Vedas and Mathematics
Sutra 3: "Vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10. i.e
7 * 8
=(10-3) * (10 -2)
= 100 - 20 -30 +6
= 56
The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply 32 by 46.
We multiply "vertically and crosswise":
Vertically 2x6=12.
so put down 2 and carry 1.
Then we multiply crosswise and add the two results: 3x6+4x2=26,
Add the carried 1 to make it 27
so put down 7 and carry 2.
Finally we multiply vertically 3x4=12 and
add the carried 2 =14.
Result: 1,472.
We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. With some practice, one can do pretty long multiplications in their heads.
Can you now do some quick calculations for multiplications with 11?
What is 28* 11? What is 375*11?
The answer for this is very simple:
28 * 11 = 308
Simply write the 2,(2+8),8 --> 2,(10),8 (1 from 10 has to be taken forward and added to 2) so answer is 308
The magic is simplified when you break 11 to (10+1): 28 * 11 = 28 * (10+1) = 280 +28 --> 2 (8+2) 8 --> 308
375 * 11 = 4125
What is happening here? 375 * 11 --> 3,(3+7),(7+5),5 --> 3,(10),(12),5 --> 4125
This Sutra is used to solve complex non-linear partial differential equations. It deals with calculation of common functions and their series expansions, and the solution of simultaneous, algebraic, transcendental and differential equations.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below. 100-4= 96 or 100-8=92. So 10000 - 1200 + 32. Applying the Sutra 2 for the calculation 10000 - 1168 = 8832.
Try this 86 * 88 = (90-4) * (90 - 2) = 8100 - 540 +8 (To get to Sutra 2, 100-540+8 is done first) = 8000-432 = (Sutra 2) 7568
This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.
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Sutras
7 * 8
=(10-3) * (10 -2)
= 100 - 20 -30 +6
= 56
The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply 32 by 46.
We multiply "vertically and crosswise":
Vertically 2x6=12.
so put down 2 and carry 1.
Then we multiply crosswise and add the two results: 3x6+4x2=26,
Add the carried 1 to make it 27
so put down 7 and carry 2.
Finally we multiply vertically 3x4=12 and
add the carried 2 =14.
Result: 1,472.
We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. With some practice, one can do pretty long multiplications in their heads.
Can you now do some quick calculations for multiplications with 11?
What is 28* 11? What is 375*11?
The answer for this is very simple:
28 * 11 = 308
Simply write the 2,(2+8),8 --> 2,(10),8 (1 from 10 has to be taken forward and added to 2) so answer is 308
The magic is simplified when you break 11 to (10+1): 28 * 11 = 28 * (10+1) = 280 +28 --> 2 (8+2) 8 --> 308
375 * 11 = 4125
What is happening here? 375 * 11 --> 3,(3+7),(7+5),5 --> 3,(10),(12),5 --> 4125
This Sutra is used to solve complex non-linear partial differential equations. It deals with calculation of common functions and their series expansions, and the solution of simultaneous, algebraic, transcendental and differential equations.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below. 100-4= 96 or 100-8=92. So 10000 - 1200 + 32. Applying the Sutra 2 for the calculation 10000 - 1168 = 8832.
Try this 86 * 88 = (90-4) * (90 - 2) = 8100 - 540 +8 (To get to Sutra 2, 100-540+8 is done first) = 8000-432 = (Sutra 2) 7568
This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.