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)