How to find square root of a number quickly
Today I am going to share with you a method to find out the square root of larger numbers or numbers that are not perfect squares. You can say this is a general method for finding the square root of any number.
The computation of square root involves the use of duplex. The concept of duplex is useful for other quicker calculations also. The duplex of a number is calculated as below:
Mental Addition Trick
How would you add 57 plus 39 in your head?
Do you start counting 9 on your fingers or visualize it in your head?
The easy way would be to add 40 to 57, (97), and take away one (1). The answer will be 96.
To make it even simpler you can add 60 and 40; the answer will be 100. We subtract the temporary increment in given number from the result. That means we subtract 3 and 1 from 100. Hence the desired result of 57+39 = 96
It is easy to add round numbers like 10 or 20 or...60 or 70 to any number mentally; for example to find 43 + 89 it's easier to find 90 + 43 = 133 and subtract to get 132 as answer. The book Trachtenberg Speed System of Basic Mathematics is filled with innumerable such techniques to improve your calculation speed in basic arithmetic operations like addition, subtraction, multiplication, etc.
Trick to Convert Miles to Kilometres
How to convert Miles into KM
The relationship between kilometres and miles can be closely represented by Fibonacci Series.
So to understand the calculation of conversion from miles to kilometres or the other way round, we first need to understand the concept of Fibonacci sequence. First few numbers of the Fibonacci Series are the following.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
If you've noticed in the above series the numbers are the sum of the previous two numbers in the series, starting with 0 and 1. As you see 0+1=1; 1+1=2; 1+2=3;…….34+55=89 and so on.
Conversion of Miles to Kilometres
Relationship of succeeding numbers in Fibonacci series closely matches the relationship between miles and km.
3 miles = 5 km
5 miles = 8 km
8 miles = 13 km (12.8748 km to be exact)
13 miles = 21 km (20.9215 km to be exact)
21 miles = 34 km (33.796224 km to be exact)
89 miles = 144 km (143.232 km to be exact)
However, what if you have to convert number (miles) which is not in Fibonacci series into km? Not to worry. There is a way out. You need to break out that number in to Fibonacci numbers. Covert the numbers as per the above scheme and add the resultant numbers to get the answer.
For example; 85 = 55+21+8+1
After the conversion: 89+34+13+1 = 137 km (136.794 km to be exact)
Finding Square of Number Ending in 5
This is the most common, yet very interesting trick of Vedic Maths. Using this technique you can find the square of any number ending in 5 very easily. Given below is the step by step explanation of this squaring technique.
Let us take a 2 digit number ending in 5,
452
We'll get our answer in 2 parts. The right hand side of the answer will always be 25. The left hand side will be the number other than 5, multiplied by it's successor (next higher integer).
= 4 x (4+1) | 25
= 4 x 5 | 25
= 2025
( | stands for concatenation}
Let us try to generalizing the above squaring technique. Say the number is a5 (=10a+5), where 'a' is the digit in ten's place
Square of a5 = a x (a+1) | 25
Similarly we can proceed for 3 digit numbers ending in 5
Square of 195
= 19 x 20 | 25
Left hand side is 19 multiplied by next higher number 20. Right hand side is as usual always 25. Hence the answer
=38025
Few more examples:
952= 9 x 10 | 25 = 9025
1252 = 12 x 13 | 25 = 15625
5052 = 50 x 51 | 25 = 255025
Test yourself
Find out the square of 85, 245, 145, 35, 15, and 995?
Using this method of squaring numbers ending in 5, post your answers below. Would love to know your feedback /suggestions too.
Multiplying Two Numbers when Sum of their Unit Digits is 10
This method of multiplication from Vedic Maths will make it very easy to multiply two numbers when sum of the last digits is 10 and previous parts are the same. For example multiplications like
23x27 : Sum of Unit digits i.e. 3+7 = 10; Remaining number i.e. 2 is same in both numbers
46x44: Sum of Unit digits i.e. 6+4 = 10; Remaining number i.e. 4 is same in both numbers
112x118: Sum of Unit digits i.e. 2+8 = 10; Remaining number i.e. 11 is same in both numbers
291x299: Sum of Unit digits i.e. 1+9 = 10; Remaining number i.e. 29 is same in both numbers
135x135: Sum of Unit digits i.e. 5+5 = 10; Remaining number i.e. 13 is same in both numbers
Solving 46 x 44
You will get the answer in two parts.
First part, to get left hand side of the answer: multiply the left most digit(s), i.e. 4 by its successor 5
Second part, to get right hand side of the answer: multiply the right most digits of both the numbers i.e. 4 and 6.
Example
First part: 4 x (4+1)
Second part: (4 x 6)
Combined effect: (4 x 5) | (4 x 6) = 2024
*| is just a separator. Left hand side denotes tens place, right hand side denotes units place
More Examples
37 x 33 = (3 x (3+1)) | (7 x 3) = (3 x 4) | (7 x 3) = 1221
11 x 19 = (1 x (1+1)) | (1 x 9) = (1 x 2) | (1 x 9) = 209
As you can see this method is corollary of "Squaring number ending in 5"
It can also be extended to three digit numbers like :
E.g. 1: 292 x 208.
Here 92 + 08 = 100, L.H.S portion is same i.e. 2
292 x 208 = (2 x 3) x 10 | 92 x 8 (Note: if 3 digit numbers are multiplied, L.H.S has to be multiplied by 10)
60 | 736 (for 100 raise the L.H.S. product by 0) = 60736.
E.g. 2: 848 X 852
Here 48 + 52 = 100,
L.H.S portion is 8 and its next number is 9.
848 x 852 = 8 x 9 x 10 | 48 x 52 (Note: For 48 x 52, use methods shown above)
720 | 2496
= 722496.
[L.H.S product is to be multiplied by 10 and 2 to be carried over because the base is 100].
Eg. 3: 693 x 607
693 x 607 = 6 x 7 x 10 | 93 x 7 = 420 / 651 = 420651.
Note: This Vedic Maths method can also be used to multiply any two different numbers, but it requires several more steps and is sometimes no faster than any other method. Thus try to use it where it is most effective
Today I am going to share with you a method to find out the square root of larger numbers or numbers that are not perfect squares. You can say this is a general method for finding the square root of any number.
The computation of square root involves the use of duplex. The concept of duplex is useful for other quicker calculations also. The duplex of a number is calculated as below:
- For a single digit number, the duplex is simply the square of the number. Duplex of 2 is 4 and 6 is 36.
- For a 2-digit number, the duplex is simply twice the product of the 2 digits of the number. Thus, the duplex of 16 is 2x1x6 = 12 and 90 is 2x9x0 = 0.
- For n-digit numbers, pair up the first digit with the nth digit of the number and find the duplex of the resulting 2-digit number. Similarly pair up the second digit with the (n-1)th digit. Continue this process until no more 2-digit pairs can be formed. If a single digit exists at the end of the process, find its duplex individually. The resulting sum of all the duplexes is the duplex of the n-digit number. Example: 89437: 2x8x7 + 2x9x3 + 4x4 = 112 + 54 + 16 = 182
- Say we have to find the square root of 119716.
Step 1
Arrange the number in two-digit groups from right to left. A single digit left if any is counted as a group by itself. The leftmost group is our starting number or SN.
11 : 9 7 1 6 :
Step 2
Consider the highest possible perfect square, say HS ≤ SN and compute its square root, S. write down S below the leftmost digit. Here, perfect square just below 11 is 9 whose square root is 3. This becomes the first digit of the required square root. Multiply the square root S by 2 and write it on the left side. This would be divisor. Here, 3X2=6.
11 : 9 7 1 6 :
6 : :
3 : :
Step 3
The difference between SN and HS (11-9=2) is carried to the next digit and written to its lower left. 29 is the gross dividend (GD) here.
29 is divided by the divisor. The quotient is written down as the next digit of square root and remainder is carried to the next digit to form the next GD. 29 / 6 gives quotient 4 and remainder 5.
11 : 9 7 1 6 :
6 2 5 :
: 3 : 4
Step 4
From this stage onwards, the duplex of all digits starting from the second digit of the square root is subtracted from each GD to get net dividend, ND. At each stage now this ND will be divided by divisor. Quotient and remainder will be taken care of as before. Duplex of 4 is 16. 57(GD)-16= 41(ND). 41/6 gives 6 as quotient and 5 as remainder.
11 : 9 7 1 6 :
6: 2 5 5 :
3 : 4 6
Step 5
51(GD)- 48(duplex of 46) =3. 3/6 gives 0 as quotient and 3 as remainder.
Next GD is 36. Subtract duplex of 460 = 36 from it. 0/6 gives 0 as quotient and 0 as remainder. This means that work has been done.
11 : 9 7 1 6 :
6: 2 5 5 3 :
3 : 4 6 . 0 0
The process continues till ND becomes 0 and no more digits are left in the number for computation. If ND does not become 0, we attach zeros to the original number and continue till the desired number of decimal places.
Once the square root is obtained or we decide to stop, the decimal point is placed after the required digits from the left. If n is even, decimal point is put after n/2 digits from the left. If n is odd, decimal point is put (n+1)/2 digits from the left.
Here we put the decimal point 6/2=3 places from the left. Hence, 346 is the square root of 119716.
It might seem a little tedious in the beginning but it is a lot easier than the conventional method.
Give it some practice instead of panicking and you will be creating magic soon. Did it finally work for you?
Mental Addition Trick
How would you add 57 plus 39 in your head?
Do you start counting 9 on your fingers or visualize it in your head?
The easy way would be to add 40 to 57, (97), and take away one (1). The answer will be 96.
To make it even simpler you can add 60 and 40; the answer will be 100. We subtract the temporary increment in given number from the result. That means we subtract 3 and 1 from 100. Hence the desired result of 57+39 = 96
It is easy to add round numbers like 10 or 20 or...60 or 70 to any number mentally; for example to find 43 + 89 it's easier to find 90 + 43 = 133 and subtract to get 132 as answer. The book Trachtenberg Speed System of Basic Mathematics is filled with innumerable such techniques to improve your calculation speed in basic arithmetic operations like addition, subtraction, multiplication, etc.
Trick to Convert Miles to Kilometres
How to convert Miles into KM
The relationship between kilometres and miles can be closely represented by Fibonacci Series.
So to understand the calculation of conversion from miles to kilometres or the other way round, we first need to understand the concept of Fibonacci sequence. First few numbers of the Fibonacci Series are the following.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
If you've noticed in the above series the numbers are the sum of the previous two numbers in the series, starting with 0 and 1. As you see 0+1=1; 1+1=2; 1+2=3;…….34+55=89 and so on.
Conversion of Miles to Kilometres
Relationship of succeeding numbers in Fibonacci series closely matches the relationship between miles and km.
3 miles = 5 km
5 miles = 8 km
8 miles = 13 km (12.8748 km to be exact)
13 miles = 21 km (20.9215 km to be exact)
21 miles = 34 km (33.796224 km to be exact)
89 miles = 144 km (143.232 km to be exact)
However, what if you have to convert number (miles) which is not in Fibonacci series into km? Not to worry. There is a way out. You need to break out that number in to Fibonacci numbers. Covert the numbers as per the above scheme and add the resultant numbers to get the answer.
For example; 85 = 55+21+8+1
After the conversion: 89+34+13+1 = 137 km (136.794 km to be exact)
Finding Square of Number Ending in 5
This is the most common, yet very interesting trick of Vedic Maths. Using this technique you can find the square of any number ending in 5 very easily. Given below is the step by step explanation of this squaring technique.
Let us take a 2 digit number ending in 5,
452
We'll get our answer in 2 parts. The right hand side of the answer will always be 25. The left hand side will be the number other than 5, multiplied by it's successor (next higher integer).
= 4 x (4+1) | 25
= 4 x 5 | 25
= 2025
( | stands for concatenation}
Let us try to generalizing the above squaring technique. Say the number is a5 (=10a+5), where 'a' is the digit in ten's place
Square of a5 = a x (a+1) | 25
Similarly we can proceed for 3 digit numbers ending in 5
Square of 195
= 19 x 20 | 25
Left hand side is 19 multiplied by next higher number 20. Right hand side is as usual always 25. Hence the answer
=38025
Few more examples:
952= 9 x 10 | 25 = 9025
1252 = 12 x 13 | 25 = 15625
5052 = 50 x 51 | 25 = 255025
Test yourself
Find out the square of 85, 245, 145, 35, 15, and 995?
Using this method of squaring numbers ending in 5, post your answers below. Would love to know your feedback /suggestions too.
Multiplying Two Numbers when Sum of their Unit Digits is 10
This method of multiplication from Vedic Maths will make it very easy to multiply two numbers when sum of the last digits is 10 and previous parts are the same. For example multiplications like
23x27 : Sum of Unit digits i.e. 3+7 = 10; Remaining number i.e. 2 is same in both numbers
46x44: Sum of Unit digits i.e. 6+4 = 10; Remaining number i.e. 4 is same in both numbers
112x118: Sum of Unit digits i.e. 2+8 = 10; Remaining number i.e. 11 is same in both numbers
291x299: Sum of Unit digits i.e. 1+9 = 10; Remaining number i.e. 29 is same in both numbers
135x135: Sum of Unit digits i.e. 5+5 = 10; Remaining number i.e. 13 is same in both numbers
Solving 46 x 44
You will get the answer in two parts.
First part, to get left hand side of the answer: multiply the left most digit(s), i.e. 4 by its successor 5
Second part, to get right hand side of the answer: multiply the right most digits of both the numbers i.e. 4 and 6.
Example
First part: 4 x (4+1)
Second part: (4 x 6)
Combined effect: (4 x 5) | (4 x 6) = 2024
*| is just a separator. Left hand side denotes tens place, right hand side denotes units place
More Examples
37 x 33 = (3 x (3+1)) | (7 x 3) = (3 x 4) | (7 x 3) = 1221
11 x 19 = (1 x (1+1)) | (1 x 9) = (1 x 2) | (1 x 9) = 209
As you can see this method is corollary of "Squaring number ending in 5"
It can also be extended to three digit numbers like :
E.g. 1: 292 x 208.
Here 92 + 08 = 100, L.H.S portion is same i.e. 2
292 x 208 = (2 x 3) x 10 | 92 x 8 (Note: if 3 digit numbers are multiplied, L.H.S has to be multiplied by 10)
60 | 736 (for 100 raise the L.H.S. product by 0) = 60736.
E.g. 2: 848 X 852
Here 48 + 52 = 100,
L.H.S portion is 8 and its next number is 9.
848 x 852 = 8 x 9 x 10 | 48 x 52 (Note: For 48 x 52, use methods shown above)
720 | 2496
= 722496.
[L.H.S product is to be multiplied by 10 and 2 to be carried over because the base is 100].
Eg. 3: 693 x 607
693 x 607 = 6 x 7 x 10 | 93 x 7 = 420 / 651 = 420651.
Note: This Vedic Maths method can also be used to multiply any two different numbers, but it requires several more steps and is sometimes no faster than any other method. Thus try to use it where it is most effective