
of a negative number is always positive (4*4 = 16).
Squares of decimals and
fractions can also be found.
Fraction 
Decimal Number 
Square of fraction 
Square of decimal 
_{}2/3 

4/9 

_{}1/10 
_{}.1 
1/100 
.01 
_{}6/10 
_{}.6 
36/100 
.36 
_{}12/10 
_{}1.2 
144/100 
1.44 
_{}2/100 
_{}.02 
4/10000 
.0004 
Square roots:
We have seen that square of
3 is 9. We say that 3 is the square root of 9.
If n^{2
} is the square of a number n then n is the ‘square root’
of the number n^{2} . Square
root is denoted by_{} or _{} and is pronounced as
“square root”.
Note:
Square root of a fraction =
(square root of numerator/square root of denominator)
_{} = _{}/ _{}
Square root of 
= Number 
_{} 
=_{}_{} 
_{} 
=_{}_{} 
_{} 
=_{}1.2 
_{} 
=_{}.02 
_{} 
=_{}5 
^{ }
1.2. Problem 1: Find the two integers between which the square
root of 147 lies?
Solution:
We know that 12^{2}=144^{
}and 13^{2}=169
We also know that
144<147<169
^{}_{} < _{}< _{}^{}
^{} 12 < _{}< 13^{}
1.2.1 Finding square root by factorisation
In this method we find all
prime factors of the given number and then group the common factors in pairs.
If some factors do not
appear in pairs then the number is not a perfect square and we stop the process
of grouping.
1.2.1 Problem 1:
Find the square root of _{}
Solution:
By successive division we
find that factors of 38025 are 5, 5,3,3,13,13
^{} 38025 = 5*5*3*3*13*13 = 5^{2}*3^{2}*13^{2}
= (5*3*13)^{2}
^{} _{} = 5*3*13=195^{}
By successive division we
find that factors of 10404 are 2, 2,3,3,17,17
^{} 10404 = 2*2*3*3*17*17 = 2^{2}*3^{2}*17^{2}
= (2*3*17)^{2}
^{} _{} = 2*3*17=102
^{= = }_{} = _{}^{}
^{ }
1.2.1 Problem 2: Find
the least number by which 2817 must be multiplied or divided to make it a
perfect square.
Solution:
By successive division we
find that factors of 2817 are
3,3,313.
^{} 2817 = 3*3*313.
We note that the factor,
313 appears only once.
If we multiply 2817 by 313,
then factors of (2817*313) are 3, 3, 313, 313 so that 2817*313= =3*3*313*313 =
3^{2}*313^{2} = (3*13)^{2}
If we divide 2817 by 313
then factors of (2817/313) are 3, 3 so that 3617/313 =3*3 = 3^{2}
Therefore 313 is the smallest number which when multiplies or divides 2817
gives us the perfect square.
1.2.1 Problem 3: How
much length of the wire is required to fence four rounds around a square garden
whose area is 3600 sq.mts?
Solution:
In order
to find the total length of wire required to fence the garden, we need to
know its length and breadth. We know
that the area of the square garden = (length)^{2}=3600 The
factors of 3600 are 2,2,2,2,3,3,5,5 ^{} 3600 = 2*2*2*2*3*3*5*5 = 2^{2}*2^{2}*3^{2}*5^{2}=(2*2*3*5)^{2} ^{} _{} = 2*2*3*5=60 Therefore
side of the square garden = 60 meters. Perimeter
of the square garden is sum of its 4 sides = 4*length of the square garden ^{} The length of wire required
to fence the square garden = 240(=4*60) meters. Since
we need to fence four rounds around the square garden, Total
length of wire required = 4*perimeter = 4*240 = 960 meters 

1.2 Summary of
learning
No 
Points studied 
1 
Perfect
squares, finding square root by factorisation method 
^{ }
Additional Points:
Diagonal Method of finding
squares (This
is an ancient Indian method used for multiplication)
As an example let us find
the square of the number 852:
Step 1:
Write the digits 8, 5 and 2 both horizontally and vertically in a table as
shown in adjacent Figure Also write
the product of each digit in the horizontal line with each digit in the vertical
line in the appropriate box 


Step 2:
Diagonals are drawn across all the
boxes where product of digits are written (box containing 64, 40, 25…) as shown
in the adjacent figure Step 3:
In each of the box write the unit’s number below the diagonal and the ten’s
digit above the diagonal (In case of 40, write 4 above the diagonal and 0
below the diagonal). If the product of digits is a single digit number, write
0 above the diagonal and the single digit number below the diagonal (4 = 04) 


Step 4
: Start adding numbers across the diagonals from the lowest diagonal, taking
into consideration the carry from previous step if any, as follows: 


The result is 725904 (Digits taken from the
‘Unit’ place in the
above table)
This method holds good even
for large numbers.
Properties of square roots:
1. If
the unit’s digit of a number is 2, 3, 7 or 8 then the number does not have a
square root. The possible square roots in case of other numbers are as follows:
Units
digits of square ==è 
0 
1 
4 
5 
6 
9 
Units
digit of square root==è 
0 
1 or 9 
2 or 8 
5 
4 or 6 
3 or 7 
2. The square root of an
even square number is even and the square root of an odd square number is odd.
3. If the number ends with
odd number of zeros then the number will not have an integer square root.
4. Negative numbers do not have real square root
number.
5. Square roots of a number
can be positive or negative (_{} = _{}5).
6. Square root of a rational number = Square
root of numerator/Square root of denominator (_{} = _{}/_{} = 8/5)