The document discusses square numbers, square roots, and related mathematical properties, including patterns in unit digits, even and odd characteristics of squares, and methods for finding square roots and perfect squares. It also introduces the Pythagorean theorem and provides examples and quizzes to illustrate these concepts. Additionally, it details methods such as repeated subtraction and prime factorization to determine squares and roots.

2. Square and square roots
Mathematics

6. Square Number
The number you get when you multiply by itself.
Example: 4 x 4 = 16, So 16 is the square
number.
0 (= 0 x 0)
4 (= 2 x 2)
9 (= 3 x 3)
16 (= 4 x 4)

8. If a number is ends with 2, 3, 7 and 8 at unit’s
place. So it must not be a square number.
Example: 132, 72 , 28
If a number ends with 1, 4, 5, 6 and 9 at unit’s
place. So it may be a square number.
Example: 1) 16 (Yes) 16 = 4 x 4
2) 15 (No) As it ends with 5.
Perfect squares

9. 1) 12
1 x 1 = 1 & 9 x 9 = 81
If a number has 1 or 9 in the unit’s place,
2) 92
Then it’s square ends with 1.
3) 22
4) 82
2 x 2 = 4 & 8 x 8 = 64
If a number has 2 or 8 in the unit’s place,
Then it’s square ends with 4.
Property of unit digit of a square number.

10. 5) 32
3 x 3 = 9 & 7 x 7 = 49
If a number has 3 or 7 in the unit’s place,
6) 72
Then it’s square ends with 9.
7) 42
8) 62
4 x 4 = 16 & 6 x 6 = 36
If a number has 4 or 6 in the unit’s place,
Then it’s square ends with 6.

12. If a number is odd so it’s square is also odd, if
a number is even so it’s square is also even.
Example: 1) 16
2) 25
3) 121
4) 144
5) 49 = 72
= 122
= 112
= 52
= 42
( Odd )
( Even )
( Odd )
( Odd )
( Even )
Even and odd squares

13. If the sum of numbers are in consecutive from 1
and odd so the square root of these numbers is
number of observation.
Example: 1) 1+3+5+7+9+11+13 = 72
2) 1+3+5+7+9+11+13+15+17
3) 1+3+5+7+9+11+13+15+17+19+21+23
= 92
= 122
Adding odd numbers

14. Odd number of zeros
If a number ends with odd number of zeros then
it is not a perfect square.
Example: 1) 30
2) 5000
4) 100
3) 400000
5) 60000
= Not perfect square
= Not perfect square
= Not perfect square
= Perfect square
= Perfect square

15. If a number contain 1 zero at the end, so 2 zeros
will have its square. It means the number
contains zeros will double zeros of its square.
Example: 1) 302
3) 4002
2) 50002
4) 100002
5) 6000002
= 2 zeros
= 6 zeros
= 4 zeros
= 8 zeros
= 10 zeros
Zeros in the square number

16. Numbers between square numbers
We can find some numbers between two
consecutive square numbers. By finding the
square numbers of any two number.
Between 12
(=1) and 22
(=4)
there are two non square numbers
Between 22
(=4) and 32
(=9)
there are four non square numbers.
= 2, 3.
= 5, 6, 7, 8, 9.

17. The non square numbers which is 1 less than
the difference of two consecutive squares.
Example: 1) 9 (=32
) and 16 (=42
)
(n - 1) – n
= (16 - 1) – 9 = 6
2) 16 (=42
) and 25 (=52
)
4) 36 (=32
) and 49 (=72
)
3) 25 (=52
) and 36 (=62
)
5) 49 (=72
) and 64 (=82
)
= (25 - 1) – 16 = 8
= (36 - 1) – 25 = 10
= (49 - 1) – 36 = 12
= (64 - 1) – 49 = 14
Non square numbers

19. Pythagoras theorem
The square of a numberis equal to the square of
two othernumbers. This property is called
Pythagoras property and the three numbers used in
this property are called Pythagoras theorem.
5cm
3cm
4cm
52
= 32
+ 42
25 = 9 + 16
25 = 25
Right angle triangle

20. The formula to finding more such triplets.
(2m2
) + (m2
-1) = (m2
+1)
Example: 1) 62
+ 82
= 102
36 + 64 = 100
100 = 100
So 6, 8, 10 are
Pythagorean triplets.
1) 32
+ 42
= 52
9 + 16 = 25
25 = 25
So 3, 4, 5 are
Pythagorean triplets.
Pythagorean triplets

21. 1) Pythagorean triplet whose smallest member is
12.
Finding Pythagorean triplets
1) 2m = 12
M = 6
2) 62
– 1
36 -1 = 35
3) 62
+ 1
36 +1 = 37
Check: 122
+ 352
= 372
144 + 1225 = 1369
1369 = 1369 So 12, 35, 37 are
Pythagorean triplets.

22. 2) Pythagorean triplet whose member is 5.
2) 2 x 3 = 61) m2
– 1 = 5
m2
= 6
m = 6/2 = 3
3) 32
+ 1
9 +1 = 10
Check: 52
+ 62
= 102
25 + 36 = 100
100 = 100
So 5, 6, 10 are
Pythagorean triplets.

24. Finding square roots
Repeated Subtraction
The subtraction of odd numbers consequently
to the number. Still we get 0. So the number of
observation we subtract to the number is the
square root of that number.

25. Repeated Subtraction
1) 64 – 1 = 63
2) 63 – 3 = 60
4) 55 – 7 = 48
3) 60 – 5 = 55
5) 48 – 9 = 39
6) 39 – 11 = 28
8) 15 – 15 = 0
7) 28 – 13 = 15
= 82
Find the square root of 64.
So the square root of 64 is
8.

26. Find the square root of 89.
1) 89 – 1 = 88
2) 88 – 3 = 85
3) 85 – 5 = 80
4) 80 – 7 = 73
5) 73 – 9 = 64
6) 64 – 11 = 53
7) 53 – 13 = 40
8) 40 – 15 = 25
9) 25 – 17 = 8
If something is remaining
from the subtraction. So,
it is not the perfect
square.

28. Prime factorization
2562
128
64
322
2
2
2 16
82
42
22
1
2 x 2256 = x
x
x
2 x 2
2 x 2
= 162 x 2

29. Property of unit digit of square root
1) 1
2) 9
If a square number has 1 or 9 in the
unit’s place, then it’s square root
ends with 1
If a square number has 2 or 8 in the
unit’s place, then it’s square root
ends with 4
3) 2
4) 8

30. 5) 3 If a square number has 3 or 7 in the
unit’s place, then it’s square root
ends with 9.6) 7
7) 4
8) 6
If a square number has 4 or 6 in the
unit’s place, Then it’s square root
ends with 6.

32. Square root of decimals
Step1) To find the square root of decimal
numbers put a bar on the internal part. Place the
bar on the decimal part.
23.04
23.04

33. Step 2) The left most bar 23 and 42
< 23 < 52
.
Take this number as the divisor and get the
remainder.
23.044
4
16
7

34. Step 3) The remainder is 7. Write the number
under the next bar to the right of this remainder,
to get 704. Double the divisor and enter it with a
blank on its right.
4
4 23.04
16
7048_

35. Step 4) We know that 88 x 8 = 704, therefore, the
new digit is 8. Divide and get the remainder.
48
4 23.04
16
70488
704
0
= 4.8

36. Finding Square by division method
Step1) To find the square of any number put
two bar on the internal part.
625
6 25

37. Step 2) Find the largest number whose square is
less than or equal to the number under the
extreme left bar ( 22
< 52
< 32
). Take this number
as the divisor. Divide and get the remainder.
6 252
2
4
2

38. Step 3) The remainder is 7. Write the number
under the next bar to the right of this remainder,
to get 225. Double the divisor and enter it with a
blank on its right.
2
2 6 25
4
2254_

39. Step 4) We know that 45 x 5 = 225, therefore, the
new digit is 5. Divide and get the remainder.
25
2 6 25
16
22545
225
0
= 25

40. Estimating Square root
In all such cases we need to estimate the
square root.
We know that 112
< 125 < 122
121 < 125 < 144
So 121 is much closer to 125 than 144, therefore
11 is the estimating square root of 125.

42. Quiz
Q1) Which one of the following number is a
perfect squares:
a) 622
d) 625
b) 393
c) 5778
Answer: d) 625

43. Q2) Check which of the following is not a
perfect square :
a) 81000
d) 6250000
b) 8100
c) 900
Answer: a) 81000

44. Q3) Which of the following perfect square
numbers, is the square of an odd number:
a) 289
d) 1600
b) 400
c) 900
Answer: a) 289

45. Q4) Which of the following perfect square
numbers, is the square of an even number:
a) 361
d) 2601
b) 625
c) 4096
Answer: c) 4096

46. Q5) How many natural numbers lie between
squares of 11 and 12.
a) 22
d) 25
b) 23
c) 24
Answer: a) 22
112
, 122
(144 - 1) - 121 = 22

47. Q6) Find the unit digit of (564)2
a) 2
d) 8
b) 4
c) 6
Answer: c) 6

48. Q7) Find the unit digit of square root of (1521)
a) 1 or 9
d) 8 or 5
b) 4 or 6
c) 2 or 7
Answer: a) 1 or 9

49. Q8) 1 + 3 + 5 +7 + 9 + 11 + 13 + 15 is a perfect
square of number:
a) 8
d) 9
b) 7
c) 6
Answer: b) 8

50. Q9) What is the formula to find Pythagorean
triplets:
a) (m) + (m2
-1) = (m2
+1)
d) (2m) + (m2
) = (m2
+1)
b) (2m) + (m2
-1) = (m2
+1)
c) (2m) + (m-1) = (m+1)
Answer: b) (2m) + (m2
-1) =
(m2
+1)

51. Q10) Can a prime number be perfect square:
GUESS
!
False!
Q10) If a number is odd so it’s square is also
odd:
True!