Vedic Mathematics — Foundation Material

Vedic Mathematics is a collection of mathematical techniques and calculation strategies associated with the Vedic-mathematics tradition. The methods taught here are presented as complementary techniques for mental and written calculation. The goal is not to replace standard mathematics, but to give learners additional efficient ways to think about numbers.

The 16 Sutras — a quick introduction: Commonly taught sutra names include Ekadhikena Purvena, Nikhilam Navatashcaramam Dashatah, Urdhva-Tiryagbhyam, Paravartya Yojayet, Shunyam Saamyasamuccaye and Yaavadunam. In this foundation booklet we focus mainly on practical number techniques.

1. Complements to 10, 100 and 1000: A complement tells us how much a number needs to reach a convenient base.

2. Nikhilam idea: subtract from the base. For numbers close to a base such as 10, 100 or 1000, work with the deficiency from that base.

3. Quick subtraction from a power of 10: For 1000 − 276, think of the complement of 276 to 1000.

Practice Worksheet:

1. Left-to-right addition: Instead of always adding from the units column, combine place values mentally.

2. Compensation: Round one number to a convenient value and compensate.

3. Subtraction by compensation:

Mental addition drill — Try to calculate without writing intermediate steps:

Multiplication by 11: For a two-digit number ab × 11, place the sum of the digits between them when the sum is below 10.

When the middle sum is 10 or more, handle the carry.

Multiplication by 12: A convenient mental method is ×12 = ×10 + ×2.

Pattern practice:

Base-100 method: When both numbers are close to 100, use their deviations from 100.

The last two digits are treated as the base-100 part. For mixed signs, be careful with borrowing/carrying.

General structure: For (100+a)(100+b): 10000 + 100(a+b) + ab.

Practice:

1. Numbers ending in 5: For a number n5, multiply n by n+1 and append 25.

2. Numbers near 100: Use (100+a)² = 10000 + 200a + a².

3. Numbers near 50: Use algebra: (50+a)² = 2500 + 100a + a².

Practice:

Useful divisibility tests:

• 2: last digit is even.
• 3: sum of digits is divisible by 3.
• 4: last two digits are divisible by 4.
• 5: last digit is 0 or 5.
• 6: divisible by both 2 and 3.
• 8: last three digits are divisible by 8.
• 9: sum of digits is divisible by 9.
• 10: last digit is 0.
• 11: alternating sum of digits is divisible by 11.
• 13: remove the last digit, multiply it by 4, and add to the remaining number; repeat as needed. The result is divisible by 13 exactly when the original number is.

Example: divisibility by 13:

HCF and LCM: HCF is the greatest common factor. LCM is the least common multiple. Prime factorisation is a reliable method, while Euclidean division is often efficient for HCF.

Practice:

1. Fraction to percentage: Remember common conversions: 1/2=50%, 1/4=25%, 1/5=20%, 1/10=10%, 3/4=75%.

2. Percentage as a fraction of 100:

3. Useful percentage reversals: 10% = divide by 10; 5% = half of 10%; 1% = divide by 100; 20% = one-fifth; 25% = one-fourth.

Fraction operations:

Practice:

1. Algebraic identities:

2. Factorisation: Look for common factors, difference of squares, and suitable quadratic patterns.

3. Linear equations: Keep both sides balanced while simplifying.

4. Mental expansion:

Practice:

Mixed Challenge — try without a calculator:

7-Day Practice Plan:

• Day 1: Complements and mental addition/subtraction
• Day 2: ×11, ×12 and mental multiplication
• Day 3: Multiplication near 100
• Day 4: Squaring techniques
• Day 5: Divisibility, HCF and LCM
• Day 6: Fractions, percentages and algebra
• Day 7: Mixed timed test and error analysis

Final Message: Vedic Mathematics is best learned through understanding, repetition and intelligent practice. Start with accuracy, then build speed. Always verify unfamiliar techniques against standard mathematical methods.

Prepared for Vedic Mathematics Training • Dr. Sai Kiran

To square any number from 51–59, split it into two digits: • Left digit = 5 • Right digit = the unit digit (call it n)

Then follow 3 simple steps: Step 1: Square the left digit → 5² = 25 Step 2: Add the right digit to 25 → 25 + n Step 3: Square the right digit and write it as two digits → n² (pad with a leading zero if needed). Suffix this to the result of Step 2.

Worked Examples:

Complete the Pattern:

Try this one on your own: 57² = ? 5² = 25, 25 + 7 = 32, 7² = 49 57² = 3249

When a number ends in 5, its square can be found very quickly using this simple Vedic Mathematics method.

Explanations:

Worksheet 1 — Basic Practice (Write down the squares):

Worksheet 2 — Intermediate Practice:

Worksheet 3 — Advanced Practice:

Worksheet 4 — Challenge:

Worksheet 5 — Speed Test (Try to solve these mentally):