ऊर्ध्व-तिर्यग्भ्याम्

Ūrdhva-Tiryagbhyāṃ

"Vertically and crosswise"

LearnStep-by-step animation PracticeInteractive problems ProofWhy it works

What this sutra solves

The all-purpose multiplication trick — multiply any two numbers in a single line, perfect for mental math and quick bill checks.

Splitting a group outing: 23 people each owe a ₹14 entry fee.

becomes

23 \times 14

⚡

Vertically & crosswise in one line → 322.

Estimating wall paint: a 43 ft by 27 ft wall area.

becomes

43 \times 27

⚡

Three crosswise columns → 1161 sq ft.

Live Demo— animated step-by-step

12 × 13

Set up Vertical & Crosswise

Write 12 and 13. We will compute column by column: vertical (same position) and crosswise (crossing positions).

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⚡ Speed Advantage

Vedic

3 steps

Traditional

6 steps

2× faster with Vedic Mathematics

Best for

• General multiplication — any two numbers
• The universal Vedic multiplier

Use when

• Any two numbers — this is the universal method

Avoid when

• Numbers very close to a base (Nikhilam is faster there)

Intuition

Multiply digits vertically, then crosswise, then vertically again — the pattern forms an expanding diamond.

Story Mode

The Diamond of Digits

Draw two numbers side by side. Now draw lines — one vertical between matching positions, then crossing diagonals. Each intersection is a product. Sum down each column. This is Urdhva-Tiryag: 'vertical and crosswise'. Every multiplication can be seen as a lattice of crossing lines. The ancient rishis had visualized what we now call polynomial multiplication centuries before algebra was formalized in Europe.

Vedic vs conventional

Conventional: $43\times 27 = 43\times 20 + 43\times 7 = 860+301 = 1161$ (2 partial products + addition). Urdhva: $4\times 2=8$ , $4\times 7+3\times 2=34$ (3 carry 3), $3\times 7=21+3=24 \to 1161$ (simultaneous mental computation).

Works for any digit count. Single mental pass with carries, no partial products written.

Applications

2-digit multiplication

The simplest crosswise pattern: three steps.

12 \times 13

23 \times 14

43 \times 27

76 \times 83

3-digit multiplication

Five-step diamond pattern — the visual beauty scales up.

123 \times 456

234 \times 567

999 \times 999

Common Mistakes to Avoid

Missing a crosswise pair in 3-digit multiplication

Wrong approach

For

123\times 456

, Step 3 needs ALL three crosswise products:

1\times 6+2\times 5+3\times 4

. Students often forget the middle term.

Correct approach

✓

Count the crosswise pairs carefully. For position k, multiply all digit pairs whose indices sum to k.

Why this happens

💡 Students treat it like 2-digit and miss the third pair in the middle column.

Why It Works

Write two 2-digit numbers in place value form:

(10a+b)(10c+d)

Expand by place value:

=100ac+10(ad+bc)+bd

Match the visual pattern:

\text{left}=ac,\quad \text{middle}=ad+bc,\quad \text{right}=bd

The vertical products give the outer positions, and the crosswise products give the middle. Carries move exactly as they do in ordinary multiplication.

For AB × CD: result = [ $A\times C$ ] [ $A\times D + B\times C$ ] [ $B\times D$ ]. This is exactly the expansion of $(10A+B)(10C+D) = 10$ $0AC + 10$ (AD+BC) + BD. The crosswise products fill the middle. For 3-digit: the diamond has 5 nodes matching 5 partial-product positions.

Explore Next

Related Sutra

Nikhilaṃ Navataścaramaṃ Daśataḥ

All from 9 and the last from 10

Related Sub-Sutra

Ānurūpyeṇa

Proportionately

Related Sub-Sutra

Ādyamādyena

The first by the first and the last by the last