“NIKHILUM NAVATAHA CHARMAM DASHTAHA” is used to simplify the process of multiplication. The meaning of this sutra is “all from nine and last from ten”. With the help of this sutra we get the complement of a number which is used for finding the product.

We classified the use of this sutra in the following classes-

i)when both the numbers are less than the nearest working base-

When both the numbers are less than the nearest working base, then first find the complements of multiplicand and multiplier, that should be in digits equal to multiplicand and multipliers i.e., for the number 97, compliment is 3 but it should be written in the form 0. Also put a negative sign before complement which shows that the number is how much less than the working base. Write corresponding complements against multiplicand and multipliers respectively.

Now, we would find the product of these complements, the result is right of resultant. Also, the sum of cross numbers is the left side of the resultant. In this process we should be very careful about carryovers. In the product of single digits numbers right side will with be single digit and for the product of the two digits numbers right side will with be two digits and so on, the remaining carryover will be with left side digit(by adding).

This process will be much clear in the following examples-

 Example- 1:     find the product of 91 &93.

Solution:           complements of 91 and 93 are 9 & 7 respectively (they are 9 & 7 respectively smaller than 100)

           91 -09
         * 93 -07
          _________
          84  / 63 

Now, complements are put against the numbers with negative sign. But we have to be careful here since the numbers are of two digits, so complements should also be of two digits which is made by putting zero before complement.

The product of (-09) and (-07) is 63, also cross sum is (93-09) or (91-07) equal to 84.

NOTE-The bar is put to keep both the sides separately. The importance of this bar will be understandable in the later segments.

Thus,        91*93=8463

Example- 2:     find the product of 995 and 997.

Solution:          complements of 995 & 997 are 005 & 003 respectively

                                                            995 -005

997 -003

___________

992 / 015

Complements are put against numbers with negative sign. The product of (-005) and (-003) is 15 but it should be of three digits, so we put one zero before 15. We too find the cross sum which is (997-005) or (995-003)  equal to 992.

Thus,   995*997=992015

ii)when both the numbers are greater than the nearest working base-

In those cases, in which numbers are greater than the nearest working base, everything is same but the complements of the multiplicand and multiplier are put against corresponding numbers with positive sign.

The process will be much clear with the help of the following examples-

Example-3:    find the product of 115 and 103.

Solution:         complements of 115 & 103 are 15 & 03 respectively.(they are 15 & 03 respectively greater than 100-the working base)

                                                             115  +015

103  +003

We put the complements against the numbers. The product of 015 and 003 is 045. Also, cross sum is 118.

Thus,          115*103=118045

iii)when one number is less than the working base and the other is greater than the working base-

In this case greater number(than working base) has a complement against that with positive sign and other smaller number has a complement against that with negative sign. These sign are used in cross sum and put a bar on the product of the right side. All other rules regarding digit surplus, digit deficit etc. will be exactly the same as before.

Example-4:      find 1013*986

Solution:           complements of 1013 and 986 are respectively 13 and -14 .

                                                          1013 +13

* 986 -14                      

     ____________

999  / -182

Here, left side = 986+13=999    or    1013-14=999

And                            right side =13*-14= -182

Thus,                        1013*986

= 999/-182

                                                    =999/(1000-182)

                                                    =999818

iv)when the numbers are not nearer to working base-

When the numbers are not nearer to working base 10, or the multiple of 10, then we have to take a base near the numbers but cross sum(left side of the resultant) is multiplied by the number (which is obtained by dividing assumed base by last working base). Here it is noted that this number is an integer.

      All other rules will be exactly the same as before.

     To understand the process follow the following examples-

Example-5:     find 65*57

Solution:         let the working base is 60 which is 6 times to the last working base 10. Complements of 65 & 57 are +05 & -03 respectively.

                        65   +05

                     *  57   -03

                       ___________

                        62  / -15

Left side= 62*6= 372

And right side= -15

Thus,     65*57 = 372/-15

                           =370/(20-15)

                           =370/5 = 3705

Example-6:     find 9561*8997

Solution:         let the base be 9000, which is 9 times to 1000. Complements of 9561 and 8997 are 561 and -3 respectively.

                           9561 +561

                         * 8997 -3

                         ______________

                         9558  / -1683

Left side= 9558*9= 86022

And right side= 561*(3)  = -1683

Thus, 9561*8997 = 86022/-1683

                                 =86020 / (2000- 1683)

                                = 86020 / 317   = 86020317

All the techniques that I have discussed till now have emphasized various methods of quick calculation. In this session, I shall discuss the digit-sum method. This method is not used for quick calculation, rather it can be used to verify the answers of various calculations such as multiplication, division, addition, subtraction, squares, square roots, cube roots,etc.

METHOD

At first, we have to convert any given number into a single digit by respectively adding up all digits of that number.

Example: a)The digit sum of 2467539 is-

2+4+6+7+5+3+9=36

Now, we take the number 36 and add its digits 3+6=9

Thus, the digit sum of 2467539 is 9.

b) The digit sum of 56768439 is-

5+6+7+6+8+4+3+9=48

4+8 =12

1+2 =3

Thus, the digit sum of 56768439 is 3.

A few more examples:-

From the above table we can see that the values of column 1&2 are same.

Note:if the digit sum of a number is 9, then we can eliminate the 9 straight away and the digit sum becomes 0.

examples

1)Verify whether 999816 multiplied by 727235 is 727101188760?

The digit sum of 999816 can be instantly found out by eliminating the three 9’s and the combination of 8+1. The remaining digit is 6, which is the digit sum.

The digit sum of 727235 can be found by the same way, i.e., it is 8.

When 8 is multiplied by 6, the answer is 48 and the digit sum of 48 is 3.

The digit sum of 727101188760 is 3.

Hence, the result is verified.

2) Verify whether 2308682040 divided by 36524 equals 63210?

In this case we can use the formula- Dividend=Divisor*Quotient+Remainder

The digit sum of the dividend is 6

The digit sum of divisor, quotient & remainder is 2,3&0 respectively.

Therefore, 6=2*3+0

Hence, we can assume that our answer is correct.

Though the digit sum method has many utility in our day to day life, it has a drawback. Even though the digit sum of the answer matches with that of the question, you cannot be 100% sure of its accuracy.

However, if the digit sum of the answer does not match with that of the question then you can be 100% sure that the answer is wrong.

In this session of my blog i shall explain you certain characteristics of dates and the corresponding days on which they fall. I shall also discuss a technique which will help you find the day on which any date fall for anyday.

TECHNIQUE

Before proceeding with the technique, I will give you a key of the months. This key will remain the same for any given year from 1901 to _

To apply this technique, you will have to memorize this months keys. To facilitate memorizing the key, there is a small verse:

“It’s the square of twelve

and the square of five

and the square of six

and one-four-six”

(Note: the square of 12 is 144; the square of 5 is 25; the square of 6 is 36 and then you have 146)

The verse will help you to memorize the key easily. I request you to be conversant with this key before proceeding ahead.

In this method of calculation, the final answer is obtained in the form in the form of a remainder. On the basis of this remainder we are to predict the day of the week.

The day key is very simple and needs no learning(I think). Next, we will do an overview of the steps used in predicting the day.

steps

i) Subtract the year with 1900(for e.g., if the year is 2002, then minus it from 1900, so the result will be 102). Then take the found result.

ii) Add the number of leap years from 1901.

iii) Add the month-key(for e.g., if the month is June, then add 5)

iv) Add the date.

v) Divide the total by 7.

vi) Take the remainder and verify it with the day-key

These are the six steps that are required to predict the day of the given date.

examples

(A) What day was 1st Jan, 1941?

we subtract the year from 1900 and take the result=41

we add the number of leap years from 1900 to 1941=10

we add the month key for January =1

we add the date =1

TOTAL =53

When we divide 53 by 7, we get the remainder as 4. From the day key, it can seen that the remainder 4 corresponding with Wednesday. Hence, 1st Jan, 1941 was Wednesday.

(B) What day will be 1st Mar, 2024?

we subtract the year from 1900 and take the result=124

we add to it the number of leap years from 1901 =31

we add the month key for march =4

we add the date =1

TOTAL =160

When we divide 160 by 7, we get the remainder as 6. Hence, the given day was Friday.

‘Magic Squares’ is a term given to squares which are fulfilled with consecutive integers and the total of whose rows, columns or diagonals are added up they reveal the same total. Many people have found these squares fascinating and so they have been regarded as magic.

Example-

We can verify the various totals:

Row 1: 4+3+8=15

Row 2: 9+5+1=15

Row 3: 2+7+6=15

Column 1: 4+9+2=15

Column 2: 3+5+7=15

Column 3: 8+1+6=15

Diagonal 1: 4+5+6=15

Diagonal 2: 8+5+2=15

RULES:

i) First, we place the first number 1 in the center-most square of the last column .

ii) Next we move to the south-east direction from this square

However, there is nothing in the south-east direction. So we have to write in a square farthest from it as shown below.

iii) Now from the square where we have written the number 2 we come to the south-east. However, it is not possible. So we have to write the number 3 farthest from it as shown below.

iv) Next, from 3 we have to come to the south-east direction to fill the number 4. But we realize that the square is already occupied by the number 1. So we have to write the number 4 in the west of the number 3. Refer to the diagram given below:

v) Next, from 4 we move to the south-east direction and write the number 5. From 5, move to the south-east direction and write 5 and hence 6.

vi) We have already written the number 6 in the last square. So we have to write the number 7 in the west of the number 6.

vii) From 7 we move to the south-east direction.But it is not possible.S we have to write the number 8 farthest from it as shown below:

viii) Now, we will put the number 9 in the remaining square.

You can now check it.

An example is given below which is also a magic square

I am back with my blogs. A long wait is now over. Now its time for intermediate level. I am going to show the base method of multiplication.

The Base method of multiplication is a wonderful contribution of Vedic Mathematics. The actual Sanskrit sutra as given to define this system is:

“NIKHILAM NAVATASCARAMAM DASATAH”, which means ‘all from 9 and the last from 10’.

The steps for multiplying by using this technique are as follows:

i) Find the BASE and the DIFFERENCE

ii)Number of digits on the RHS= Number of of zeroes in the base

iii) Multiply the difference on the RHS

iv) Put the Cross Answer on the LHS

Example-1:

100(Base): Multiplication of 97 & 99

   97-3

   99-1_

96|03 =9603

Example-2:

1000(Base):Multiplication of 997 & 991

997-3

991-9

988|027 =988027

We can also use this trick in the following examples:

9989*9995=99840055; 99*96=9504; 998*996=994008; 9993*9995=99880035; 995*987=982065.

This is part-1 of this technique. Part-2 will be ready soon.

Hoping to have a nice response in this post.

Today in this part, I am going to show a trick based on multiplication, which a very easy but interesting and it can help you a lot in future.

Before starting of this I want to thank all my subscribers, followers and the viewers and visitors for giving me support with my blog.

Squaring of the numbers between 50 and 60;-

51*51=2601

This trick can be performed by taking two simple steps;-

i) Add 25 to the digit in the units place and put it as the left-hand part of the answer. In this situation our answer is 25_

ii) Square the digits in the units place and put it as the right hand part of the answer. (If it is single digit then convert it into two digits; e.g., 1*1=01)

Hence, the required answer is 2601.

We can also take some more examples –

52*52=2704

53*53=2809

54*54=2916

55*55=3025

56*56=3136

57*57=3249

58*58=3364

59*59=3481

In my blog’s second part, I will introduce a new trick which would be amazing and interesting like the previous one. I hope that you all will enjoy this trick and could it in your future.

Before starting, i would like to thank to all my viewers and visitors for giving a good response to my first blog. I hope that you do the same with this one.

Multiplication of those numbers in which the sum of the one’s digits is 10 and of the same ten’s digit;-

Multiplication of 45 & 45

45*45=2025

The steps of this trick are as follows;-

i) In the number 45, separate the numbers 4 & 5

ii) We will start from the right hand side. After 4 comes 5, so will multiply 4 by 5 and write the answer as 20_

iii) Next, we multiply the last numbers, i.e., (5*5) and write down 25 to the right of 20 and complete our multiplication.

Hence, our answer is 2025.

We can take more examples like:-

15*15=225 103*107=11021

85*85=7225 72*78=5616

66*64=4224 23*27=621

51*59=3009 84*86=7224

Mathematics is an easy subject. India is a country which is known for its mathematicians. All great mathematicians starting from the great Aryabhatta to great Ramanujan were born in India. But it is a very shameful fact that the present generation of students find it “wasteful” and “fearsome” topic. In my first blog I am going to introduce some easy tricks which could be beneficial for those who have fear in it.

At first I will show you an easy process of multiplying and number with the repetition of the digit “9′.So lets start this by taking an example:-

Multiplication of 64585 with 99999:-

64585*99999=6458435415

The steps of solving this critical multiplication are as follows;-

i) We subtract 1 from 64585 and write half of the answer as 64584. Answer at this stage is 64584_

ii)Now we will be dealing with 64584. Subtract each of the digits 6,4,5,8 and 4 from 9 and write the one by one.

9-6=3;9-4=5;9-5=4;9-8=1;9-4=5

iii)The answer already obtained was 64584 and now we suffix to it the digits 3,5,4,1 and 5. The complete answer is 6458435415.

We can take various examples like 3656,5644,9821,3645,2548,etc for doing this technique.

This is an example post, originally published as part of Blogging University. Enroll in one of our ten programs, and start your blog right.

You’re going to publish a post today. Don’t worry about how your blog looks. Don’t worry if you haven’t given it a name yet, or you’re feeling overwhelmed. Just click the “New Post” button, and tell us why you’re here.

Why do this?

• Because it gives new readers context. What are you about? Why should they read your blog?
• Because it will help you focus you own ideas about your blog and what you’d like to do with it.

The post can be short or long, a personal intro to your life or a bloggy mission statement, a manifesto for the future or a simple outline of your the types of things you hope to publish.

To help you get started, here are a few questions:

• Why are you blogging publicly, rather than keeping a personal journal?
• What topics do you think you’ll write about?
• Who would you love to connect with via your blog?
• If you blog successfully throughout the next year, what would you hope to have accomplished?

You’re not locked into any of this; one of the wonderful things about blogs is how they constantly evolve as we learn, grow, and interact with one another — but it’s good to know where and why you started, and articulating your goals may just give you a few other post ideas.

Can’t think how to get started? Just write the first thing that pops into your head. Anne Lamott, author of a book on writing we love, says that you need to give yourself permission to write a “crappy first draft”. Anne makes a great point — just start writing, and worry about editing it later.

When you’re ready to publish, give your post three to five tags that describe your blog’s focus — writing, photography, fiction, parenting, food, cars, movies, sports, whatever. These tags will help others who care about your topics find you in the Reader. Make sure one of the tags is “zerotohero,” so other new bloggers can find you, too.