Multiply 98 by 97.
If you reached for a pen, or started stacking the numbers in your head, there is a faster way. It takes about three seconds once you know it, and it needs no written working at all.
The answer is 9506. Here is how to get there.
The sutra, and what it is telling you to do
The rule is called Nikhilam Navatashcaramam Dashatah, usually shortened to Nikhilam. It translates as all from 9 and the last from 10.
That sounds cryptic until you see what it is for. It is an instruction for working out how far a number sits below a base such as 10, 100 or 1000.
Take 97, with a base of 100. Each digit from 9, the last from 10: nine minus nine is zero, ten minus seven is three. So 97 sits 3 below the base. That number, the deficiency, is what the whole method runs on.
The method, on 98 times 97
Step by step
- Pick the base. Both numbers are near 100.
- Find each deficiency. 98 is 2 below, 97 is 3 below.
- Left-hand part: take either number and subtract the other's deficiency. 98 minus 3 is 95. Or 97 minus 2 is 95. Either way, the same answer, and that is a useful check.
- Right-hand part: multiply the deficiencies. 2 times 3 is 6.
- Write them side by side. The base has two zeros, so the right-hand part gets two digits: 06.
95 | 06 = 9506
Two more, quickly.
96 times 93. Deficiencies 4 and 7. Left: 96 minus 7 is 89. Right: 4 times 7 is 28. Answer 8928.
88 times 98. Deficiencies 12 and 2. Left: 88 minus 2 is 86. Right: 12 times 2 is 24. Answer 8624.
Why it works, which matters more than it sounds
Most of what is written about Vedic Maths presents these methods as magic. That is a mistake, and it is the main reason people distrust the subject. It is not magic. It is three lines of algebra.
Call the base B, and let the two numbers be B minus a and B minus b. Multiply them out and you get:
(B − a)(B − b) = B² − Ba − Bb + ab
= B(B − a − b) + ab
The bracket, B − a − b, is the left-hand part. And notice it equals (B − a) − b, which is the first number minus the second's deficiency. That is exactly step 3. The ab on the end is step 4.
So the method is not a trick at all. It is a rearrangement, chosen because subtracting a small number and multiplying two small numbers is much easier in your head than multiplying two large ones.
This is worth teaching to a child rather than hiding. A student who has seen the algebra can work out for themselves when the method will help and when it will not.
When the right-hand part does not fit
The base decides how many digits the right-hand part is allowed. A base of 100 has two zeros, so two digits. A base of 1000 allows three.
Sometimes the product overflows. Take 87 times 89. Deficiencies 13 and 11. Left: 87 minus 11 is 76. Right: 13 times 11 is 143, which is three digits where only two are allowed.
You carry, exactly as in ordinary arithmetic. The extra 1 moves left: 76 plus 1 is 77, and 43 stays behind.
77 | 43 = 7743
Above the base, and one of each
Nothing changes except the signs.
103 times 104. Both sit above 100, by 3 and 4. Now you add instead of subtracting: 103 plus 4 is 107. Right: 3 times 4 is 12. Answer 10712.
103 times 98. One above by 3, one below by 2. Left: 103 minus 2 is 101. Right: 3 times minus 2 is minus 6. A negative right-hand part means you subtract it from the whole: 10100 minus 6 gives 10094.
That last case is where most people stumble, and it is worth practising a few times, because a mixed pair looks exactly like the others until the sign bites.
Other bases
The base does not have to be 100. 994 times 998, against a base of 1000: deficiencies 6 and 2. Left: 994 minus 2 is 992. Right: 6 times 2 is 12, padded to three digits as 012. Answer 992012.
It works downwards too. 9 times 8 against a base of 10: deficiencies 1 and 2. Left: 9 minus 2 is 7. Right: 1 times 2 is 2. Answer 72. Which is a small illustration of a large point: this is the same arithmetic children already know, arranged differently.
What it is not good for
Here is the honest limit, and it is worth saying plainly because a great deal of writing about Vedic Maths will not say it.
Nikhilam only helps when both numbers sit near the same base. Try it on 37 times 84 and the deficiencies against 100 are 63 and 16. You now have to multiply 63 by 16 in your head, which is harder than the sum you started with.
Knowing when not to reach for a technique is part of knowing the technique. A child taught only the trick will use it everywhere, including where it costs them time. A child taught the algebra will see immediately that large deficiencies defeat the point.
Where it earns its keep
Arithmetic-heavy sections of competitive exams, where marks depend on speed as much as accuracy, and where written working is time you do not have. Squares of numbers near a base fall out of the same rule, since a square is just a number multiplied by itself.
It is also, quite simply, satisfying. A child who can do 98 times 97 faster than an adult with a pen tends to develop an appetite for the rest of the sixteen sutras, and appetite is most of the battle in mathematics.
Learning the rest of the sutras
Ms. Ishita Parikh holds the International Vedic Maths Teachers' Training certification at Grade A, a six-month programme approved under Skill India and the NSDC. Classes cover all sixteen sutras with the reasoning underneath each one, not as tricks to memorise.
See how the course is structured, or book the free first class and try a few of these live.
Questions people ask
What does the Nikhilam sutra actually mean?
Nikhilam Navatashcaramam Dashatah translates as all from 9 and the last from 10. It is an instruction for finding how far a number sits from a base like 10, 100 or 1000. For 97 against a base of 100, you take each digit from 9 and the final digit from 10, which gives 3. That 3 is the deficiency, and the whole method runs on it.
Does the Nikhilam method work for any two numbers?
No, and anyone who says otherwise is overselling it. It works when both numbers sit close to the same base. For something like 37 times 84, neither is near 10, 100 or 1000, and the deficiencies are so large that the method gives you a harder sum than the one you started with. Knowing when not to use a technique is part of knowing the technique.
Why does the Nikhilam multiplication trick work?
It is ordinary algebra. If the two numbers are base minus a and base minus b, multiplying them out gives base squared, minus base times a, minus base times b, plus a times b. Factor the first three terms and you get base times the quantity base minus a minus b, plus a times b. The left-hand part of the answer is base minus a minus b, which is just one number minus the other's deficiency, and the right-hand part is a times b.
What happens if the right-hand part has too many digits?
You carry, exactly as in ordinary arithmetic. The base decides how many digits the right-hand part is allowed: two for a base of 100, three for 1000. If the product of the deficiencies overflows that, the excess carries into the left-hand part. For 87 times 89 the deficiencies multiply to 143, which is one digit too many, so the 1 carries and you are left with 77 and 43, giving 7743.
Is Vedic Maths useful for competitive exams?
For the arithmetic-heavy sections, yes, because the marks there depend on speed as much as accuracy and these methods remove written working. It is worth being realistic about the scope: they help most with a specific family of calculations, and they are a supplement to a solid grasp of ordinary arithmetic rather than a replacement for it.