Sūtra 3 of 16

Ūrdhva-Tiryagbhyām

Vertically and crosswise

General multiplication for any two numbers — the 'Vedic multiplication' people know.

When to use

Any multiplication; especially efficient for 2- and 3-digit numbers done mentally.

How it works

For two 2-digit numbers ab × cd: rightmost digit = b·d; middle = a·d + b·c; leftmost = a·c. Carry as needed.

Memory formula

2-digit ab × cd: ac | ad + bc | bd, carrying from right to left.

Worked examples

Problem 1

23 × 41

0/4

Problem 2

62 × 78

0/4

Try it yourself

Try to solve, then reveal the steps one at a time.

easy

Compute 21 × 32

medium

Compute 56 × 47

Modern applications

Where this ancient technique still lives in today's world.

Coding

Schoolbook to Karatsuba

Vertically-and-crosswise is the schoolbook multiplication that big-integer libraries (GMP, Java BigInteger) optimise via Karatsuba and Toom–Cook.

AI / ML

Matrix dot products

The crosswise pattern is conceptually a 1D convolution / inner product — the building block of every neural network layer.

Code in your favourite language

Multiply any two numbers using vertical-and-crosswise digit by digit.

Python

def urdhva(a: int, b: int) -> int:
    sa = list(map(int, str(a)))[::-1]
    sb = list(map(int, str(b)))[::-1]
    n = len(sa) + len(sb)
    out = [0] * n
    for i, x in enumerate(sa):
        for j, y in enumerate(sb):
            out[i + j] += x * y
    for k in range(n - 1):
        out[k + 1] += out[k] // 10
        out[k] %= 10
    return int(''.join(map(str, out[::-1])).lstrip('0') or '0')

print(urdhva(56, 47))  # 2632

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