Vedic Maths has three main ways to square a number. Two are shortcuts that only work for certain numbers: numbers ending in 5, and numbers close to 10, 100 or 1,000. The third, the duplex method, works for any number. This guide explains each one step by step, shows why it works, and ends with a practice set so you can try them yourself.
All three build on the squares you already know. If the squares from 1 to 20 aren't automatic yet, start with our squares 1 to 20 guide.
The sutra Ekadhikena Purvena means "by one more than the one before". To square a number ending in 5:
For 85², the digit in front is 8. One more than 8 is 9, and 8 × 9 = 72. Write 25 after it to get 7,225.
Why it works: 85 is halfway between 80 and 90, and 85 × 85 is 80 × 90 plus 5 × 5. The 80 × 90 gives 7,200, which is where 8 × 9 comes from, and 5 × 5 adds the 25. The same idea works for any two numbers with the same front digits whose last digits add up to 10, such as 33 × 37. That version is in our Vedic Maths fast multiplication tricks.
Yavadunam means "whatever the extent of its deficiency". It works for numbers close to a base: 10, 100, 1,000 and so on. Find how far the number is from the base, then:
Take 97². The base is 100 and 97 is 3 below it. Take the 3 away again: 97 − 3 = 94. Square the difference: 3 × 3 = 9, written as 09 because the base 100 has two zeros. Put the two parts side by side to get 9,409.
Above the base, add instead. For 104², 104 is 4 above 100, so the left part is 104 + 4 = 108 and the right part is 4 × 4 = 16. The answer is 10,816.
If the square of the difference has too many digits, carry the extra digit into the left part. For 88², the difference is 12, so the left part is 88 − 12 = 76. But 12 × 12 = 144, which is three digits when only two are allowed. Keep the 44 and carry the 1: the left part becomes 77, and the answer is 7,744.
Base 10 works the same way, with one digit on the right. For 14², the left part is 14 + 4 = 18 and the right part is 4 × 4 = 16. Keep the 6 and carry the 1, so the answer is 196.
Near 1,000, the right part has three digits. For 988², the difference is 12 below, so the left part is 988 − 12 = 976, and 12 × 12 = 144. The answer is 976,144. For 1,007², the left part is 1,007 + 7 = 1,014 and the right part is 7 × 7 = 49, written as 049. The answer is 1,014,049.
Why it works: for a number d above the base, (base + d)² multiplies out to base × (base + 2d) + d². The number moved another d away is base + 2d, so multiplying it by the base gives the left part, and d² is the right part. Below the base, the same algebra works with d taken away. The same base idea speeds up multiplying two different numbers; see the Nikhilam sutra.
The duplex method squares any number, column by column, without writing out a long multiplication. It uses the duplex of a group of digits, written D:
For a two-digit number, there are three columns. From the left, they are the duplex of the first digit, the duplex of both digits, and the duplex of the last digit. Then write the answer from the right, carrying as you go.
Take 43². The columns are 16, 24 and 9. From the right: write 9. Next, 24: write 4 and carry 2. Last, 16 + 2 = 18. The answer is 1,849.
Take 67². The columns are 36, 84 and 49. Write 9 and carry 4. Then 84 + 4 = 88: write 8 and carry 8. Last, 36 + 8 = 44. The answer is 4,489.
A three-digit number has five columns. For 213, they are the duplexes of 2, 21, 213, 13 and 3: that is 4, 4, 13, 6 and 9. From the right, write 9, then 6, then 3 from the 13 and carry 1. The next column is 4 + 1 = 5, and the last is 4. The answer is 45,369.
The duplex method is slower than the shortcuts for numbers they suit, but it never fails, and with practice you can do two-digit squares entirely in your head.
| The number | Use | Example |
|---|---|---|
| Ends in 5 | Ekadhikena Purvena | 75² is 5,625 |
| Close to 10, 100 or 1,000 | Yavadunam | 96² is 9,216 |
| Anything else | Duplex | 54² is 2,916 |
Two quick checks catch most slips. First, the last digit: a number ending in 7 always has a square ending in 9, and one ending in 4 has a square ending in 6. Second, the digit sum. For 67², the digits of 67 add up to 13, then 4, and 4 × 4 = 16, which reduces to 7. The digits of 4,489 add to 25, which also reduces to 7, so the answer passes. Our guide to digital roots explains this check in full.
Answers: (1) 225, from 1 × 2 = 2. (2) 2,025, from 4 × 5 = 20. (3) 9,216: the difference is 4, 96 − 4 = 92, and 4 × 4 = 16. (4) 10,609: 103 + 3 = 106, and 3 × 3 = 9, written 09. (5) 984,064: 992 − 8 = 984, and 8 × 8 = 64, written 064. (6) 2,916: the columns are 25, 40 and 16. (7) 5,041: the columns are 49, 14 and 1. (8) 97,344: the columns are 9, 6, 13, 4 and 4.
Squaring works in both directions: once you can square quickly, finding square roots of perfect squares gets much easier. For the non-Vedic patterns, see squaring numbers fast. Squares are multiplication facts, so keep them sharp with the multiplication game or the times tables chart. For the other methods in this series, browse all our Vedic Maths guides.
It depends on the number. For numbers ending in 5, Ekadhikena Purvena is fastest. For numbers close to 10, 100 or 1,000, use Yavadunam. For any other number, the duplex method works every time.
The right part must have exactly as many digits as the base has zeros. When the square of the difference is longer, as with 88², where 12 × 12 is 144, the extra digit belongs in the left part, so you add it there.
Yes. A number with more digits just has more columns. A three-digit number has five, and each column's duplex pairs digits from the outside in, so the method scales to any length.