Let us study some basic Vedic Maths formulas and Vedic Maths tricks
Squaring a number less than Base:
You can save time in an exam if you calculate squares of numbers mentally. It is easy to calculate squares of numbers ending in 0. For example: square of 100 is 10,000. But, what if we have to find the square of say, 92. Let us use elementary algebra to arrive at an answer. We can write 92 as 100 – 8. If we square, we get three terms, = 1002 - 2 x 100 x 8 + 64. Now we have the last two digits of the square as 64, because the other two terms have got 2 or 4 zeroes in them. For the remaining terms 10,000 – 1600, can also be written as 9200 – 800 = 8400. Now, these will be the first two digits of the answer because of the two zeroes present. So the answer comes out to be 8400 + 64 = 8464.
When squaring a number, we need a base. Any of the powers of 10 can be taken as a base. So the method that we can develop is start by taking the nearest complete base, in this case 100. The difference between the base 100 & the number 92 is 8. The square of 8 is 64, which will be written on right hand side. Because it is having two digits, so it will be placed on the right side. Now the difference 8 is to be subtracted from the number given i.e. 8 should be subtracted from 92, so we get 84. 84, therefore becomes the left hand side. Therefore, the square of 92 is 8464.
In case, the square of the number is having fewer digits than required, then, in order to compensate for that, we can put 0’s. Example: in case you square a number, let us say, 98, difference is 2. The right side in this case will become 04, because 4 is a single digit number & therefore we have to put a zero before it to make it a two digit number. The left hand side in this case will be 98-02= 96. Therefore, the square is 9604.
Just in case, the number of digits is more than needed number of digits, the extra digit is carried to the left side. For example: let us say, we need to find the square of 87. The difference is 13 and the square of this difference is 169, in this case it is a three digit number, so the third extra digit is 1 which will be carried to the left side. Therefore, the left side is 87-13 + 1 = 75. Therefore, the square of this number is 7569.
The square of a number will have two parts, the left part and the right part. There is no limit for the left side, but the right side must have as many digits as the number of zeroes in the base i.e. if 100 is taken as base there should be 2 zeroes on the right side and if 1000 is taken as base then the number of digits on RHS should be 3.
To conclude, to get the square of a number less than a base, find the difference. The important thing is the left hand digits. These will be the number minus the base, and the right hand digits will be the square of the difference.
Squaring a Number More than the Base:
In order to square a number, a base is needed. All those numbers can be taken as a base that have a 1 and the rest number of zeroes with them (i.e. the complete round numbers like 100, 1000, 10000 etc.). The square of a number will have two parts, the left part and the right part.
There is no limit for the left side, but the right side must have as many digits as the number of zeroes in the base i.e. if 100 is taken as base there should be 2 zeroes on the right side and if 1000 is taken as base then the number of digits on RHS should be 3.
In case, the number to be squared is greater than the base taken, then the difference between the number & the base taken are to be added in the number instead of subtracting it from the same.
Example: Take a number 106. It’s base will be 100. The difference is 6. The right side will have a square of difference i.e. 62 = 36. The left side will be 106 + 6 = 112. Therefore the number required is 11236.
Just in case, the square of the number is a three digit number, then the third digit shall be carried over & added to the left hand side.
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Example: Take a number 112. Its base is 100. The difference is 12. The square of 12 is 144. 1 will be carried to the left hand side. On the left hand side, we have 112 + 12 + 1 (carried over) = 125. Therefore the square of 112 is 12544.
Similarly for the higher numbers, you can take the larger bases like 1000, 10000, 100000 etc. and the squares of the numbers can be found out using a similar technique.
We would learn a simple technique of multiplying two 2-digit numbers and two 3-digit numbers in a much faster way.
- Multiplying two 2-digit numbers
Let the 2 numbers be PQ and RS. Their product would be calculated as follows:
Step 1: QS (Write only the unit’s digit and carry the rest to the next step)
Step 2: PS + QR + carry over (Cross multiply and add, write a single digit and carry rest to next step)
Step 3: PR + carry over (Write the complete number as this is the final step)
Step 1: 2*5 = 10 (Write 0 and 1 is carried forward to the next step)
Step 2: 2*3 + 6*5 + 1(from carry over) = 37 (Write 7 and 3 is carried over to next step)
Step 3: 6*3 + 3 (from carry over) = 21 (Write 21 as this is the final step)
Thus, the answer is 2170.
- Multiplying two 3-digit numbers
Let the 2 numbers be XYZ and PQR. Their product would be calculated as follows:
Step 1: ZR (Write only the unit’s digit and carry the rest to the next step)
Step 2: YR + ZQ + carried over (Write only the unit’s digit and carry the rest to the next step)
Step 3: XR + ZP + YQ + carried over (Write only the unit’s digit and carry the rest to the next step)
Step 4: XQ + YP + carried over (Write only the unit’s digit and carry the rest to the next step)
Step 5: XP + carried over (Write the complete number as this is the last step)
Step 1: 6*7 = 42 (Write 2 and 4 is carry 4 to next step)
Step 2: 6*6 + 7*7 + 4 = 89 (Write 9 and carry 8 to next step)
Step 3: 6*2 + 3*7 + 6*7 + 8 = 83 (Write 3 and carry 8 to next step)
Step 4: 3*6 + 7*2 + 8 = 40 (Write 0 and carry 4 to next step)
Step 5: 3*2 + 4 = 10 (Write 10 as this is the final step)
Thus the answer is 100392.
This way you can save time in actual exam and become more efficient.
3. Base Multiplication
a) When both the numbers are below the base
When we need to multiply both the numbers below the base of powers of 10 i.e. whether it is 10, 100, 1000 etc., we need to follow the steps mentioned below :
Step 1: Find the deficits of the numbers from the given base.
Step 2: Now, we need to subtract the difference between one multiple with the deficit of the other
(this has to be subtracted from the base). This becomes the first part of the answer.
Step 3: The remaining part of the answer is the product of deviations of the numbers from the base taken. This part contains the number of digits equal to the number of zeroes in the base. For example: in case base is 10 we will fill 1 place. In case base is taken to be 100, we need to fill 2 places & so on.
Let P and Y be two numbers close to a given base in the powers of 10, and d1 and d2 are their respective deviations from the base. Then P X Y can be represented as:
(P+d2) OR (Y+d1) / (d1xd2)
Suppose we want to multiply 93 by 97
Both the numbers are very close to 100, so the base here is taken as 100.
93 is 7 below the base and 97 is 3 below.
We can cross-subtract either way: 93-3=90 or 97-7=90. This is the first part of the answer and multiplying the "differences" vertically 3x7=21 gives the second part of the answer.
b) When both the numbers are above the base
The technique works equally well for numbers above the base. In this case, we add the differences in step-2.
The general form of the multiplication:
Find 1175 * 1003.
(1175 + 003) or (1003 + 175) / (175 X 003)
1178 / 525
Since the base is 1000, we write down 100 and carry 1 over to the left giving us 1178 / 1100 = (1178) / 525
So, the answer is 1178525.
c) Base multiplication with base other than 10x
Let's multiply 43x47, both are very close to 50.Here we make use of a theoretical base and a working base. The theoretical base should always be a power of 10. In this case, it is 100. 50 is our working base in this case since both the numbers are close to 50. (Note: 50=100/2). Your working base should be a factor or multiple of your theoretical base.
43 is 7 less than 50
47 is 3 less than 50
We multiply 7 x 3 = 21, which becomes the right part of the answer. Then, we calculate 50-(7+3) =40. Now, we take 40 and divide it by 2 which is the ratio of our theoretical base to working base = 100/50 = 2. So, 40/2 = 20, becomes the left part of our answer. Thus, the answer is 2021.
Let us take one more example: 406 * 417
Here the working base is 400
= 423 | 102
= 423 * 400 | 102
= 169200 | 102
= 169200 + 102