Vedic Math Guru

Meaning: "Transpose and apply" or "Transpose and adjust".

Primary Applications:

Algebraic Division: This sutra provides an elegant method for division of polynomials, especially when the divisor is a simple linear binomial (like x - a or x + a) or can be easily related to one. It's also applicable for division of numbers when the divisor is slightly greater or slightly less than a power of 10.
Example: Divide 123 by 11. Base is 10. Divisor 11 is 1 more than base. Transposed digit is -1.
```
  1 | 2  3 
    | -1 -1 
  ----------- 
  1   1 | 2
```
(Quotient 11, Remainder 2).
Bring down 1. Multiply 1 by -1 = -1. Add to 2 -> 1. Multiply 1 by -1 = -1. Add to 3 -> 2.
Solving Linear Equations: It's used to solve linear equations by transposing terms. When a term moves from one side of the equation to the other, its sign changes. For example, in ax + b = c, transposing b gives ax = c - b. Then transposing 'a' (as a divisor) gives x = (c - b) / a.
Solving Specific Types of Equations: The sutra is particularly powerful for equations of the form ax + b = cx + d. By transposing, we get ax - cx = d - b, so x(a-c) = d-b, leading to x = (d-b)/(a-c). More complex forms like (x+a)(x+b) = (x+c)(x+d) can also be solved by first expanding and then applying transposition, or by noticing specific relationships between a, b, c, d that simplify using Paravartya principles.

Core Idea: The sutra embodies the principle of "changing sides, changing signs" and using the "adjusted" or "transposed" values in subsequent operations. In division, the digits of the divisor (relative to a base) are effectively "transposed" (signs changed) and used in a simplified multiplication-addition process instead of conventional subtraction.

Let's divide 123 by 9 using the Paravartya Yojayet method. This method is great for divisors near a base.

Dividend: 123

Divisor: 9

Nearest Base for divisor 9: 10

Deficiency of divisor from base: 10 - 9 = 1.

"Transpose and Apply": The working digit (transposed deficiency) is +1 (since 9 is 1 *less* than 10, we use the positive value).

Setting up the Division

Since the divisor (9) has one digit (and base 10 has one zero), we separate one digit from the right of the dividend for the remainder.

1 2 | 3 (Dividend separated)

Working Digit (Transposed): 1

The Division Process

Bring down the first digit of the dividend (1). This is the first digit of our quotient.
```
  1  2 | 3
           |
  -------
  1       |
```
Quotient starts with: 1
Multiply this quotient digit (1) by the working digit (1).
1 × 1 = 1.
Place this result (1) under the next dividend digit (2).
```
  1  2 | 3
     1   |
  -------
  1       |
```
Add vertically: 2 + 1 = 3. This is the next digit of our quotient.
```
  1  2 | 3
     1   |
  -------
  1  3 |
```
Quotient so far: 13
Multiply this new quotient digit (3) by the working digit (1).
3 × 1 = 3.
Place this result (3) under the next dividend digit (which is 3, in the remainder part).
```
  1  2 | 3
     1   | 3
  -------
  1  3 |
```
Add vertically in the remainder part: 3 + 3 = 6. This is our remainder.
```
  1  2 | 3
     1   | 3
  -------
  1  3 | 6
```

The Result!

Quotient: 13

Remainder: 6

So, 123 ÷ 9 = 13 with a remainder of 6. 🎯