Wallis wrote in a letter in 1669: "In a dark night in bed,
without pen, ink or paper or anything equivalent, I did
by memory extract the square root of 3,00000,00000,
00000,00000,00000,00000,00000,00000 which I found
to be 1,73205,08075,68077,29353, feré, and did the next
day commit to writing." "February 18th 1670, Joannes
Georgius Pelshower giving me a visit, and desiring an
example of the like, I did that night propose to myself
in the dark without help to my memory a number in 53
places: 2468135791011121411131516182017192122
242628302325272931 of which I extracted the square
root in 27 places: 157103016871482805817152171
proxima."
John Wallis (16161703) was a brilliant mathematician and
a
contemporary of Isaac Newton. In addition to his works on pure
mathematics he also wrote on astronomy, the tides, the laws of
motion, botany, physiology, music, geology etc.
CHAPTER 5
ALL FROM 9 AND THE LAST FROM 10
This chapter shows a surprisingly easy way of multiplying numbers near a base, near different bases, or near multiples of a base, and has a considerable range. The use of negative numbers which can enormously simplify calculations is introduced, and applications in addition and subtraction are also included.
In the conventional system of mathematics a sum like 88 × 98
is considered especially difficult because of the large figures, 8 and 9.
But since the numbers 88 and 98 are close to the base of 100 we may think that there ought to be an easy way to find such a product.
In the Vedic system this kind of sum is extremely easy however.
We simply note that both numbers are close to 100, and that 88 is 12 below
100, and 98 is 2 below 100, and we just give the answer:
Example 1 The deficiencies (12 and 2) have been written above the numbers (on
the flag), the minus signs indicating that the numbers are below
100. and the 24 is simply the product of the deficiencies: 12 × 2
= 24.
Example 2 The differences from 100 are 7 and 4,
Example 3 Note the zero inserted here: the numbers being multiplied are near
to 100, so two digits are required on the right, as in the other examples. 
Example 4 
So the most efficient mental procedure is to take one number and subtract the other deficiency from it. Then multiply the deficiencies together, mentally adjusting the first part of the answer if there is a carry figure.
Find 92 × 196 by
a) thinking of 196 as1/96,
b) using proportionally and All from 9 and the Last from 10,
c) using proportionally and By One More than the One Before
Example 5 We see here that the numbers 568 and 998 are conveniently close to
1000, so we allow 3 figures on the right. The method is just the same: 
However in the case of 568 here the deficiency is not so obvious as in the
previous examples, and this is exactly where the Sutra of the present chapter
comes in.
If All From 9 and the Last From 10 is applied to the digits of 568
we get 432:
we
take the 5 from 9 and get 4,
we
take the 6 from 9 and get 3,
and
we take the 8 from 10 and get 2.
This formula gives the deficiency of any number from the next highest base.
It could have been applied in the previous examples too: for 88, in the first
example, we take the first 8 from 9 to get 1, and the last from 10 to get
2. This gives the deficiency of 12 below the base of 100. For 98 we get 02,
or just 2.
Thus All From 9 and the Last From 10 provides us also with a very effective method of subtraction from a base number:
1000  587 = 413,
10000  785 = 10000  0785 = 9215, and so on.
Also 7000  111 = 6889, the 7 is reduced to a 6 because 111 is to be taken from one of the 7 thousands, so only 6 thousands are left, and the Sutra is applied to 111 to get 889.
Thus frequent subtraction problems involving money etc. are quickly solved by this method:
for example £70.00  £1.11 = £68.89.
We may also write 7000  111 = 7 in which we put a bar (called a "vinculum") over the 111 to show that it is negative.
This subtraction method is completely general, covering all types of subtraction:
7654  1928 = 63 (since 71= 6, 69 = , 52 = 3, 48 = )
and 63 = 6/3 = 5726 (since 603 = 57, 304 = 26).
We will see subtractions like this coming up in some of the later multiplication
devices.
Example 6 58776  2 = 58774, and the formula gives the deficiencies 41224, 2
Example 7 where 851 = 857  6, and 858 = 143 × 6.
Example 8 Multiplication tables above 5 × 5 are not essential in the Vedic system. Here the deficiencies are 3 and 2 as we take 7 and 8 from 10 (...Last
from 10): 
Division by numbers near a base (above or below the base) includes the use
of the Sutra Vertically and Crosswise as well as the Sutra of the present
chapter.
It is therefore shown at the end of the next chapter.
We now extend this simple multiplication technique in several different directions.
First let us suppose that the numbers being multiplied are both above a base, rather than below it.
Example 9 This is even easier than the previous examples, but the method is just the same. The deficiencies are +3 and +4: positive now because the numbers are above the base. 103 + 4 = 107 or 104 + 3 = 107, and 4 × 3 = 12.
Example 10
Example 11
Example 12 
ONE NUMBER ABOVE AND ONE NUMBER BELOW THE BASE
Example 13 Here the base is 100 and the deficiencies from 100 are +24 and 2. Applying the usual procedure we find 124  2 = 122
Example 14 Similarly, we first get 100313 = 990 or 987+3 = 990, Here we have a minus one to carry over to the left so that the 112 is reduced by 2 altogether. 
Here we bring in the formula from Chapter 2.
Example 16 We observe here that the numbers are not near any of bases used
Example 17 The base is 30 (3×10), and the deficiencies are 1 and 2.
Example 18 Here the numbers are above and below 300: we multiply the lefthand 
Thus the Proportionately formula extends considerably the range of
the method.
The only additional step being the multiplication of the lefthand part of
the answer. One further application of this formula may also be noted:
Example 19 
Example 20 The bases here are 10,000 and 100 and the deficiencies are 2 and
6. 
Note that the number of figures in the righthand part of the answer corresponds to the base of the lower number (94 is near 100, therefore there are 2 figures on the right).
Example 21 Lining the numbers up: 10007
Example 22 Note here that because 98 = 1 the deficiency 2 is deducted from the 3 to give 1012 on the left. 
MULTIPLYING THREE NUMBERS SIMULTANEOUSLY
Example 23 These numbers are all close to 100, their deficiencies being 2, 3,
4. Then multiply the deficiencies together in pairs and add the results
up:

This is an especially easy case under the present formula, which is described by the subformula Reduce (or increase) by the Deficiency and also set up the square.
Example 25 96 is 4 below 100, so we reduce 96 by 4, which gives us the first part of the answer. The last part is just 4^{2 }= 16 as the formula says.
Example 26 Here 1006 is increased by 6 to 1012, and 6^{2} = 36: but with a base of 1000 we need 3 figures on the right, so we put 036.
Example 27 This is the same but because our base is 300 the lefthand part of
the 
Note the following alternative method: if we look at the number split so that
304^{2} = 9/24/16, then we may see that 9 = 3^{2},
24 = twice 3 × 4, and 16 = 4^{2}.
It is worth noting this case, which also comes under the above formula.
Example 28 Since 50^{2} = 2500 and we have 54^{2}, which is 4 more,
Example 29 25  2 = 23 (25, as before, minus the deficiency, 2) 
The Vedic formula By One Less Than the One Before, which is the converse of the formula of the previous chapter, comes in here in combination with All From 9 and the Last From 10.
Example 30 The number being multiplied by 9's is first reduced by 1:7631 = 762,
Example 31 Here, as 1867 has 4 figures, and 99999 has 5 figures, we suppose 1867
to be 01867. This is reduced by 1 to give 1866, 
Example 32 If asked to add 19 to a number we would probably add 20 and take
Example 33 In adding 177 we might add 200 and subtract 23. 
Example 34 Similarly here, the natural thing to do is to subtract 20 and add 1 back on.
We subtract 2000 and add 312: the All From 9... value of 688.
Example 36 Here the natural thing to do is probably to observe that we can easily
subtract 222 of the 333, leaving 111 to be subtracted from 7000. 
WRITING ADDITION AND SUBTRACTION SUMS
Earlier in this chapter we made use of negative numbers. We have written
for 48, for example. This bar on the top of a number is called a vinculum.
The use of this vinculum enormously simplifies many calculations.
Below we give, without explanations , the equivalent written forms of examples
32 to 36.
32. 7 7 33. 3 6 5 34.
7 7 35. 5 0 4 0 36.
7 2 2 2
2 +
2 +
2 
2 
3
3 3 
9 6 5
4 2 5
8 3
3 5 2 7
6
8 8 9
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