1. If you want to multiply any number with 9, 99, 999,…just use this rule: eg. 999×343=342657 (second number -1)(subtract 9 from each of these new digits)ie. 342 and (9-3)(9-4)(9-2)
2. If you want to square a number that is closer to 10, 100, 1000, …: eg. 988^2=976144 (base number-(nearest ’0′ number – the base number))(square of the inner difference) ie. 988-(1000-988)and(1000-988)^2. I am at loss of words here, hope you get the idea.
3. If you want to square a number (that is not closer to 10, 1000,…): eg. 43^2=1849 (first digit x (the base number + second digit))(square of second digit) ie. (4 x (46))(3^2)=>1849
4. The square of any number ending in a five: (x5)^2 = [(x)(x+1)]25 Examples: 15^2 = ((1)(1+1))25 = 225 95^2 = (9(9+1))25 = 9025
5. The result of one over any number ending in a 9 can be computed in two very simple ways. I will give the easier example. 1/x9 ->
a. Take the digit x. Increment it by one.
b. Start with a 1 one on the left.
c. Multiply that number by (x+1) and write that number to the left of it.
d. Now take this new number. Multiply this number by (x+1).
e. Write the last digit of the result to the left of the result so far.
f. Any carry over, remember it.
g. Take the last digit of result entered. Multiply it with (x+1) and add the carry over remembered in step f.
h. Write the last digit of the result to the left of the result so far.
i. Any carry over, remember it.
j. Repeat the steps g, h, and i untill the result pattern starts repeating or you hit a zero as the result digit..
k. In most cases the pattern will repeat after every (x9 – 1) digits of result.
6. Here are examples, again for numbers closer to 10, 100, 1000… If you want to multiply two of these numbers, here are two cases: (Since folks are getting confused with syntax, I am just giving examples)
a. 95 x 97 = <97-(100-95)><(100-95)x(100-97)> = <92><5×3> = 9215 It is actually much easier when you write it on a piexe of paper.
b. if one of the numbers is over the ’0′ number: 94 x 104 = <104-(100-94)><(100-94)x(100-104)>. Here is where you use your intuition. The next step is (9800)-(24)=9776
This logic above cannot be used for numbers that are not close to 10, 100,1000 etc.
Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.
The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: “by one more than the one before” and “all from nine and the last from 10″. These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra “all from nine and the last from 10″. Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.
This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques – and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.
The sutra “vertically and crosswise” has many uses. One very useful application is helping children who are having trouble with their tables above 5×5. For example 7×8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.
The sutra “vertically and crosswise” is often used in long multiplication. Suppose we wish to multiply 32 by 44. We multiply vertically 2×4=8.
Then we multiply crosswise and add the two results: 3×4+4×2=20, so put down 0 and carry 2.
Finally we multiply vertically 3×4=12 and add the carried 2 =14. Result: 1,408.
We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a “3 by 2″ long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.
All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either “special”, in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.
We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the “differences” vertically 4×8=32 gives the second part of the answer.
This works equally well for numbers above the base: 105×111=11,655. Here we add the differences. For 205×211=43,255, we double the first part of the answer, because 200 is 2×100.
When the children learn about Pythagoras’s theorem in Year 9 we do not use a calculator; squaring numbers and finding square roots (to several significant figures) is all performed with relative ease and reinforces the methods that they would have recently learned.
Dr. David Gray writes:
“The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the “Western” scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages.”
Dr Gray goes on to list some of the most important developments in the history of mathematics that took place in India, summarizing the contributions of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and Maadhava. He concludes by asserting that “the role played by India in the development (of the scientific revolution in Europe) is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of its greatest contributions to world civilization.”
Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae from the Atharvaveda after assiduous research and ‘Tapas’ (austerity) for about eight years in the forests surrounding Sringeri. Obviously these formulae are not to be found in the present recensions of Atharvaveda. They were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda.
Use the formula ALL FROM 9 AND THE LAST FROM 10 to
perform instant subtractions.
For example 1000 – 357 = 643
We simply take each figure in 357 from 9 and the last figure from 10.
So the answer is 1000 – 357 = 643
And thats all there is to it!
This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.
Similarly 10,000 – 1049 = 8951
For 1000 – 83, in which we have more zeros than
figures in the numbers being subtracted, we simply
suppose 83 is 083.
So 1000 – 83 becomes 1000 – 083 = 917
Using VERTICALLY AND CROSSWISE you do
not need the multiplication tables beyond 5 X 5.
Suppose you need 8 x 7
8 is 2 below 10 and 7 is 3 below 10.
Think of it like this:
The answer is 56.
You subtract crosswise 8-3 or 7 – 2 to get 5,
the first figure of the answer.
And you multiply vertically: 2 x 3 to get 6,
the last figure of the answer.
That’s all you do:
SO far the numbers are below 10, subtract
one number’s deficiency from the other number,
and multiply the deficiencies together.
7 x 6 = 42
Here there is a carry: the 1 in the
12 goes over to make 3 into 4.
Here’s how to use VERTICALLY AND CROSSWISE
for multiplying numbers close to 100.
Suppose you want to multiply 88 by 98.
Not easy,you might think. But with
VERTICALLY AND CROSSWISE
you can give the answer immediately,
using the same method as above
Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.
You can imagine the sum set out like this:
As before the 86 comes from subtracting crosswise:
88 – 2 = 86 (or 98 – 12 = 86: you can subtract either way,
you will always get the same answer).
And the 24 in the answer is just 12 x 2: you
So 88 x 98 = 8624
Multiplying numbers just over 100.
103 x 104 = 10712
The answer is in two parts: 107 and 12,
107 is just 103 + 4 (or 104 + 3),
and 12 is just 3 x 4.
Similarly 107 x 106 = 11342
107 + 6 = 113 and 7 x 6 = 42
The easy way to add and subtract fractions.
Use VERTICALLY AND CROSSWISE
to write the answer straight down!
Multiply crosswise and add to get the top of the answer:
2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13.
The bottom of the fraction is just 3 x 5 = 15.
You multiply the bottom number together.
Subtracting is just as easy: multiply
crosswise as before, but the subtract:
A quick way to square numbers that end in 5 using
the formula BY ONE MORE THAN THE ONE BEFORE.
752 = 5625
75² means 75 x 75.
The answer is in two parts: 56 and 25.
The last part is always 25.
The first part is the first number, 7, multiplied
by the number “one more”, which is 8:
so 7 x 8 = 56
Similarly 852 = 7225 because 8 x 9 = 72.
Method for multiplying numbers where the first
figures are the same and the last figures add up to 10.
32 x 38 = 1216
Both numbers here start with 3 and the last figures (2 and add up to 10.
So we just multiply 3 by 4 (the next number up)
to get 12 for the first part of the answer.
And we multiply the last figures: 2 x 8 = 16 to
get the last part of the answer.
And 81 x 89 = 7209
We put 09 since we need two figures as in all the other examples.
An elegant way of multiplying numbers using a simple pattern
21 x 23 = 483
This is normally called long multiplication but actually
the answer can be written straight down using the
VERTICALLY AND CROSSWISEformula.
We first put, or imagine, 23 below 21:
There are 3 steps:
a) Multiply vertically on the left: 2 x 2 = 4.
This gives the first figure of the answer.
b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8
This gives the middle figure.
c) Multiply vertically on the right: 1 x 3 = 3
This gives the last figure of the answer.
And thats all there is to it.
Similarly 61 x 31 = 1891
6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1
Multiply any 2-figure numbers together by mere mental arithmetic!
If you want 21 stamps at 26 pence each you can
easily find the total price in your head.
There were no carries in the method given above.,/p>
However, there only involve one small extra step.
21 x 26 = 546
The method is the same as above
except that we get a 2-figure number, 14, in the
middle step, so the 1 is carried over to the left
(4 becomes 5).
So 21 stamps cost £5.46.