Class 08 GENERAL SQUARING BY VEDIC MATHEMATICS

The Duplex 

We will use the term Duplex, D, as follows: 

For 1 digit number D is its square, e.g., D (4) = 42 = 16 

 For 2 digit number D is twice the product of two digits

 e.g., D (43) = 2 x 4 x 3 = 24 

Now, find the Duplex of: 5, 23, 55, 26,90

The square of any number is just the total of its Duplexes, combined in the way we have been using for mental multiplication. 

43² = 1849.

Working from left to right there are three duplexes in 43: 

D (4), D (43) and D (3).

D (4) = 16, D (43) = 24, D (3) = 9,

Combining these three results in the usual way we get: 16 

16,24 = = 184

184,9 = 1849. 

2. 64² = 4096. D (6) = 36, D (64) = 48, D (4) = 16, 

Combining these results we get: 36 

36,48 = 408

408,16 = 4096 

Now, find the square of: 31, 14, 41, 32, 66, 81, 91, 56, 63

 Class 08 Activity – Happy Numbers

A happy number is one for which the sum of the squares of its digits ends in 1 after repeated squaring and adding as shown below. Is 13 a happy number? 

1² + 3² = 10 and then 1² + 0² = 1. 

Yes, 13 is a happy number. 

Sometimes many repetitions are necessary.

Is 44 a happy number?

4² + 4² = 3² and 3² + 2² = 13 and 1² + 3² = 10 and 1² + 0² = 1. Yes.

Is your house number a happy number?

Is your telephone number a happy number?

Is your birthdate a happy number?

Is today's date a happy number?

A happy number name, or word, is found by giving each letter of the alphabet a number, i.e.,

A B C D …... Z

1 2 3 4 …… 26

Is MATHS a happy number word?

MATHS is 13 1 20 8 19.

13² + 12²+ 202+8² + 19² = 995

 and 9² +9²+ 5² = 187 etc. 

Continue to find whether MATHS is a happy number word.

Which days of the week are happy number days? 

Which months of the year are happy number months?

Is your name a happy number word? 

Is the name of your city or town or district a happy number name?

 Class 08 Activity – Dividing Square roots

Objective:

 Dividing square roots.

Procedure: 

Let the children note the following geometrical patterns. and deduce       

                                        1 = 1²

    1 + 3 = 2²

1 + 3 + 5 = 3²

     1 + 3 + 5 + 7 = 4²

1 + 3 + 5 + 7 + 9 = 5²

  1 + 3 + 5 + 7 + 9 + 11 = 6²

     1 + 3 + 5 + 7 + 9 + 11 + 13 = 7²

This geometrical pattern verifies that the sum of the first n odd numbers is n²? 

Conversely, any square number can be decomposed into the sum of a number of odd natural numbers. If a given number can be expressed as the sum of first n natural odd numbers, then the number must be the square of the number n, which will accordingly be the square root of the given number. This fact can be used to find the square root of small numbers. 

To find the square root of any number, we subtract from it consecutively 1,3,5,7,9,11, …..

The number times we have to subtract to get 0, gives the square root of the given number. 

Here the procedure is shown for 81 and 110. 

From 81, we have to subtract the first 9 odd natural numbers, to get 0.Therefore, square root of 81 is 9. 

From 110, we have to subtract first 10 odd natural numbers and we cannot subtract the first 11 odd natural numbers. 

Thus, 110 is not a square number and its square root lies between 10 and 11.

Class 08 Activity – Square root of 2 (approximate)

 Activity – Square root of 2 (approximate)

Observations:

OA = AB (1 unit each)

2. ∆OAB is a right triangle.

OB2 = OA2 + AB2 (By Pythagoras Theorem)

OB2 = (1)2 + (1)2

OB2 = 2 units

 OB = √2 units.

3. From the number line, OC = √2 units (∵OB = OC).

4. On reading from the number line, OC = 1.4 units.

5. = 1.4 (correct to one place of decimal).