Vedic Maths Sutra 03 : Ūrdhva-Tiryagbhyām

Sūtra 3 of 16
ऊर्ध्वतिर्यग्भ्याम्
Ūrdhva-Tiryagbhyām
“Vertically and Crosswise”

Meaning & Essence

This Sūtra means “Vertically and Crosswise”. It is the most general and powerful multiplication technique in Vedic Mathematics. Unlike the previous two Sūtras which work best for special cases, this method works for any two numbers of any size.

The method multiplies digits vertically and then crosswise, making it ideal for mental calculation once you get used to the pattern. It can be extended from 2-digit numbers to 3-digit, 4-digit and beyond.

Mathematical Principle

For two 2-digit numbers written as \( 10a + b \) and \( 10c + d \):

\[ (10a + b)(10c + d) = 100(ac) + 10(ad + bc) + bd \]

For three-digit numbers the same idea expands:

\[ \begin{align*} &\text{Units: } && c_1 \times c_2 \\ &\text{Tens: } && (b_1 \times c_2) + (c_1 \times b_2) \\ &\text{Hundreds: } && (a_1 \times c_2) + (b_1 \times b_2) + (c_1 \times a_2) \\ &\text{Thousands: } && (a_1 \times b_2) + (b_1 \times a_2) \\ &\text{Ten-thousands: } && a_1 \times a_2 \end{align*} \]

In simple words (working from right to left):

  • Vertical on the units place
  • Crosswise on successive groups of digits
  • Continue moving left, combining vertical and crosswise products
  • Handle carries whenever a result has more than one digit

Worked Examples

Example 1 → Find \( 23 \times 12 \) (2-digit)
  • Units (vertical): \( 3 \times 2 = 6 \)
  • Crosswise: \( 2 \times 2 + 3 \times 1 = 4 + 3 = 7 \)
  • Tens (vertical): \( 2 \times 1 = 2 \)
Answer: 276
Example 2 → Find \( 48 \times 37 \) (2-digit with carries)
  • Units: \( 8 \times 7 = 56 \) → write 6, carry 5
  • Crosswise: \( 4 \times 7 + 8 \times 3 = 28 + 24 = 52 \) + carry 5 = 57 → write 7, carry 5
  • Tens: \( 4 \times 3 = 12 \) + carry 5 = 17
Answer: 1776
Example 3 → Find \( 123 \times 456 \) (3-digit)
  • Units: \( 3 \times 6 = 18 \) → write 8, carry 1
  • Tens cross: \( 2 \times 6 + 3 \times 5 = 12 + 15 = 27 \) + 1 = 28 → write 8, carry 2
  • Hundreds: \( 1 \times 6 + 2 \times 5 + 3 \times 4 = 6 + 10 + 12 = 28 \) + 2 = 30 → write 0, carry 3
  • Thousands: \( 1 \times 5 + 2 \times 4 = 5 + 8 = 13 \) + 3 = 16 → write 6, carry 1
  • Ten-thousands: \( 1 \times 4 = 4 \) + 1 = 5
Answer: 56088
Interactive Demo – Multiply 2-digit or 3-digit numbers
Please enter two valid numbers between 10 and 999

    When to Use This Sūtra

    • Multiplication of any two numbers (2-digit, 3-digit, 4-digit and larger)
    • Especially powerful for mental math with 2-digit and 3-digit numbers
    • Can also be extended to algebraic multiplication and division
    • The most versatile multiplication method in Vedic Mathematics

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