Vedic Math Guru

Description

Meaning: "All from 9 and the last from 10". This sutra is a cornerstone for subtraction from powers of 10 (like 100, 1000) and for multiplying numbers near these bases.

Subtraction from Powers of 10:

To subtract a number from a power of 10 (e.g., 1000 - 457):

Take the number being subtracted (minuend, here 457). Starting from its leftmost digit, subtract each digit from 9.
(9 - 4 = 5), (9 - 5 = 4).
Subtract the very last digit of the minuend from 10.
(10 - 7 = 3).
Combine these results: 543. So, 1000 - 457 = 543.

Note: The number of digits in the result should match the number of zeros in the power of 10. If the minuend has fewer digits than zeros in the base, pad it with leading zeros (e.g., 1000 - 57 becomes 1000 - 057).

Multiplication Near a Base:

For numbers close to a base (e.g., 98 x 97, base 100):

Find the 'deficiencies' from the base:
98 is 2 less than 100 (deficiency = -2 or simply 2).
97 is 3 less than 100 (deficiency = -3 or simply 3).
The answer has two parts:
First part (LHS): Subtract one deficiency from the other number (or cross-subtract). E.g., 98 - 3 = 95 (or 97 - 2 = 95).
Second part (RHS): Multiply the deficiencies: (-2) x (-3) = 6. Since the base 100 has two zeros, the RHS should have two digits. So, write 06.
Combine: 9506. So, 98 x 97 = 9506.

If numbers are above the base (e.g., 103 x 104), the 'deficiencies' are 'surpluses' (+3, +4). The process is similar:
LHS: 103 + 4 = 107 (or 104 + 3 = 107).
RHS: 3 x 4 = 12.
Answer: 10712.

Why it works (Conceptual):

For subtraction, 1000 - N is like 999 - N + 1. Subtracting each digit of N from 9 (except the last) and the last from 10 achieves this. For multiplication, (Base - a)(Base - b) = Base² - Base(a+b) + ab. The Nikhilam method cleverly arranges this.

Example 1: Calculate 1000 - 457

Let's use the Nikhilam Sutra to subtract 457 from 1000. The sutra says: "All from 9 and the last from 10".

Number to subtract: 457

Base: 1000 (three zeros, so our answer will have 3 digits)

Applying "All from 9, Last from 10" to 457

First digit (4): Subtract from 9.
9 - 4 = 5
Second digit (5): Subtract from 9.
9 - 5 = 4
Last digit (7): Subtract from 10.
10 - 7 = 3

Result: Combine the Digits!

543

So, 1000 - 457 = 543! Simple, right? 🎉

Example 2: Calculate 96 x 98

Let's multiply 96 × 98 using the Nikhilam Sutra. These numbers are close to the base 100.

First Number: 96

Second Number: 98

Chosen Base: 100

Step 1: Find Deficiencies from Base 100

For 96: Deficiency = 100 - 96 = 4 (It's 4 less than 100)
For 98: Deficiency = 100 - 98 = 2 (It's 2 less than 100)

We can write this as:
96 -4 (deficiency)
98 -2 (deficiency)

Step 2: Calculate the Two Parts of the Answer

Part 1 (Left Hand Side - LHS):

Cross-subtract one number and the other's deficiency:

96 - 2 (deficiency of 98) = 94

OR 98 - 4 (deficiency of 96) = 94

So, LHS = 94.

Part 2 (Right Hand Side - RHS):

Multiply the deficiencies:

(-4) × (-2) = 8

Since our base (100) has two zeros, the RHS must have two digits. So, 8 becomes 08.

So, RHS = 08.

Step 3: Combine LHS and RHS!

LHS | RHS

94 | 08

9408

Therefore, 96 × 98 = 9408! Cool, isn't it? 😎