Vedic Addition, Subtraction, Multiplication & Division

1. Quick Addition & Subtraction Techniques

A. Dot Method of Addition (Shuddha Method)

Conventional vertical column addition involves holding large running totals in memory. In the Vedic Shuddha method:

Numbers in a column are added from top to bottom.
Whenever the progressive sum reaches or exceeds 10, drop 10 by placing a small dot (Rekha or Shuddha) next to the number, and carry forward only the remaining units digit.
Count the total number of dots and add them directly to the next column.

B. Nikhilam Subtraction (“All from 9 and Last from 10”)

Subtracting from large powers of 10 (such as 1,000 or 10,000) often requires multiple borrows. The Vedic rule eliminates borrowing completely:

Subtract every digit from 9, and the last non-zero digit from 10.
Example: Calculate 10,000 - 6,437
- First digit: 9 - 6 = 3
- Second digit: 9 - 4 = 5
- Third digit: 9 - 3 = 6
- Last digit: 10 - 7 = 3
- Result = 3,563 (calculated mentally from left to right in 2 seconds).

2. Fast Multiplication Algorithms

A. Base Multiplication using Nikhilam Navatashcaramam Dashatah

Used when both numbers are close to a common base (e.g., 10, 100, 1000).

Case 1: Numbers Below Base (Example: 96 × 94, Base = 100)

Step 1: Left-Hand Part (LHS)

Write deviations from base 100:

96 has deviation: -4
94 has deviation: -6

Cross-subtract: 96 - 6 = 90 (or 94 - 4 = 90).
LHS = 90

Step 2: Right-Hand Part (RHS)

Multiply the deviations:

(-4) × (-6) = +24

Since base 100 has two zeros, RHS takes two digits: 24.
Combined Answer: 9,024

Case 2: Numbers Above Base (Example: 108 × 107, Base = 100)

Step 1: Cross-Addition (LHS)

Deviations: +8 and +7.
Cross-add: 108 + 7 = 115 (or 107 + 8 = 115).
LHS = 115

Step 2: Multiply Deviations (RHS)

Multiply: (+8) × (+7) = 56.
Two digits match two zeros of base 100.
Combined Answer: 11,556

B. General Multiplication: Urdhva-Tiryakbhyam (Vertically and Crosswise)

The universal master method for any two numbers without restriction of base:

Three-Step Flow of 2-Digit Urdhva-Tiryakbhyam (Example: 43 × 26)

Step 1: Vertical Units 3 × 6 = 18 (Write 8, Carry 1)

➔

Step 2: Crosswise & Sum (4 × 6) + (3 × 2) + 1 = 31 (Write 1, Carry 3)

➔

Step 3: Vertical Tens (4 × 2) + 3 = 11

➔

Final Product Combine: 1,118

3. Fast Division using Paravartya Yojayet

Paravartya Yojayet (“Transpose and Apply”) transposes divisor digits slightly below or above base into their sign-inverted counterparts, converting tedious long divisions into simple vertical multiplication and addition.

Example: Divide 1,234 by 112 (Base = 100):

Transpose divisor digits after base 1: +1 and +2 become -1 and -2.
The quotient digits multiply the transposed digits and are added down column-by-column, yielding:
- Quotient = 11
- Remainder = 2
- Verification: (112 × 11) + 2 = 1,232 + 2 = 1,234.

Vedic Addition, Subtraction, Multiplication & Division | B.Com Notes