Table of Contents
| 1. | Introduction |
| 2. | Some simple Vedic Maths Tricks |
| 3. | Squaring Tricks! Square numbers faster |
| 4. | Square root Tricks! Learn how to find square roots |
| 4. | Summary |
| 4. | FAQs |
12 November 2020
Reading Time: 10 Minutes
Introduction
Do you know how to square big numbers?
Do you find it hard or take too long to square numbers?
Bharthi Krishna Tritha, an Indian monk, wrote the book "VEDIC MATHEMATICS", which has a list of mathematical tricks to solve maths calculations faster than traditional methods. It gives the easiest way to solve addition, subtraction, multiplication, division, and related problems. Vedic mathematics can applied to pneumatics, statics, astronomy, and the financial domain.
Here we will learn about how to find the square root of a number.
Also read:
- Features of Vedic Math
- Vedic Math Examples
- Addition and Subtraction Tricks
- Multiplication and Division Tricks
- Vedic Math Tricks and their Importance
- 15 Math tricks for kids
Downloadable PDF
If you ever want to read it again as many times as you want, here is a downloadable PDF to explore more.
| 📥 | Squaring and Square Roots |
Some simple Vedic Maths Tricks
Tricks to Subtract any number from 10000, 1000 and 100
Everyone finds it difficult to subtract numbers from 10000 and 100.here is a simple technique stated in Vedic mathematics, subtract all numbers from 9 and last number by 10.
| Number :1000 - 589 | Steps |
| 9 - 5 9 - 8 10 - 9 |
Subtract the first two numbers 5 and 8 by 9 and the last number 9 by 10 |
| Answer = 411 | |
Tricks to Multiply 3 digit numbers
| Number: 208 × 206 | Steps |
| 208 - 8 = 200 206 - 6 = 200 |
1.Subtract the number in units place by the number itself |
| 206 + 8 = 214 | 2.Select number in units place among given two number and add it to another number |
| 214 × 200 = 42800 | 3.Multiply the above-obtained number with the number obtained in step 1. |
| 8 × 6 = 48 | 4.Multiply the numbers in units place of the given question |
| 42800 + 48 = 42848 | 5.add the answers from step 1 and Step 4. This is the answer. |
| Answer: 42848 | |
Trick to Divide any large number with 5
| Number: 275 ÷ 5 | Steps |
| 275 × 2 = 550 | 1.Multiply the given number with 2 |
| 55.0 or 55 | 2.Move one decimal point |
| Answer: 55 | |
Squaring Tricks! Square numbers faster
A number, when multiplied by the number itself the product obtained, is called "SQUARE OF THAT NUMBER ". Here are some examples followed by Square tricks.
|
Example |
| Number | Square of the number |
| 62 | 6 × 6 = 36 |
| 82 | 8 × 8 = 64 |
| 122 | 12 × 12 = 144 |
Now we shall learn few basic Vedic mathematics methods or Vedic Maths Squaring Tricks:
Type 1: Squaring of number ending with 5.
Sutra is "BY ONE MORE THAN PREVIOUS ONE"
Step 1: Add 1 to the first digit from the left and multiply by the number itself.
Step 2: Add 52 (25) at the end to the number obtained from step 1.
| Number | Steps |
| 65 | |
| 6 × (6 + 1) = 6 × 7 = 42 |
Add 1 to the left number (6) and multiply by the number itself and |
| =4225 | Add 52 (25) at the last of 42 |
| Answer : 652 = 65 × 65 = 4225 | |
| 85 | |
| 8 × (8 + 1) = 8 × 9 = 72 |
Add 1 to the left number (8) and multiply by the number itself and |
| = 7225 | add 52 (25) at last of 72 |
| Answer : 852 = 85 × 85 = 7225 | |
| 155 | |
| 15 × (15 + 1) = 15 × 16 = 240 |
Add 1 to the left number (15) and multiply by the number itself and |
| = 24025 | add 52 (25) at last of 240 |
| Answer : 1552 = 155 × 155 = 2402 | |
Type 2: Squaring of numbers less than 50 and numbers not ending with 5.
| Number | Steps |
| 34 | 50 - 16 = 34 |
| 52 = 25 = 25 + (-16) = 9 |
Square the first digit (5) of first part (50) then add part (-16) |
| 162 = 256 | Square the second part of number (16) |
| 9 + 256 = 1156 | Add the answers got in step 1 (9)and step 2 (256) |
| Answer : 342 = 34 × 34 = 1156 | |
| 28 | 50 - 22 = 28 |
| 52 = 25 = 25 + (-22) = 3 |
Square the first digit (5) of first part (50) then add second part (-22) |
| 222 = 484 | Square the second part of number (22) |
| 3 + 484 = 784 | Add the answers got in step 1 (3)and step 2 (484) |
| Answer : 282 = 28 × 28 = 784 | |
Type 3: Squaring of numbers less than 50 and numbers not ending with 5.
| Number | Steps |
| 74 | 50 + 24 = 74 |
| 52 = 25 | Square the first digit (5) of first part (50) then add |
| = 25 + 24 = 49 |
second part (24) |
| 242 = 576 | Square the second part of number 24 |
| 49 + 576 = 5476 | Add the answers got in step 1 (49)and step 2 (576) |
| Answer : 742 = 74 × 74 = 5476 | |
| 57 | 50 + 7 = 57 |
| 52 = 25 = 25 + 7 = 32 |
Square the first digit (5) of first part (50) then add second part 7 |
| 72 = 49 | Square the second part of number 7 |
| 32 + 49 = 3249 | Add the answers got in step 1 (32)and step 2 (49) |
| Answer : 572 = 57 × 57 = 3249 | |
Type 4: Squaring of number near to their base 10,100,1000, and so on:
| Number | Steps |
| 105 | 100 + 5 = Divide the given number to their base and number |
| 105 + 5 = 110 | Add the second part of number 5 to the given number (105) |
| 52 = 25 | Square the second part of the 52 |
| 11025 | Combine the numbers from step 1 and step 2 |
| Answer : 1052 = 105 × 105 = 11025 | |
| 986 | 1000 - 986 = 14 |
| 986 - 14 = 972 |
The given number 986 is less than 14 from its base value 1000, so the deficient number 14 should be subtracted by the given number 986 |
| 142 = 196 | Square of deficient number 211 |
| 972196 | Combine the numbers from step 1 and step 2 |
| Answer : 9862 = 986 × 986 = 972196 | |
- If the number is lesser than its nearest base number then the deficient number is reduced from the given number.
- If the given number is greater than its nearest base number then the surplus number is added to the given number.
Type 5: Squaring of a number near to their sub base:
| Number | Steps |
| 306 | 300 + 6 = Divide the given number to their sub base and number |
|
3 × (306 + 6) = 3 ×312 |
Add the second part of number 6 to the given number (306) and multiply it by 3 |
| 62 = 36 | Square the second part of the 62 |
| 93636 | Combine the numbers from step 1 and step 2 |
| Answer : 3062 = 306 × 306 = 93636 | |
| 480 | 500 - 480 = 20 |
| 480 - 20= 5 ×(480 -20) =5 × 460 = 2300 |
The given number 480 is less than 20 from its sub base value 500, so the deficient number 20 should be subtracted by the given number 480 and multiplied by 5 |
| 202 =400 | Square of deficient number 211 |
| 230400 | Combine the numbers from step 1 and step 2 |
| Answer : 4802 = 480 × 480 = 230400 | |
GENERAL METHOD TO FIND SQUARE OF NUMBER OR DUPLEX METHOD :
Duplex combination or Dwanda yoga is term used in terms for squaring and multiplication,denoted as D. Following are the basic methods used in duplex:
- D (a) = a2
- D (ab) = 2(ab)
- D (abc) = 2 (ac) + b2
- D (abcd) = 2 (ad) + 2(bc)
- D (abcde) = 2 (ae) + 2(bd) + c2
| Example |
Square root Tricks! Learn how to find square roots
The square root of a number is the inverse operation of square of number.
| Example |
| Number | Square of the number | Square root of the number |
| 62 | 6 × 6 = 36 | \(\begin{align}\sqrt {36} = 6\end{align}\) |
| 82 | 8 × 8 = 64 | \(\begin{align}\sqrt {64} = 8\end{align}\) |
| 122 | 12 × 12 = 144 | \(\begin{align}\sqrt {144} = 12\end{align}\) |
Points to be remembered before finding the square root of any number :
- The given number is arranged as a group of two numbers from right to left. If a single number has remained at the left, then it is also considered as one group.
- The number of groups derived for the given number will be the number of digits of the square root of that number.
Example : \(\begin{align}\sqrt {25} = 5\end{align}\), here there are two digits so one group. So, the square root of 25 is 5.
\(\begin{align}\sqrt {169} = 13\end{align}\), the number forms two groups (1, 69), so the square root is two-digit number 13.
3. If the square of a number is an odd number, then the square root is also an odd number, and if the square is an even number, then the square root is also an even number.
4.
| Number | Square |
| 12 | 1 |
| 22 | 4 |
| 32 | 9 |
| 42 | 16 |
| 52 | 25 |
| 62 | 36 |
| 72 | 49 |
| 82 | 64 |
| 92 | 81 |
| Logic to find the square root |
|
|
| Square Number ending with following number | Square root ends with following number |
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
Sutra is "FIRST BY THE FIRST AND LAST BY THE LAST"
Let's solve few examples and learn how this technique works :
Type 1: Square root of perfect square number:
| Number:\(\begin{align}\sqrt {3969}\end{align}\) | Step |
| 39 | 69 | Grouping: from right to left |
| 38 is greater than 36, the perfect square number of 62, and 69 is lesser than 64, which is a perfect square number of 82 | |
| 3600 | 6400 602 | 802 |
Therefore the given number lies between 3600 and 6400 that means there square number will be between 60 and 80 |
| So the first number in the square root will be 6 | |
| The number in the units places is 9 so the square number should be ending with 3 or 7 (as discussed above) | |
| The square root of the given number should be either 63 or 67 | |
| Applying digit sum method to find the square root | |
| 63 = 3969 (6 + 3) = 3+9+6+9 9 = 2 + 7 9 = 9 |
So the square root of number 3969 is 63 |
| Answer : 3969 = 63 | |
Type 2: Square root of imperfect square number :
| Number | Steps |
| \(\begin{align}\sqrt {2}\end{align}\) | The given number is not a perfect square |
| 1 + 1 = 2. or 4 - 2 = 2 |
The nearest square number is 1 and 4 |
| 1 + 1/2 1 + 0.5 1.5 |
Square root of 1 is 1, add (1 ÷ [ 1 + 1]) i.e 1 divide by twice the perfect square number |
| 1.5 | The same when calculated in calculator gives 1.414 which is approximately equal to 1.5 |
| \(\begin{align}\sqrt {20}\end{align}\) | \(\begin{align}\sqrt {2}\end{align}\) |
| 25 - 5 = 20 | 25 is the perfect square of 5 subtract it from (1/10) 1 divide by twice the perfect square |
| 25 - 5/10 5 - 0.5 4.5 |
The same when calculated in calculator gives 4.47 which is approximately equal to 4.5 |
Summary
After learning a few of the tricks to find square and square roots of the number, one will be able to easily find the square numbers and their square roots without any help from calculators. These tricks are very helpful in solving aptitude problems related to square and square roots in competitive exams.
Written by Nethravati C, Cuemath Teacher
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FAQs
What is the square of a number?
If a number is multiplied by the same number itself then it is called a square of the number.
Example: square of 6 = 62
= 6 × 6 = 36
What is the square root of a number?
A number which when multiplied by the number itself gives a square of that number the inverse operation of this is known as the square root of the number.
Example \(\begin{align}\sqrt {25}=5\end{align}\)
How to find the square root of a number?
Step 1 : group the given number as two digits in one group from right to left.
Step 2 : find the nearest perfect square number of the grouped numbers and analyse between which numbers the given number lies.
Step 3 : depending on the number in units place decide which number will be the possible square number
Step 4 : apply the digit sum method to find the possible square number.
How to find the square of a number?
Square of a number can be determined by multiplying the number by the same number .
Example: 162 = 16 × 16 = 256
What is the square root of 5?
The nearest perfect square number is 4, so 5 can be written has
\(\begin{align}\sqrt {5}=\sqrt {4}+1\end{align}\)
Adding two times of 4 i.e 8 to the denominator of second part
\(\begin{align}\sqrt {5}=\sqrt {4}+1/8\end{align}\)
Simplifying
\(\begin{align}\sqrt {5}=2 + 0.125\end{align}\)
\(\begin{align}\sqrt {5}= 2.125\end{align}\)
What is the square root of 3?
The nearest perfect square number is 4, so 3 can be written has
\(\begin{align}\sqrt {3}=\sqrt {4}-1\end{align}\)
Adding two times of 4 i.e 8 to the denominator of second part
\(\begin{align}\sqrt {3}=\sqrt {4}-1/8\end{align}\)
On simplifying
\(\begin{align}\sqrt {3}=2 - 0.125\end{align}\)
\(\begin{align}\sqrt {3} = 1.875\end{align}\)
What is the square root of 8?
The nearest perfect square number is 9, so 8 6can be written has
\(\begin{align}\sqrt {8} = \sqrt {9}-1\end{align}\)
Adding two times of 9 i.e 18 to the denominator of second part
\(\begin{align}\sqrt {8} = \sqrt {9}-1/18\end{align}\)
On simplifying
\(\begin{align}\sqrt {8} = 3 - 0.055 \end{align}\)
\(\begin{align}\sqrt {8} = 2.945 \end{align}\)
What is the square root of 2?
The nearest perfect square number is 1, so 2 can be written has
\(\begin{align}\sqrt {2} = \sqrt {1}+1\end{align}\)
Adding two times of 1 i.e 2 to the denominator of second part
\(\begin{align}\sqrt {2} = \sqrt {1}+1/2\end{align}\)
On simplifying
\(\begin{align}\sqrt {2} = 1 + 0.5 \end{align}\)
\(\begin{align}\sqrt {2} = 1.5\end{align}\)