Vedic maths › Vertically and crosswise
Vertically and crosswise: multiplying any two numbers
Ūrdhva-tiryagbhyām, “vertically and crosswise”, is the general multiplication method of Vedic maths: it works for any two numbers, not just special ones. It builds the answer one place at a time, so the working fits on one line, or in your head.
The idea is much older than the sutra: Indian mathematicians such as Śrīdhara and Mahāvīra taught this kind of cross-multiplication more than a thousand years ago.
The method
- Write one number above the other.
- Vertically on the left: multiply the first digits. That is the first part of the answer.
- Crosswise: multiply the first digit of each number by the last digit of the other, and add. That is the middle part.
- Vertically on the right: multiply the last digits. That is the last part.
- Each part keeps one digit: carry the rest to the part on its left, starting from the right.
Worked examples
23 × 41
- Vertically on the left: the first digits.2 × 4 = 8
- Crosswise: first by last, and last by first.2 × 1 + 3 × 4 = 2 + 12 = 14
- Vertically on the right: the last digits.3 × 1 = 3
- Put the parts side by side.8 | 14 | 3
- Carry from the right, keeping one digit in each part.3; 14: keep 4, carry 1; 8 + 1 = 9 → 943
23 × 41 = 943
67 × 58
- Vertically on the left: the first digits.6 × 5 = 30
- Crosswise: first by last, and last by first.6 × 8 + 7 × 5 = 48 + 35 = 83
- Vertically on the right: the last digits.7 × 8 = 56
- Put the parts side by side.30 | 83 | 56
- Carry from the right, keeping one digit in each part.56: keep 6, carry 5; 83 + 5 = 88: keep 8, carry 8; 30 + 8 = 38 → 3886
67 × 58 = 3886
213 × 132
- Vertically on the left: the first digits.2 × 1 = 2
- Crosswise on the left: the first two digits of each.2 × 3 + 1 × 1 = 6 + 1 = 7
- The big cross: first by last, middle by middle, last by first.2 × 2 + 1 × 3 + 3 × 1 = 4 + 3 + 3 = 10
- Crosswise on the right: the last two digits of each.1 × 2 + 3 × 3 = 2 + 9 = 11
- Vertically on the right: the last digits.3 × 2 = 6
- Put the parts side by side.2 | 7 | 10 | 11 | 6
- Carry from the right, keeping one digit in each part.6; 11: keep 1, carry 1; 10 + 1 = 11: keep 1, carry 1; 7 + 1 = 8; 2 → 28116
213 × 132 = 28116
Why it works
A two-digit number is 10a + b. So (10a + b)(10c + d) = 100 × ac + 10 × (ad + bc) + bd: the vertical and crosswise products are the hundreds, tens and units of the answer. A part bigger than 9 spills into the next place, which is the carry.
With three digits there are five places, and each one collects every product whose two digits’ places add up to it. That is why the middle place has three products: the big cross.
Watch out
- Most slips are in the carries. Write the parts first, then carry from the right.
- For numbers of different lengths, put zeros in front of the shorter one: 123 × 45 is 123 × 045.
- The special methods are quicker when they fit: near a base, last digits adding to 10, times 11.
Try these
Work each one out, then tap it to see the answer.
32 × 21
6 | 7 | 2 → 672
46 × 37
12 | 46 | 42 → 1702
84 × 26
16 | 56 | 24 → 2184
123 × 211
2 | 5 | 9 | 5 | 3 → 25953
312 × 124
3 | 7 | 16 | 8 | 8 → 38688
Practise it in the game: Maths, the multiplication levels. Free, with no sign-up.
Source: This method is from Bharati Krishna Tirtha’s Vedic Mathematics (1965). The explanation, examples and questions here are our own.
More Vedic maths
- Squares ending in 5: 75²: 7 × 8 = 56, then 25: 5625.
- All from 9, the last from 10: 1000 − 357: 9 − 3, 9 − 5, 10 − 7: 643.
- Multiplying near 100: 97 × 96: 97 − 4 = 93, 3 × 4 = 12: 9312.
- Squaring near 100: 96²: 96 − 4 = 92, 4² = 16: 9216.
- Multiplying by 9, 99, 999: 437 × 999: 436, then 999 − 436 = 563: 436563.
- Last digits adding to 10: 63 × 67: 6 × 7 = 42, 3 × 7 = 21: 4221.
- Multiplying by 11: 253 × 11: 2 | 2 + 5 | 5 + 3 | 3: 2783.
- Checking with digit sums: 437 × 26 = 11362? 5 × 8 → 4, and 11362 → 4: it passes.
- Dividing by 9: 2312 ÷ 9: 2, 2 + 3, 5 + 1: 256, remainder 6 + 2 = 8.
- Squaring with the duplex: 43²: 4² | 2 × 4 × 3 | 3² = 16 | 24 | 9: 1849.
Where Vedic maths comes from, all 16 sutras, and our sources: Vedic maths, explained.