5 Methods to Square Two Digit Numbers in Seconds.

Learn how to Square two digit numbers in seconds.

 

 

 

 

 

 

 

In this article you will learn easy ways to calculate square of two digit numbers quickly.After reading this article you will be able to square two digit number mentally within seconds.

Before I proceed further make sure you remember square of 1 digit numbers.

What you should already know?

0^2=0,1^2=1,2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49,8^2=64,9^2=81

Okay! Now Get start with different methods of finding square of two digit numbers.later on you will see special cases and easy formulae.

1st method to square two digit numbers:Based on ab^2= (10a+b)^2=a^2*100 +b^2 +2*10a*b

where “ab” means  a two digit numbers like 34, 16 45 etc.

Unit place digit=b^2 ,any carry added to middle or tenth place digit

Middle or tenth place digit= 2*a*b ,any carry added to hundredth place digit

Hundredth place digit=a^2

Example: 24^2=?

4*4=16, 1 will be taken as carry,6 as unit place digit

2*4*2 +1=17, again 1 will be taken as carry, 7 as tenth place digit.

2*2+1=5 will be taken as hundredth place digit in square.

24^2=576

Example 76^2=?

First step  6^2=36, carry 3 & 6 as unit place digit.

2*7*6=84,84+3=87, carry 8 & 7 as tenth place digit.

7^2=49, 49+8=57

therefore 76^2=5776

You can also solve the same as follows:

4    9     3     6

+    8     4

5   7     7    6

Square of first digit=49

square of second digit=36

now add 2*7*6=84 in the middle.

84^2

64     16

+6   4

70    5 6

2nd method to square two digit numbers:

49*49=49*(50-1)=49*50-49=2450-49=2401

99*99=99(100-1)=9900-99=9801

51*51=51*(50+1)=2550+51=2601

22*22=22*(20+2)=440+44=484
You can use this method for numbers close to 100 or 50.However this is very productive & beneficial method if you practice this much. As this method is also used in multiplying any two digit numbers together.

For more details on this method read my article: How to memorize Multiplication table up to 99*99 super fast

Third method to square two digit numbers:

Based on (a+b)^2= a^2+b^2+2ab or on (a-b)^2= a^2+b^2-2ab

24^2=(20+4)^2=20^2 +4^2+2*20.4=400+16+160=576

29^2=(30-1)^2 = 30^2+1^2-2.30.1=900+1-60=901-60=841

Fourth method to square two digit numbers:

Use special cases to find square of two digit number. In this method you will have shortcut formula for each special cases. Remembering formula will give you extra speed in finding square of two digits numbers.Formulas are derived based on the certain pattern we observe in the squares.

Special cases to square two digit numbers:

Case 1: Square of two digit numbers ending with 0 like:10,20,30,40,50,60,70,80,90

You may already know how to calculate square of numbers 10,20,30,40,50,60,70,80,90.Just square 1st digit and add two zeroes after that.

Example:40^2

4^2= 16

Therefore 40^2=1600

Case 2: Square of two digit numbers ending with 1 like: 11,21,31,41,51,61,71,81,91

11^2= 10^2+2*10+1=121

21^2=20^2+2*20+1=441

31^2=30^2+2*30 +1=961

41^2=40^2+2*40+1=1681

51^2=50^2+2*50+1=2601

61^2=60^2+2*60+1=3600+120+1=3721

71^2=70^2+2*70+1=5041

81^2=80^2+2*80+1=6400+160+1=6500+60+1=6561

91^2=90^2+2*90+1=8100+180+1=8281

This is simply based on (a+b)^2=a^2 +2.a.b+b^2

Notice the pattern above, You can write square of these numbers in seconds as follows.

31^2= 3^2  3*2  1=961

41^2= 4^2   4*2 1=1681

51^2=5^2 5*2 1=25 10 1=2601  —- 1 is taken carry from middle number and added to 25.

and so on.

Notice the pattern in the squares.

Unit place digit= 1 always

middle number or Tenth place digit= 2*first digit 

hundredth place number= square of first digit +carry from middle number,

or you can directly write First three digits in squaresquare of first digit*10+2*first digit

or First three digits in square =first digit*(first digit+1)*10- 8* first digit

You will see 2, 4,6,10,12,14,16,18 as the middle number which are double of first digit.

1*2=2, 2*2=4,3*2=6,4*2=8,5*2=10,6*2=12,7*2=14, 8*2=16,9*2=18.

In case of 10,12,14,16 and 18 we take carry 1 from middle digit and add to the hundred place number.

Case 3: Square of two digit numbers ending with 9 like:19, 29,39,49,59,69,79,89,99

This method is based on (a-b)^2=a^2+b^2-2ab

19^2= 20^2+1-2*20=400-40 +1=361

29^2=900-60+1=841

39^2=40^2+1-2*40=1600-80+1=1521

49^2=2500-100+1=2401

59^2=3600-120+1=3481

69^2=4900-140+1=4761

79^2=6400-160+1=6241

89^2=8100-180+1=7921

99^2=10000-200+1=9801

Notice the pattern in the squares:

First three digits can be written easily by:

square of (first digit+1) *10 – 2*(first digit+1)  ,

or you can also write it as First three digits=first digit* (first digit+1)*10 +8*(first digit +1)

series added in the middle  16,24,32,40,48,56,64, 72 ,80

and unit place digit =1

Like 89^2=  Square of (8+1)* 10- 2*(8+1) 1= 810-18 1=7921

So first write the square as  8101 then subtract 2*9= 18 from 810.

Case 4: Square of two digit numbers ending with 5 like: 15,25,35,45,55,65,75,85,95

Apply (a+b)^2=a^2+b^2+2*a*b to find squares.

15^2=(10+5)^2=100+25+2*10*5=225

25^2=625

35^2=1225

45^2=2025

55^2=3025

65^2=4225

75^2=5625

85^2=7225

95^2=9025

Notice the pattern in the squares.

Last two digit is always 25.

First two digit in the square = first digit*(first digit+1)

Like in 85^2 first two digits=  8*(8+1)=8*9=72

Similarly, for cases below first write the squares by using any of the first three methods explained by me, then find the shortcut by finding the pattern in the squares.I’m writing the pattern along with squares for all the cases below.

Case 5: Square of two digit numbers ending with 6 like: 16,26,36,46,56,66,76,86,96

16^2=256

26^2=676

36^2= 1296

46^2= 2116

56^2=3136

66^2=4356

76^2=5776

86^2=7396

96^2=9216

You will see the series 5,7,9,11,13,15,17,19,21 is used in the middle  or tenth place in squares of 16,26,36,46,56,66,76,86,96 respectively.

Where carry from 11,13,15,17,19,21 added to the hundred place.

You can write the squares  easily by using.

Unit place digit=6

tenth place number= first digit*2 +3

Hundred place number or first two digits in square= first digit*(first digit+1)  +carry from tenth place number.

or you can directly write first three digit = first digit*(first digit+1)*10+First digit*2 +3

Case 6: Squares of two digit numbers ending with 4 like:14,24,34,44,54,64,74,84,94

14^2=196

24^2= 576

34^2= 1156

44^2=1936

54^2=2916

64^2=4096

74^2= 5476

84^2=7056

94^2=8856

Notice the pattern in the above squares.

First three digits in squares= first digit (first digit+1)*10 – x= first digit (first digit+1)*10 – 2*first digit+1

where x is 1, 3, 5,7,9,11,13,15 for Squares of  14,24,34,44,54,64,74,84,94 respectively.

x=2*first digit -1

Case 7: Square of two digit numbers ending with 3 like:13,23,33,43,53,63,73,83,93

13^2=169

23^2=529

33^2=1089

43^2=1849

53^2=2809

63^2=3969

73^2=5329

83^2=6889

93^2=8649

Pattern:

Unit place digit= 9

First three digits in squares=(Square of first digit)*10 +6*first digit

You can use below pattern also.

unit place digit=9

First three digits in squares=  first digit*(first digit+1)*10 -4*first digit

Case 8:  Square of two digit numbers ending with 2 like :12,22,32,42,52,62,72,82,92

12^2=144

22^2=484

32^2=1024

42^2=1764

52^2=2704

62^2=3844

72^2=5184

82^2=6724

92^2=8464

Unit place digit=4

tenth place digit= 4,8,2,6,0,4,8,2,6 respectively.

First three digits=square of First digit*10 +4*first digit

or First three digits= first digit*(first digit+1)*10 – 6* first digit

Case 9: Square of two digit numbers ending with 7 like: 17,27,37,47,57,67,77,87,97

17^2=2 89

27^2=729

37^2=1369

47^2= 2209

57^2=3249

67^2= 4489

77^2=  5929

87^2= 7569

97^2= 9409

Pattern:

Unit place digit=9

Series in the middle  8,12,16,20,24,28,32,36,40

First three digit= first digit(first digit+1)*10 +4*first digit +4

Case 10: Square of two digit numbers ending with 8 like:18,28,38,48,58,68,78,88,98

18^2=324

28^2=784

38^2=1444

48^2= 2304

58^2= 3314

68^2=4624

78^2= 6084

88^2=  7744

98^2= 9604

Pattern:

Series used in the middle: 12,18,24,30,36,42,48,54,60

Unit place digit=4

First three digit= first digit*(first digit+1)*10 +6*(first digit+1)

Note:While we write first three digits with the help of formula, we don’t add carry from unit place, as it is already added .We simply neglect carry number.

Case 11:Square of two digit numbers like: 11,22,33,44,55,66,77,88,99

To solve this first learn my method to multiply any number by 11 mentally.

Example:11*25= 2 (2+5)  5= 2 75

Now 33^2= 11^2 *3^2= 11*11*9=11*99=9 (18) 9 = 1089

44^2 = 11^2* 4^2= 11*11*16=11*176=18 (13) 6 = 1936

55^2= 11*11*25=11*275=29 (12) 5 =3025

66^2= 11*11*36=11*396=3 (12) (15)  6=4356

Fifth method: To square number between 80 to 100 if you remember square of numbers from 1 to 20 within seconds.

first two digit= number-(100-number)

last two digits= square of (100-number) any carry added to first two digit.

Like suppose we take 87^2

100-87=13,

87-13=74,

13^2=169

last two digit=69 and first two digit in square=74+1=75

therefore square of 87=7569

In fact this method can be applied to square of any number. You will read about this method in more details in my article how to square of three digit number quickly will be published soon…

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Niraj Tulsyan
Btech in Electronics & Communication.I love Maths, Science & Technology.