Vedic Math - Squaring of numbers near '50'

In this article, we pick another special case of squaring i.e. squaring numbers which are near 50. It can have two cases, which are:

• Case 1: Numbers greater than 50.
• Case 2: Numbers lesser than 50

In both the cases, we need to take 50 as the base value. First let us take 'Case 1' i.e. 'Numbers greater than 50'.

Case 1 -
Take an example: Say, 54²=2916 
Consider this answer in two parts: 29 (first part) and 16(second part). Now let us study, how Vedic Math can help us to achieve this answers or both of these parts.

As we are taking '50' as base, so the number presentation will be like 50 + 4. So the first part is 50, and second part is 4.

For the first part of the answer:

1. Pick the first part i.e. 50.
2. Pick the first digit i.e. 5
3. Square this digit i.e. 5² > 25
4. Add 4(second part) to it
5. And we get our first part of the answer i.e. 29 (25 + 4).

For the second part of answer, following are the steps:

1. Pick second part i.e. 4
2. Square this digit i.e. 4² > 16
3. And we get our second part of the answer i.e. 16

So the answer is 2916

Let us understand it with another example to make it more clear. Say, 61²=3721
As we are taking '50' as base, so the number presentation will be like 50 + 11. So the first part is 50, and second part is 11.

For the first part of the answer:

1. Pick the first part i.e. 50.
2. Pick the first digit i.e. 5
3. Square this digit i.e. 5² > 25
4. Add 11(second part) to it
5. And we get our first part i.e. 36 (25 + 11).

For the second part of answer, following are the steps:

1. Pick second part i.e. 11
2. Square this digit i.e. 11² > 121
3. Now the result is of three digit. So the first digit (1) will be added to the first part i.e. 1 + 36 = 37
4. So first part becomes 37 now
5. And second part of answer will be 21 

So the answer is 3721. Refer to image below for visual representation.

Now, let us study 'Case 2' i.e. Numbers lesser than 50.

Case 2 -
Take an example: Say, 48²=2304
As we are taking '50' as base, so the number presentation will be like 50 - 2. So the first part is 50, and second part is (-2). For numbers below 50, we take the deficiency from 50 (2 in this case), to get the number (48 in this case); and use the square of the deficiency (2²= 4 in this case) for calculation.

For the first part of the answer:

1. Pick the first part i.e. 50.
2. Pick the first digit i.e. 5
3. Square this digit i.e. 5² > 25
4. Add (-2) (i.e. second part) to'25'
5. And we get our first part i.e. 23 (25 + (-2)).

For the second part of answer, following are the steps:

1. Pick second part i.e. (-2)
2. Square this digit i.e. (-2)²  =  4
3. And second part of answer will be 04

So the answer is 2304. Refer to image below for visual representation.

In the second case, we explore the sub-sutra "Whatever the deficiency lessen by that amount and set up the square of the deficiency"

The Algebra behind this method is:

• (50 + a)² = 100 (25 + a) + a²   (if number is above 50)
• (50 - a)² = 100 (25 - a) + a²    (if number is below 50)

Here are few exercises for your practice:

43² = ?

49² = ?

56² = ?

52² = ?

62² = ?


Try it. I hope by now you would have understood the method. Even then if you have any difficulty, post your doubts here. Enjoy!!

Please do share your views that would be having great value for us and will encourage us.

Vedic Math - Squaring Of Numbers Ending with '5'

In this article, we shall discuss a very common and interesting trick to square those numbers quickly which are having '5' as last digit. For example, what is the result of 65², 85², 125² ?

Let us start with an example:- 35 x 35. How will you multiply?

The conventional approach is-

     35
   x 35
   -------
    175
   105
 --------
   1225
 --------
 In above problem, we followed the following steps:

1. In first step, we multiply 5 by 35, get 175 and wrote it below the line.
2. In second step, we multiply 3 by 35, get 105, wrote it below the first step and leave one space from right.
3. In last, we add results from both the steps and get 1225 as answer.

Now here is the magical trick or quicker way to do this calculation using Vedic Math (to square any number with a 5 on the end). Let us have a look on the same example once again, following 'Vedic Math' steps to solve it.

1. In 35, the last digit is 5 and other number is 3.
2. Add 1 to the top left digit 3 to make it 4 (i.e. 3+1=4) (See the image below).
3. Then multiply original number '3' with increased number i.e. '4'. Like 3 x 4, and we get 12.
4. Now you can see that this is the left hand side of the answer.
5. Next, we multiply the last digits, i.e 5 x 5 and write down 25 to the right of 12.
6. And here we come up with a desired answer, 1225
7. Visual representation is given below.

 Let us do one more exercise. Find the square of 105:
 
    105
  x 105
   -------
   11025
   -------

I'm going to explain the magical trick method once more.

1. In 105, the number apart from 5 the digits are 1 and 0, that is, 10
2. Add 1 to the top left digit 10 to make it 11(i.e. 10+1=11).
3. Multiply the original left hand side number with its successor number i.e. 10x11, and we get 110.
4. Write this on the left hand side.
5. Multiply the last digits, i.e 5 x 5 and write down 25 to the right of 110.
6. And here is the answer > 11025

So you can see how easy it is to square the numbers which are ending with a 'five' (5) digit! In fact, if you memorize this technique and practice it, eventually you would be able to perform these calculations verbally (in mind). Just multiply the non-five number (left side number) with its successor number and put outcome on the left side. Multiply the last digits (5 x 5) and put 25 on right side of the previous multiplication.

A few more examples/exercises are given below:

25² = 625

45² = 2025

65² = 4225

95² = 9025

115² = 13225  (11 x 12 = 132)

You try the above examples and I'm sure that you will solve these problems fast or might be orally.

The technique of squaring numbers ending with 5 is a very popular technique. With this technique, you have used the formula (sutra) "By One More Than the One Before" , which provides a beautiful and simple way of squaring numbers that end by 5.

The ALGEBRA behind this method is (ax+5)²= a(a+1)x² + 25, where x=10. 

Now here the question arises in your mind that: Is this formula applicable to a number that ends with 5 only?
Answer: It is not applicable to all kind of multiplications. But you will be happy to know that the above formula can be applied to the multiplication of numbers whose last digits add to 10 and the remaining first digits are the same. But remember, same rule is not applied on the vice-versa i.e. if last figures are the same and the first figures add up to 10. See the image below.



So same rule is applicable for case 1, but not for case 2.

We will take one more example of the same kind.
     6 9
   x 6 1
    --------
    42 09
    --------
Here we can see that right digits sum is 10 i.e.(9+1) and left side digits are same. So we can now apply the same method.

1. First, multiply the right side numbers(1 x 9) and the result is 09.
2. Second, multiply 6 by the number that follows it, i.e.7, so the result of (6 x 7) is 42.
3. And now the final output is 4209.

Now, following are some of the problems for your practice.

1. 76 x 74
2. 33 x 37
3. 91 x 99
4. 85 x 85
5. 55 x 55

So this is all for today. Hope you have enjoyed the 'Vedic Math' tricks. We shall come up with more tricks soon.

What is Vedic Math?

We are going to start a new and very interesting section and that is 'VEDIC MATH'. Many of us are interested in increasing our productivity with calculations. This is where ‘Vedic Math’ helps us. It teaches us many ways to do the calculations quickly and if practiced correctly then all the calculations can be done in mind. Hence it helps us not only in our work, but routine works also. Vedic Math is also very useful for students to get rid of math phobia and improve grades. With these techniques one could be able to solve the mathematical problems 15 times faster. It improves mental calculations, concentration and confidence. Isn’t this great!

Once you are aware of the basics of Vedic Math, you can practice and make yourself a human calculator. Vedic Mathematics is magical. Let us take a simple example of multiplication to feel what Vedic Math is and what it can do.

So, let’s try 14 times 11.

•     Split the 14 apart, like:
•     1    4
•     Add these two digits together
•     1 + 4 = 5
•     Place the result, 5 in between the 14 to have 154
•     And the result is
•     14 X 11 = 154

This is a very basic example to show the magical power of Vedic Math. Once you learned all the techniques, you will be able to do various complex calculations very fast as mentioned above. Before we proceed towards the different techniques of Vedic mathematics in detail, we first give you brief background of Vedic Mathematics history.

'Vedic Mathematics' is the name given to the ancient system of mathematics derived from ancient treasure of knowledge called ‘Veda’. ‘Veda’ means knowledge. Vedic Mathematics believes to be a part of ‘Atharva Veda’. It a unique technique of calculations based on simple rules and principles, using which any mathematical problem related to arithmetic, algebra, geometry or trigonometry can be solved quickly and possibly orally (once you master it).

Vedic Mathematics was devised probably thousands of years back; however it was rediscovered again from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras or word-formulae. For example, 'Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in solving the problems.

Following are 16 Sutras and 14 Sub-Sutras:

Sutras (Formulae)

• By one more than the one before
• All from 9 and the last from 10
• Vertically and crosswise
• Transpose and apply
• If the Samuccaya (i.e. both sides of the equation) is the same it is zero
• If one is in ratio the other is zero
• By addition and by subtraction
• By the completion or non-completion
• Differential calculus
• By the deficiency
• Specific and general
• The remainders by the last digit
• The ultimate and twice the penultimate
• By one less than the one before
• The product of the sum
• All the multipliers

    Sub-Sutras (Sub-Formulae)

• Proportionately
• The remainder remains constant
• The first by the first and the last by the last
• For 7 the multiplicand is 143
• By osculation
• Lessen by the deficiency
• Whatever the deficiency lessen by that amount and set up the square of the deficiency
• Last totaling 10
• Only the last terms
• The sum of the products
• By alternative elimination and retention
• By mere observation
• The product of the sum is the sum of the products
• On the flag

-Excerpt from Sri Bharati Krsna Tirthaji's "Vedic Mathematics"

We will discuss these sutras and subsutras in greater detail later. First we will go with some of the tips and tricks on addition, subtraction, multiplication and division. And side by side, you realize that eventually we shall come across these sutras in different aspects of mathematical calculations, as all those calculations utilize these sutras (formulae) in one form or other. Then you shall also learn and understand the meaning of these sutras.

A simple example,

If we wished to subtract 378 from 1,000; we simply apply the sutra "all from nine and the last from 10". Each figure in 378 is subtracted from 9 and the last figure is subtracted from 10, yielding 622.

1000  -  378      =    622

1000  -     3                  7                8
            subtract    subtract    subtract
            from 9       from 9       from 10
               |                |                  |
               6               2                 2   

Vedic Mathematics is quite simple in calculations, and that means you can do simple to complex calculations orally. Imagine how much value that can add to different procedures in day to day life. Further, this whole system was devised considering utmost flexibility and extensibility. Pupils were encouraged to learn from practiced techniques and devised their new techniques. It is quite beautiful approach to mathematics. However, you can appreciate this beauty only by practicing it yourselves.

So, we shall start learning more about Vedic Math with coming articles. I hope you had a great time while reading brief history and few basic concept of Vedic Math. And wish, you will enjoy more with coming sessions. Then the phobia of math will disappear as you move forward with reading and eventually it fills you with lots of confidence. I’ll come up with more and more information soon.

Do you know:- 
What is the name of Veda, from which Vedic Mathematics comes from?