Week 18: Squares Introduction

Intermediate Level • Estimated: 75 minutes

Lesson 18 of 48

Vedic Mathematics: Squares

Numbers Ending in 5 Near Base Method Duplex Method General Squaring Mental Calculation

The Power of Vedic Squaring

Welcome to Week 18 of your Vedic Mathematics journey! This week introduces one of the most impressive areas of Vedic Math: squaring numbers mentally. You'll learn techniques that make squaring faster than multiplication!

Why Learn Vedic Squaring?

Speed: Square 2-digit numbers in seconds
Accuracy: Systematic methods reduce errors
Mental Math: Impress with mental calculations

Pattern Recognition: See mathematical patterns
Foundation: Essential for square roots and cubes
Real Applications: Geometry, physics, statistics

Did You Know? The traditional method of squaring 65 would involve 65 × 65 = 4225 (multiple steps). With Vedic Math, you can calculate 65² = 4225 in one mental step using the "ending in 5" rule!

The 4 Vedic Squaring Techniques

"Yavadunam, Ekadhikena Purvena, Anurupyena"

Technique 1: Ending in 5

Numbers ending with 5

Easy

For 25², 35², 95²...

Multiply n by (n+1), append 25

Technique 2: Near Base

Numbers near 10, 100, 1000

Intermediate

For 98², 104², 997²...

Use surplus/deficit method

Technique 3: Duplex

General squaring method

Advanced

For any number

D(a) = a², D(ab) = 2ab, etc.

Technique 4: Ekadhikena

One more than previous

Intermediate

For consecutive numbers

(n+1)² = n² + 2n + 1 pattern

Technique 1: Squaring Numbers Ending in 5

Calculate: 65² Ending in 5 Method

65² = 65 × 65 = ?

Traditional Multiplication:

× 65

-----

325 (65 × 5)

3900 (65 × 60)

-----

4225

Multiple steps, carry operations!

Vedic Method (Ending in 5):

Rule: For any number ending in 5:

1. Take the number before 5 (n)

2. Multiply n by (n+1)

3. Append "25" to the result

That's it! One mental calculation.

(n5)² = n × (n+1) | 25

Where "|" means concatenation

For 65:

n = 6 (number before 5)

6 × (6+1) = 6 × 7 = 42

Append 25 → 4225

65² = 4225 ✓

More Examples:

25²

n = 2

2 × 3 = 6

Append 25 → 625

25² = 625

75²

n = 7

7 × 8 = 56

Append 25 → 5625

75² = 5625

105²

n = 10

10 × 11 = 110

Append 25 → 11025

105² = 11025

Why does this work? Algebraically: (10n+5)² = 100n² + 100n + 25 = 100n(n+1) + 25. The Vedic method is this formula simplified!

Technique 2: Near Base Method

Calculate: 98² Near Base Method

98² = 98 × 98 = ?

Near Base Method Step-by-Step:

Step 1: Identify the base

98 is close to 100

Base = 100

100

-2

Step 2: Find surplus/deficit

98 is 2 less than 100

Deficit = -2

Step 3: Apply formula

For numbers near a base:

Number² = (Number + Surplus/Deficit) | (Surplus/Deficit)²

Adjust for base place value

Step 4: Calculate for 98²

1. 98 + (-2) = 96 (left part)

2. (-2)² = 4 (right part)

3. Since base is 100 (2 zeros), right part needs 2 digits

4. Write 04 (not just 4)

5. Answer = 96 | 04 = 9604

Step 5: Verify

98² = 9604

Check: 100 × 96 = 9600, 9600 + 4 = 9604 ✓

More Near Base Examples:

Number Base Surplus/Deficit Calculation Square 104 100 +4 104+4=108, 4²=16 → 10816 10816 997 1000 -3 997-3=994, 3²=9 → 994009 994009 1006 1000 +6 1006+6=1012, 6²=36 → 1012036 1012036

Base Adjustment Rule: The right part must have the same number of digits as zeros in the base. For base 100 (2 zeros), right part needs 2 digits (04, not 4).

Technique 3: Duplex Method (General Squaring)

Calculate: 43² Duplex Method

43² = 43 × 43 = ?

Understanding Duplex

Duplex of a number is calculated as:

D(a) = a² (for single digit)
D(ab) = 2 × a × b (for two digits)
D(abc) = 2 × a × c + b² (for three digits)
And so on...

To square any number, we compute duplexes of its subsets.

Duplex Method for 43²:

Step 1: Write number and identify digits

43 has digits: 4 and 3

Step 2: Compute duplexes

We need duplexes of:

1. Single digits: D(4) and D(3)

2. Pair: D(43)

Step 3: Calculate each duplex

D(4) = 4² = 16

D(3) = 3² = 9

D(43) = 2 × 4 × 3 = 24

Step 4: Arrange results

For 2-digit number (n₁ n₂):

Square = D(n₁) | D(n₁ n₂) | D(n₂)

= D(4) | D(43) | D(3)

= 16 | 24 | 9

Step 5: Write with proper place value

16 (hundreds place)

24 (tens place) - but 24 has 2 digits!

9 (units place)

We need to handle carries...

Step 6: Handle carries

Starting from right:

9 (units) = 9, write 9, carry 0

24 (tens) + carry 0 = 24

Write 4, carry 2

16 (hundreds) + carry 2 = 18

Write 18

Result: 1849

43² = 1849

Check: 40² = 1600, 3² = 9, 2×40×3 = 240

1600 + 240 + 9 = 1849 ✓

Duplex Method Example: 27²

Digits: 2 and 7

Duplexes:

D(2) = 2² = 4

D(7) = 7² = 49

D(27) = 2 × 2 × 7 = 28

Step Duplex Value Place Value Working 1 D(7) 49 Units Write 9, carry 4 2 D(27) 28 Tens 28 + carry 4 = 32, write 2, carry 3 3 D(2) 4 Hundreds 4 + carry 3 = 7, write 7

27² = 729

Check: 20² = 400, 7² = 49, 2×20×7 = 280

400 + 280 + 49 = 729 ✓

Duplex Insight: The duplex method works for any number of digits! It's systematic and follows the algebraic expansion (a+b)² = a² + 2ab + b².

Squaring Strategy Guide

Choosing the Right Method

Number ends in 5 (25, 75, 105...): → Ending in 5 method (n×(n+1)|25)

Number near base (98, 104, 997...): → Near base method

2-digit number (43, 27, 68...): → Duplex method

Consecutive from known square: → Ekadhikena (n+1)² = n² + 2n + 1

3+ digit number (123, 4567...): → General duplex method

This Week's Mastery Goals

Master squaring numbers ending in 5
Apply near base method for numbers close to 10/100/1000
Understand duplex method for 2-digit numbers
Choose appropriate method based on number pattern
Square any 2-digit number mentally

Squaring Badge

Unlocks after mastering 3 squaring techniques

Squaring Practice Arena

Squaring Mastery Challenge

Test your squaring skills:

Ending in 5 Challenge

85² = ?

(Use ending in 5 method)

Near Base Challenge

96² = ?

(Use near base method)

Duplex Challenge

38² = ?

(Use duplex method)

Method Identification

Which method is best for each?

115²

Ending in 5

Near Base

Duplex

Your Progress: 0/4 correct

Squares Introduction Review

This week you learned:

Ending in 5 Method: n×(n+1)|25 for numbers like 25, 75, 105...
Near Base Method: For numbers close to 10, 100, 1000 (like 98, 104)
Duplex Method: General method using D(a)=a², D(ab)=2ab, etc.
Strategy Selection: How to choose the best method for each number
Pattern Recognition: Seeing mathematical patterns in squares

Squaring Foundation Built! You now have powerful tools for squaring numbers mentally. With practice, you'll be able to square 2-digit numbers faster than using a calculator!