Page 42 - ICSE Math 8

3. The square of an even number is always even.
2
2
For example, (4) = 16; (12) = 144 and so on.
4. The square of an odd number is always odd.
2
2
2
For example, (3) = 9; (7) = 49; (19) = 361 and so on.
5. The square of a proper fraction is always smaller than the fraction.
1 
3 
For example,    2 = 11 < 1    2 = 9 ; 9 < 3
;
 2  4 4 2  4  16 16 4
6. The units digit of the square of a natural number is the square of the units
digit of the given number. Units digit Units digit of
In other words, there is a relationship between the units digit of a number of the the square of
and that of its square. number the number
7. For every natural number n, 0 0
2
2
(n + 1) – n = {(n + 1) + n}{(n + 1) – n} = (n + 1) + n 1 or 9 1
In other words, the difference in squares of any two consecutive natural 2 or 8 4
numbers is equal to their sum. 3 or 7 9
2
2
For example, 12 – 11 = 144 – 121 = 23 = 12 + 11. 4 or 6 6
2
8. For every natural number n, the sum of first n-odd natural numbers = n . 5 5
In other words, the square of a natural number n is equal to the sum of the
2
first n-odd natural numbers, i.e., n = 1 + 3 + 5 + ... + (2n – 1).
2
2
For example, 2 = 4 = 1 + 3; 3 = 9 = 1 + 3 + 5 and so on.
Maths Info

Pythagorean Triplet
2
2
2
If three natural numbers a, b, c satisfy the condition a + b = c , then the three natural numbers are said to form a
2
2
2
2
2
2
Pythagorean triplet. For example, 3 + 4 = 5 , 8 + 15 = 17 and so on.
2
2
For any natural number n (n > 1), there exists a Pythagorean triplet (2n, n – 1, n + 1).
For example, let the natural number n be 4 (4 > 1).
2
2
2
2
Now, 2n = 2 × 4 = 8 n – 1 = 4 – 1 = 16 – 1 = 15 n + 1 = 4 + 1 = 16 + 1 = 17
So, Pythagorean triplet is 8, 15, 17.
If a and b are relatively prime (or co-prime) natural numbers (a > b) and one of them is even and the other is odd then
2
2
2
2
the Pythagorean triplet is formed by (2ab, a – b , a + b ). For example, if a = 8, b = 5, (8 > 5), then
2
2
2
2
2
2
2
2
2ab = 2 × 8 × 5 = 80; a – b = 8 – 5 = 64 – 25 = 39 and a + b = 8 + 5 = 64 + 25 = 89.
Hence, the Pythagorean triplet is 80, 39, 89.
Patterns of Perfect Squares
I. The numbers which can be arranged as dot patterns in squares are called square numbers.
1 4 9 16
Similarly, the numbers that can be arranged as dot patterns in triangles are called triangular numbers.
1 3 6 10 15
( nn + 1)
th
n triangular number =
2
30