Here
are three techniques to square any number ending in 9. All the techniques can
be applied to any number, no matter how many digits it has, so long as the
unit’s digit is 9. Of course, as the number gets larger,
multiplying the two numbers of the first step will become cumbersome unless a
short cut can be used. However, most two- and three digit numbers ending in 9
can be readily squared, once a facility with the other short-cut methods has
been achieved.
Technique-1
To square a number ending in 9; e.g.
29, 59, 129...., first square the next number that is ending in zero, and then
subtract both the number and the number next to it.
292 =?
The number to be squared is 29 and the number next to it is 30. Finding square of 30 is simple as it ends in ‘0’.
So, 292 = 302 -
(29 + 30) = 900 - 59 = 841
So, 792 =?
The number to be squared is 79 and the number
next to it is 80.
So, 792 = 802 -
(79 + 80) = 6400 - 159 = 6241
Further example, 1292 =?
1292 = 1302 -
(129 + 130) = 16900 - 259 = 16641
Technique-2
Step-1: Write down 1.
Step-2: Add 1 to the ten’s digit and multiply by 8.
Write this number down, carry if necessary.
Step-3:
Multiply the number in the ten’s digit by that number plus 1. Write this
result.
Example:
692 =?
1.
Write down 1.
2.
8 X (6 + 1) = 56. Write
6, carry *5.
3.
6 X (6 + 1) = 42 + *5 =
47. Write 47.
The
answer is 4761.
Another
example: 2492 =?
1.
Write down 1.
2.
8 X (24 + 1) = 200. Write 0, carry *20
(see ‘Multiplying by 25’).
3.
24 X (24 + 1) = 600 + *20 = 620. Write
620.
The answer is 62001.
Technique-3
Multiply
the number to the left of the 9 by two more than itself. Affix an 8 to the
result and subtract twice the number to the left of the 9. Affix a 1 to the
result.
Example:
792=?
The
number to the left of the 9 is 7. Two more than this is 9. Multiply 7 by 9.
7
x 9 = 63
Affix
8 to get 638.
Subtract
twice the number to the left of the 9,
638
- (2 x 7) = 638 - 14 = 624
Affix
a 1 to obtain the final answer 6,241.
So,
792 = 6,241
Example:
2492=?
The
number to the left of the 9 is 24. Two more than this is 26. Multiply 24 by 26.
(Multiplication short cut method can be used).
24
x 26 = 625 - 1 = 624
Affix
8.
6,248
Subtract
twice the number to the left of the 9,
6,248
- (2 x 24) = 6,248 - 48 = 6,200
Affix
a 1 to obtain the final answer 62,001.
So,
2492 = 62,001
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