Urdhva-Tiryagbhyam: The Vertically and Crosswise Multiplication Method

What is Urdhva-Tiryagbhyam?

Urdhva-Tiryagbhyam means "Vertically and Crosswise" in Sanskrit. It is one of the most powerful sutras in Vedic Mathematics, capable of multiplying any two numbers — regardless of digit length — in a single, unified process.

Two-Digit Multiplication

Let's multiply 23 × 45 using this method:

• Step 1 (Vertical) — Multiply the units digits: 3 × 5 = 15. Write down 5, carry 1.
• Step 2 (Crosswise) — Cross multiply: (2 × 5) + (3 × 4) = 10 + 12 = 22. Add the carry 1 = 23. Write down 3, carry 2.
• Step 3 (Vertical) — Multiply the tens digits: 2 × 4 = 8. Add the carry 2 = 10. Write down 10.
• Result — 1035 ✓

Three-Digit Multiplication

For 123 × 456, the pattern extends naturally:

• Step 1: Vertical (units × units)
• Step 2: Crosswise (tens×units + units×tens)
• Step 3: Crosswise (hundreds×units + tens×tens + units×hundreds)
• Step 4: Crosswise (hundreds×tens + tens×hundreds)
• Step 5: Vertical (hundreds × hundreds)

Why It's So Fast

Traditional multiplication requires writing intermediate rows and adding them. Urdhva-Tiryagbhyam eliminates this entirely — you compute each digit of the answer directly, moving from right to left.

With practice, students can multiply 4-digit numbers mentally in under 10 seconds using this technique.

Practice Tips

Start with 2-digit numbers and master the crosswise pattern before moving to 3-digit. Speed comes naturally with consistent practice.