Chapter 1

πŸ•‰οΈ Introduction to Vedic Mathematics

⚑ Can You Solve This in 3 Seconds?

98 Γ— 97 = ?

Traditional method: Multiple steps of multiplication, carry-overs, addition…

Vedic method: 98βˆ’3 = 95 | 2Γ—3 = 06 β†’ Answer: 9506 ✨

That's the magic of Vedic Mathematics! You'll learn this trick in Chapter 4.

πŸ“– What is Vedic Mathematics?

Vedic Mathematics is a collection of techniques and sutras (formulas) that simplify mathematical calculations. These methods were rediscovered from the Atharva Veda by Swami Bharati Krishna Tirthaji (1884–1960), one of India's greatest mathematicians and the Shankaracharya of Govardhan Math, Puri.

After years of deep study of ancient Sanskrit texts, Tirthaji identified 16 Sutras (formulas) and 13 Sub-sutras (corollaries) that cover virtually all of mathematics β€” from simple arithmetic to algebra, geometry, calculus, and beyond.

Swami Bharati Krishna Tirthaji wrote his findings in a book called "Vedic Mathematics", published posthumously in 1965. The book has since been translated into dozens of languages and has inspired millions of students worldwide.

πŸ“œ The 16 Sutras of Vedic Mathematics

#Sutra (Sanskrit)Meaning (English)
1Ekadhikena PurvenaBy one more than the previous one
2Nikhilam Navatashcaramam DashatahAll from 9 and the last from 10
3Urdhva-TiryagbhyamVertically and crosswise
4Paraavartya YojayetTranspose and adjust
5Shunyam SaamyasamuccayeWhen the sum is the same, that sum is zero
6Anurupye ShunyamanyatIf one is in ratio, the other is zero
7Sankalana-vyavakalanabhyamBy addition and by subtraction
8PuranapuranabhyamBy the completion or non-completion
9Chalana-KalanabhyamDifferences and similarities
10YaavadunamWhatever the extent of its deficiency
11VyashtisamanshtihPart and whole
12Shesanyankena CharamenaThe remainders by the last digit
13SopaantyadvayamantyamThe ultimate and twice the penultimate
14Ekanyunena PurvenaBy one less than the previous one
15GunitasamuchyahThe product of the sum is equal to the sum of the product
16GunakasamuchyahThe factors of the sum is equal to the sum of the factors

πŸš€ Why Learn Vedic Maths?

10–15Γ— Faster Calculations
Solve complex problems mentally in seconds that would take minutes with traditional methods.
Boosts Concentration
Mental math exercises sharpen your focus, memory, and analytical thinking.
Competitive Edge
Essential for JEE, NTSE, Olympiads, SSC, Banking exams β€” speed is everything!
Builds Confidence
When you can calculate faster than a calculator, math becomes fun, not fear.

🎯 Real-World Applications

  • Competitive Exams: JEE, NEET, NTSE, Olympiads, SSC, CAT β€” save 15–20 minutes per paper
  • Daily Life: Quick shopping calculations, tip estimation, discount computation
  • Programming: Optimize algorithms, understand number theory, bit manipulation
  • Mental Fitness: Keep your brain sharp β€” like yoga for the mind!
"Like the Sun, mathematical knowledge illuminates everything it touches." β€” Ancient Vedic Saying
Throughout this book, we'll use specific Vedic Sutras for each technique. Don't worry about memorizing all 16 sutras now β€” you'll learn them naturally as we explore each chapter!
Chapter 2

βœ–οΈ Multiplication by 11, 12, and 13

ΰ€ΰ€•ΰ€Ύΰ€§ΰ€Ώΰ€•ΰ₯‡ΰ€¨ ΰ€ͺΰ₯‚ΰ€°ΰ₯ΰ€΅ΰ₯‡ΰ€£
"Ekadhikena Purvena" β€” By One More Than the Previous One

πŸ”’ Multiplying Any Number by 11

This is one of the most elegant tricks in Vedic Mathematics. To multiply a 2-digit number AB by 11:

1
Write down the first digit A
2
Add the two digits: A + B (this goes in the middle)
3
Write down the last digit B
4
Result: A, (A+B), B. If A+B β‰₯ 10, carry 1 to A.
Take digits A, B
β†’
Calculate A+B
β†’
Write A, (A+B), B
β†’
Carry if sum β‰₯ 10
β†’
Answer! ✨

βœ… 10 Solved Examples β€” Multiply by 11

Example 1: 23 Γ— 11

Digits: 2, 3 β†’ Middle: 2+3 = 5 β†’ Answer: 253

Example 2: 45 Γ— 11

Digits: 4, 5 β†’ Middle: 4+5 = 9 β†’ Answer: 495

Example 3: 67 Γ— 11

Digits: 6, 7 β†’ Middle: 6+7 = 13 (carry 1) β†’ 6+1, 3, 7 β†’ Answer: 737

Example 4: 89 Γ— 11

Digits: 8, 9 β†’ Middle: 8+9 = 17 (carry 1) β†’ 8+1, 7, 9 β†’ Answer: 979

Example 5: 56 Γ— 11

Digits: 5, 6 β†’ Middle: 5+6 = 11 (carry 1) β†’ 5+1, 1, 6 β†’ Answer: 616

Example 6: 78 Γ— 11

Digits: 7, 8 β†’ Middle: 7+8 = 15 (carry 1) β†’ 7+1, 5, 8 β†’ Answer: 858

Example 7: 99 Γ— 11 (with carry!)

Digits: 9, 9 β†’ Middle: 9+9 = 18 (carry 1) β†’ 9+1, 8, 9 β†’ 10, 8, 9 β†’ Answer: 1089

Example 8: 85 Γ— 11

Digits: 8, 5 β†’ Middle: 8+5 = 13 (carry 1) β†’ 8+1, 3, 5 β†’ Answer: 935

Example 9: 37 Γ— 11

Digits: 3, 7 β†’ Middle: 3+7 = 10 (carry 1) β†’ 3+1, 0, 7 β†’ Answer: 407

Example 10: 94 Γ— 11

Digits: 9, 4 β†’ Middle: 9+4 = 13 (carry 1) β†’ 9+1, 3, 4 β†’ 10, 3, 4 β†’ Answer: 1034
For 3-digit numbers Γ— 11: Use the same idea but work pair by pair!
123 Γ— 11 β†’ Write: 1, (1+2), (2+3), 3 = 1, 3, 5, 3 = 1353
246 Γ— 11 β†’ Write: 2, (2+4), (4+6), 6 = 2, 6, 10, 6 β†’ carry β†’ 2706
Don't forget the carry! When the sum of two adjacent digits is 10 or more, you must carry 1 to the left. For example, in 78 Γ— 11: 7+8 = 15, write 5 and carry 1 to make 7+1 = 8. Answer: 858, not 7158!

πŸ”’ Multiplying by 12

The trick: 12 Γ— n = 10n + 2n (double and add)

1
Multiply the number by 10 (just add a zero)
2
Multiply the number by 2 (double it)
3
Add both results

Example 1: 34 Γ— 12

34 Γ— 10 = 340, 34 Γ— 2 = 68 β†’ 340 + 68 = 408

Example 2: 56 Γ— 12

56 Γ— 10 = 560, 56 Γ— 2 = 112 β†’ 560 + 112 = 672

Example 3: 78 Γ— 12

78 Γ— 10 = 780, 78 Γ— 2 = 156 β†’ 780 + 156 = 936

Example 4: 125 Γ— 12

125 Γ— 10 = 1250, 125 Γ— 2 = 250 β†’ 1250 + 250 = 1500

Example 5: 99 Γ— 12

99 Γ— 10 = 990, 99 Γ— 2 = 198 β†’ 990 + 198 = 1188

πŸ”’ Multiplying by 13

The trick: 13 Γ— n = 10n + 3n (triple and add)

Example 1: 25 Γ— 13

25 Γ— 10 = 250, 25 Γ— 3 = 75 β†’ 250 + 75 = 325

Example 2: 42 Γ— 13

42 Γ— 10 = 420, 42 Γ— 3 = 126 β†’ 420 + 126 = 546

Example 3: 67 Γ— 13

67 Γ— 10 = 670, 67 Γ— 3 = 201 β†’ 670 + 201 = 871

Example 4: 88 Γ— 13

88 Γ— 10 = 880, 88 Γ— 3 = 264 β†’ 880 + 264 = 1144

Example 5: 150 Γ— 13

150 Γ— 10 = 1500, 150 Γ— 3 = 450 β†’ 1500 + 450 = 1950
⏱️ Can you solve these in 5 seconds each?
72 Γ— 11   |   45 Γ— 12   |   30 Γ— 13
Answers: 792, 540, 390

πŸ“ Practice Problems β€” Multiply by 11, 12, 13

PRACTICE 1

36 Γ— 11 = ?

36 Γ— 11 = 3, (3+6), 6 = 3, 9, 6 = 396
PRACTICE 2

54 Γ— 11 = ?

54 Γ— 11 = 5, (5+4), 4 = 5, 9, 4 = 594
PRACTICE 3

72 Γ— 11 = ?

72 Γ— 11 = 7, (7+2), 2 = 7, 9, 2 = 792
PRACTICE 4

88 Γ— 11 = ?

88 Γ— 11 = 8, (8+8=16, carry 1), 8 β†’ 9, 6, 8 = 968
PRACTICE 5

63 Γ— 11 = ?

63 Γ— 11 = 6, (6+3), 3 = 6, 9, 3 = 693
PRACTICE 6

47 Γ— 11 = ?

47 Γ— 11 = 4, (4+7=11, carry 1), 7 β†’ 5, 1, 7 = 517
PRACTICE 7

81 Γ— 11 = ?

81 Γ— 11 = 8, (8+1), 1 = 8, 9, 1 = 891
PRACTICE 8

234 Γ— 11 = ?

234 Γ— 11 = 2, (2+3), (3+4), 4 = 2, 5, 7, 4 = 2574
PRACTICE 9

45 Γ— 12 = ?

45 Γ— 10 = 450, 45 Γ— 2 = 90 β†’ 450 + 90 = 540
PRACTICE 10

63 Γ— 12 = ?

63 Γ— 10 = 630, 63 Γ— 2 = 126 β†’ 630 + 126 = 756
PRACTICE 11

85 Γ— 12 = ?

85 Γ— 10 = 850, 85 Γ— 2 = 170 β†’ 850 + 170 = 1020
PRACTICE 12

27 Γ— 12 = ?

27 Γ— 10 = 270, 27 Γ— 2 = 54 β†’ 270 + 54 = 324
PRACTICE 13

36 Γ— 13 = ?

36 Γ— 10 = 360, 36 Γ— 3 = 108 β†’ 360 + 108 = 468
PRACTICE 14

55 Γ— 13 = ?

55 Γ— 10 = 550, 55 Γ— 3 = 165 β†’ 550 + 165 = 715
PRACTICE 15

72 Γ— 13 = ?

72 Γ— 10 = 720, 72 Γ— 3 = 216 β†’ 720 + 216 = 936
PRACTICE 16

91 Γ— 11 = ?

91 Γ— 11 = 9, (9+1=10, carry 1), 1 β†’ 10, 0, 1 = 1001
PRACTICE 17

111 Γ— 11 = ?

111 Γ— 11 = 1, (1+1), (1+1), 1 = 1, 2, 2, 1 = 1221
PRACTICE 18

48 Γ— 12 = ?

48 Γ— 10 = 480, 48 Γ— 2 = 96 β†’ 480 + 96 = 576
PRACTICE 19

75 Γ— 13 = ?

75 Γ— 10 = 750, 75 Γ— 3 = 225 β†’ 750 + 225 = 975
PRACTICE 20

96 Γ— 11 = ?

96 Γ— 11 = 9, (9+6=15, carry 1), 6 β†’ 10, 5, 6 = 1056
Chapter 3

πŸ”² Squaring Numbers Ending in 5

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"Ekadhikena Purvena" β€” By One More Than the Previous One

This is perhaps the most famous Vedic Mathematics trick. To square any number ending in 5:

1
Take the digit(s) before 5. Call it n.
2
Multiply n Γ— (n+1). This gives the left part.
3
Append 25 at the end. Done!
Formula: (n5)Β² = n Γ— (n+1) | 25
Just multiply the left digit by one more than itself, and tack on 25. It works for ANY number ending in 5!
Number ending in 5
(e.g., 75)
β†’
Take left part
n = 7
β†’
n Γ— (n+1)
7 Γ— 8 = 56
β†’
Append 25
56 | 25
β†’
Answer!
5625

βœ… 12 Solved Examples

15Β² = ?

1 Γ— 2 = 2 | 25 β†’ 225

25Β² = ?

2 Γ— 3 = 6 | 25 β†’ 625

35Β² = ?

3 Γ— 4 = 12 | 25 β†’ 1225

45Β² = ?

4 Γ— 5 = 20 | 25 β†’ 2025

55Β² = ?

5 Γ— 6 = 30 | 25 β†’ 3025

65Β² = ?

6 Γ— 7 = 42 | 25 β†’ 4225

75Β² = ?

7 Γ— 8 = 56 | 25 β†’ 5625

85Β² = ?

8 Γ— 9 = 72 | 25 β†’ 7225

95Β² = ?

9 Γ— 10 = 90 | 25 β†’ 9025

105Β² = ?

10 Γ— 11 = 110 | 25 β†’ 11025

115Β² = ?

11 Γ— 12 = 132 | 25 β†’ 13225

125Β² = ?

12 Γ— 13 = 156 | 25 β†’ 15625

⚑ Traditional vs Vedic β€” Speed Comparison

ProblemTraditional MethodVedic MethodTime Saved
25Β²25 Γ— 25 = write, multiply, carry, add β†’ ~30 sec2Γ—3 = 6 | 25 = 625 β†’ ~3 sec~90%
75Β²75 Γ— 75 = long multiplication β†’ ~45 sec7Γ—8 = 56 | 25 = 5625 β†’ ~3 sec~93%
115Β²115 Γ— 115 = very long β†’ ~90 sec11Γ—12 = 132 | 25 = 13225 β†’ ~5 sec~94%
This technique works because of algebra: (10n + 5)Β² = 100n(n+1) + 25. The Vedic mathematicians discovered this pattern thousands of years before modern algebra was formalized!
⏱️ Speed Round β€” Square these in 3 seconds each!
35Β²   |   65Β²   |   95Β²   |   145Β²
Answers: 1225, 4225, 9025, 21025

πŸ“ Practice Problems β€” Squaring Numbers Ending in 5

PRACTICE 1

15Β² = ?

1 Γ— 2 = 2, append 25 β†’ 225
PRACTICE 2

45Β² = ?

4 Γ— 5 = 20, append 25 β†’ 2025
PRACTICE 3

55Β² = ?

5 Γ— 6 = 30, append 25 β†’ 3025
PRACTICE 4

85Β² = ?

8 Γ— 9 = 72, append 25 β†’ 7225
PRACTICE 5

105Β² = ?

10 Γ— 11 = 110, append 25 β†’ 11025
PRACTICE 6

135Β² = ?

13 Γ— 14 = 182, append 25 β†’ 18225
PRACTICE 7

145Β² = ?

14 Γ— 15 = 210, append 25 β†’ 21025
PRACTICE 8

155Β² = ?

15 Γ— 16 = 240, append 25 β†’ 24025
PRACTICE 9

175Β² = ?

17 Γ— 18 = 306, append 25 β†’ 30625
PRACTICE 10

195Β² = ?

19 Γ— 20 = 380, append 25 β†’ 38025
PRACTICE 11

205Β² = ?

20 Γ— 21 = 420, append 25 β†’ 42025
PRACTICE 12

225Β² = ?

22 Γ— 23 = 506, append 25 β†’ 50625
PRACTICE 13

250Β² = ?

Treat as 25 Γ— 10, so (25)Β² Γ— 100 = 625 Γ— 100 = 62500. Or: left part 25 β†’ 25Γ—26 = 650 | 25 β†’ but that gives a 5-digit ending. Better: 25 Γ— 26 = 650, append 25 β†’ 62500. βœ…
PRACTICE 14

995Β² = ?

99 Γ— 100 = 9900, append 25 β†’ 990025
PRACTICE 15

1005Β² = ?

100 Γ— 101 = 10100, append 25 β†’ 1010025