Vedic maths › Squaring near 100

Squaring numbers close to 100 or 1000

यावदूनं तावदूनीकृत्य वर्गं च योजयेत् Yāvadūnaṃ Tāvadūnīkṛtya Vargañca Yojayet “whatever the deficiency, lessen by it, and set up its square”, a sub-sutra of Vedic maths. The book also calls it the Yāvadūnam sutra, after sutra 10, “by the deficiency”.

A square is a number multiplied by itself, so the base method works for squares too. Vedic maths gives squares a rule of their own: lessen the number by its deficiency from the base, and set the square of the deficiency beside it.

The method

Find how far the number is from the base: 96 is 4 below 100; 107 is 7 above.
Left part: the number, lessened by the deficiency (96 − 4 = 92), or grown by the excess (107 + 7 = 114).
Right part: the square of the difference: 4² = 16, 7² = 49.
The right part has as many digits as the base has zeros. Fill it with a 0 in front, or carry the extra to the left part.

Worked examples

96²

96 is 4 below 100.96: −4
Left part: take the difference away from the number.96 − 4 = 92
Right part: square the difference.4² = 16

96² = 92 | 16 = 9216

107²

107 is 7 above 100.107: +7
Left part: add the difference to the number.107 + 7 = 114
Right part: square the difference.7² = 49

107² = 114 | 49 = 11449

89²

89 is 11 below 100.89: −11
Left part: take the difference away from the number.89 − 11 = 78
Right part: square the difference.11² = 121
The right part can only have 2 digits: carry 1 to the left part.78 | 121 = 79 | 21

89² = 79 | 21 = 7921

988²

988 is 12 below 1000.988: −12
Left part: take the difference away from the number.988 − 12 = 976
Right part: square the difference.12² = 144

988² = 976 | 144 = 976144

Why it works

Write the number as B + d, where B is the base and d the difference (negative below the base). Then (B + d)² = B² + 2Bd + d² = B × (B + 2d) + d². And B + 2d is the number plus its difference once more: (B + d) + d.

Watch out

The right part needs exactly as many digits as the base has zeros: 103² is 106 | 09, not 106 | 9.
Far from the base, the square of the difference carries: 89² gives 78 | 121, which is 7921.
For numbers far from any base, the duplex works for every number.

Try these

Work each one out, then tap it to see the answer.

97²
97 − 3 = 94, 3² = 9: 9409
94²
94 − 6 = 88, 6² = 36: 8836
103²
103 + 3 = 106, 3² = 9: 10609
112²
112 + 12 = 124, 12² = 144: 12544
996²
996 − 4 = 992, 4² = 16: 992016

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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.
Vertically and crosswise: 23 × 41: 2 × 4 | 2 × 1 + 3 × 4 | 3 × 1: 943.
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.