Duplex Method (For Two-Digit Numbers)

For N = ab (where a is tens, b is units place)

N² = a²∣2ab∣b²

Example: 23²

2²∣2(2)(3)∣3²

4∣12∣9 = 529

Understanding How 4∣12∣9 = 529

We computed:

23² = 2²∣2.2.3∣3² = 4 ∣ 12 ∣ 9

Since each section represents different place values, we need to properly align them in the hundreds, tens, and units places.

Place Value Concept

Each part of the result corresponds to different place values:

• 4 (hundreds place, from 2²)
• 12 (tens place, from 2 × 2 × 3)
• 9 (units place, from 3²)

However, 12 is a two-digit number, meaning that the "1" belongs to the hundreds place, not the tens place.

Why Shift ?

• In our decimal system, each place can only hold a single digit (0-9).
• If a part of the calculation exceeds 9, it must be carried over to the next place.
• 12 in the tens place means:
• The "2" stays in the tens place.
• The "1" should be added to the hundreds place.

Applying the Shift

We initially have: 4 ∣ 12 ∣ 9

• The "9" remains in the unit place.
• The "2" stays in the tens place.
• The "1" is carried over and added to 4 (in the hundreds place).

Now:

(4+1) ∣ 2 ∣ 9

5 ∣ 2 ∣ 9=529

Thus, 23² = 529

Whenever get a two-digit number in any section, carry its leftmost digit to the next higher place.

Other Types

1. Duplex Method (General Universal Method)

For any two-digit number N = (10a+b) use:

N² = a² ∣ 2ab ∣ b²

Example: 23²

• a=2, b=3
• 2²=2² ∣2(2)(3)+3²
• 4 ∣ 12 ∣ 9
• Adjust carry → 529

    2. Base Method (For Numbers Near 10, 100, etc.)

If N is near a power of 10, use:

(N+x)²=N²+2Nx+x²

Example: 98² = (100-2)²

• N=100, x=−2
• 100²+2.10.-2+(-2)²
• 10000−400+4=9604

   3. Squaring Numbers Ending in 5

For any number ending in 5, use:

N² = (N//10) × (N//10 + 1) × 100 + 25

Example: 75²

• 7×8 = 56
• 75² = 5625

   4. Special Case for Numbers Between 50-60

If N = 50 + x, use:

N² = 50²+2.50.x+x²

N² = 2500+100x+x²

Example: 56² = (50+6)²

• 50²+2.50.6+6²
• 2500+600+36=3136

   5. Vedic Math Sutra (Ekadhikena Purvena for 9s series)

For N = 9X, use:

N² = (N−1)(N+1)+1

Example: 99² = (98+1)²

• (99-1)(99+1)+1
• 98×100+1= 9801

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