Speed Math: Find Square of Any Number | Mental Mathematics for Class 8 PDF Download

Base Method of Calculating Squares

Generally, we can calculate the square of a number by the formula (a2 + b2 + 2ab). This formula applies by splitting a number into a and b. For eg. (49)2 can be calculated using this formula by splitting into 40 and 9. But calculating with this method always is a lengthy process and not recommended to follow in the exam. 

The base method of calculating squares is an easy method to square 2 and 3 digits numbers easily. The major thing is to find the Base here. The base is referred to as n x 10x where n is 1, 2, 3, and so on. You have to take the value of base i.e, 10x which is nearer to the number.

Let us see the steps to find the square of a number using the Base Method. 

Step 1: Find Base i.e, n x10^x.
Step 2: 

• Case 1: Square the (Number - Base) if Number > Base. 
• Case 2: Square the (Base - Number) if Base > Number.

Make sure the number of digits in this part = number of zeroes in the base, add zeroes if it is of fewer digits, or carry forward extra digits in case of more than the required digits.  

Step 3: 

• Case 1: Add (Number - Base) to Number if Number > Base and then multiply by n.
• Case 2: Subtract (Base - Number) from Number if Base > Number and then multiply by n. 

Step 4: Merge results of Steps 2 and 3. Keep Step 2 result on the right side. 

Suppose you have to find the square of 104.

• Step 1: Now, here the base is 100 or 10².  
• Step 2: Here Number > Base or 104 > 100 so, (104 - 100)² = 16  
• Step 3: 1 x (104 + 4) = 108
• Step 4: 10816

In this way, you have to calculate the square using the base method. Let us take more examples to have a clear understanding. 

Find the square of 99.

• Step 1: Here, the base is 100 or 10².  
• Step 2: 100 > 99 (100- 99)² = 01 (Here we added one zero to the left side of square number to make it a 2 digit number)
• Step 3: 1 x (99 - 1) = 98
• Step 4: 9801

Find the square of 119.

• Step 1: Base is 100 or 10².  
• Step 2: (119- 100)² = 3 61 (Here 3 will be carried forward as we need only 2 digits at this side)
• Step 3: 1 x (119 + 19) = 138 + 3 carry = 141
• Step 4: 14161

Find the square of 198

• Step 1: Now, here the base is 200 or 2 x 10².  
• Step 2: (200- 198)² = 04 (Here we added one zero to the left side of square number to make it a 2 digit number)
• Step 3: 2 x (198- 2) = 392 
• Step 4: 39204

Find the square of 482

• Step 1: Now, here the base is 500 or 5 x 10².  
• Step 2: (500- 482)² = 3 24 (Here 3 will be carried forward as we need only 2 digits on this side)
• Step 3: 5 x (482- 18) = 2320 + 3 carry = 2323 (Here we had multiplied it by 5 as the base was 5 x 10²)
• Step 4: 232324

Find the square of 1012

• Step 1: Now, here the base is 1000 or 1 x 10³.  
• Step 2: (1012- 1000)² = 144 (Here we will keep three digits as the number of zeroes in the base is 3).
• Step 3: (1012 + 12) = 1024
• Step 4: 1024144

In the exam, you have don't have to write the individual steps. Just start calculating mentally and write the final value. After practicing, you will be able to solve this quickly.

Calculate the following squares using the base method and answer in the comment section.
(A) (93)²
(B) (216)²
(C) (1001)²
(D) (3014)²

The base method is useful when numbers are nearer to the base as it will become more calculating when the difference between numbers and base becomes larger. To overcome such calculation, we are introducing one more method i.e., the Duplex Method. 

Duplex Method to Calculate Squares

To understand this method, at first, you must know how to calculate the duplex of the numbers.
Dup (a) = a²
Dup (ab) = 2 x a x b
Dup (abc) = 2 x a x c + b²
Dup (abcd) = 2 x a x d + 2  x b x c

Now let`s start calculating squares using this method.

(A) Method of calculating the square of a number ab.
It will be Dup a | Dup ab | Dup b

Calculate the square of 42
Dup 4 | Dup 42 | Dup 2
4²    |2 x 4 x 2|   2²
16   |   1 6       |    4   (Keep only one digit in each part except the first one)

           1764

(B) Method of calculating the square of a number abc.
Dup a | Dup ab |    Dup abc  | Dup bc   | Dup c

Find the square of 145

(C) Method of calculating the square of a number abcd.
Dup a | Dup ab |    Dup abc   |   Dup abcd  |    Dup bcd    | Dup cd | Dup d

Find the square of 1234

This is the way to solve squares using the duplex method. But in exams, don`t write steps. Just start calculating the duplex from the right-hand side and write the final answer. Practice solving squares using this method without steps. You can calculate any number of digits from this method. The only need for this method is to know how to calculate the duplex of the numbers. 

Calculate the following squares using the duplex method and answer in the comment section.
(A) (87)²     
(B) (529)²         
(C) (1991)²       
(D) (5018)²

Let us see some more tricks of solving squares of the numbers ending with digits 1, 5, 6, and 9.

Trick to Calculate Squares of Numbers ending with Digit 5

• Step 1: We all know 5² is 25. In the squares of numbers ending with 5, just write 25 on the right-hand side of the answer. 
• Step 2: First digit of the number x Successive digit of the first number.
  Calculate 25² = 2 x 3 | 25 = 625
  Calculate 55² = 5 x 6 | 25 = 3025
  Calculate 85² = 8 x 9 | 25 = 7225
  Calculate 125² = 12 x 13 | 25 = 15625
  Calculate 325²= 32 x 33 | 25 = 105625

In the same way, you can calculate the square of other numbers ending with digit 5. You just need to strengthen your multiplication skills. 

Trick to Calculate Squares of Numbers ending with Digit 6

Suppose you have to find the square of 76.
Step : (Number - 1)² + Number + (Number - 1)
(76)² = (75)² + 75 + 76 = 5625 + 151 = 5776 (Calculate square of number ending with digit 5 using the above method.)
(146)² = (145)² + 145 + 146 = 21025 + 291 = 21316
(1006)² = (1005)² + 1005 + 1006 = 1010025+ 2011= 1012036

To calculate squares of numbers ending with digit 6, you must know how to calculate squares of numbers ending with digit 5 and fast multiplication skills as it is the foremost requirement.

Trick to Calculate Squares of Numbers ending with Digit 4

Suppose you have to find the square of 64.
Step : (Number + 1)² - Number - (Number + 1)
(64)² = (65)² - 65 - 64 = 4225 - 129 = 4096(Calculate square of number ending with digit 5 using the above method.)
(294)² = (295)² - 295 - 294 = 87025 - 589= 86436
(3004)² = (3005)² - 3005 - 3004 = 9030025 - 6009 = 9024016

Again, you must know how to calculate squares of numbers ending with digit 5 with strong multiplication skills to calculate squares of numbers ending with digit 4.

Trick to Calculate Squares of Numbers ending with Digit 1

Suppose you have to calculate the square of 91.

Step : (Number - 1)² + Number + (Number + 1)

(91)² = (90)2 + 90 + 91 = 8100 + 181 = 8281
(161)² = (160)² + 160 + 161 = 25600 + 321 = 25921
(921)² = (920)² + 920 + 921 = 846400 + 1841 = 848241 (Calculate 92² mntally using base method)
(1201)² = (1200)² + 1200 + 1201 = 1440000 + 2401 = 1442401

In the same way, you can calculate the square of other numbers ending with digit 1. For this, you must remember squares up to 20 and strong addition calculation mentally.

Trick to Calculate Squares of Numbers ending with Digit 9

Suppose you have to calculate the square of 79.
Step : (Number + 1)² - Number - (Number + 1)
(79)² = (80)² - 80 - 79 = 6400 - 159 = 6241
(159)² = (160)² - 160 - 1659 = 25600 - 319 = 25281
(579)² = (580)² - 580 - 579 = 336400 - 1159 = 335241 (Calculate 92² mntally using base method)
(1239)² = (1240)² - 1240 - 12039 = 1537600 - 2479 = 1535121 (Calculate 124² using duplex method)

For this also, you must possess strong multiplication skills.

Trick to Calculate Squares of Numbers in the range of 51 to 70

Step : [25 + (Number - 50) | (Number - 50)²]
(51)² = 25 + 1 | 01 = 2601
(57)² = 25 + 7 | 49 = 3249
(66)² = 25 + 16 | 256 = 41 | 2 56 = 43 | 56 = 4356 (Keep only 2 digit on right side)
(69)² = 25 + 19 | 361 = 44 | 3 61 = 47 | 61 = 4761 (Keep only 2 digit on right side)

Trick to Calculate Squares of Numbers in the Range of 31 to 50

Step : [25 -  (50 - Number) | (50 - Number)²]
(49)² = 25 - 1 | 01 = 2401
(43)² = 25 - 7 | 49 = 1849
(37)² = 25 - 13 | 169 = 12 | 1 69 = 13 | 69 = 1369 (Keep only 2 digit on right side)
(31)² = 25 - 19 | 361 = 6 | 3 61 = 9 | 61 = 961 (Keep only 2 digit on right side)

Another method: (a + b) (a - b) + b²

(88)² : it is 100 - 12 or a = 100 and b  = 12
So, (88)² = (88 + 12) (88 - 12) + 12² = 76 x 100 + 144 = 7744
(104)² = (104 + 4) (104 - 4) + 4² = 108 x 100 + 146 = 10816
(438)² = (438 + 38) (438- 38) + 38² = 476 x 400 + 1444 = 191844

Calculate the following squares using any of the above mentioned method:
(A) (59)² 
(B) (876)²
(C) (1441)² 
(D) (995)² 
(E) (1441)² 
(F) (78)²