Vedic Mathematics/Techniques/Addition And Subtraction

Introduction
Other than counting, addition is perhaps the most basic mathematical operation. As such there are relatively few techniques to improve its efficiency, those techniques that exist are basically ways to organise the calculation to make it simpler to compute mentally. Subtraction is similar but in this case there are also some techniques to help with borrow process which unlike addition can run across multiple digits. You will no doubt already use most of the addition and subtraction techniques described here intuitively without thinking about them, many of the techniques may seem to be so basic that they don't deserve explanation, but it is worth re-iterating them even if just to make you aware that you are probably already using techniques that can be expanded upon to improve your arithmetical ability.

Memory versus Calculation
It is a basic fact of arithmetic that the more you memorise the less you have to calculate. The first thing you learn about numbers is how to count. You can think of this as the ability to add one to any number, e.g. 0+1=1, 4+1=5, 7+1=8, 9+1=10, 10+1=11, 19+1=20, 56+1=57. You instinctively know the answer to these sums, they have been memorised from an early age by repeatedly counting from 1 to 100.

Once you have learnt to count, you learn to add by counting, that is, repeatedly adding one (at first often using your fingers). Indeed addition is implicitly defined as the repeated addition of one e.g. $$ \begin{align} 6+3 & = 6+1+1+1 \\       & = 7+1+1 \\        & = 8+1 \\        & =9    \end{align} $$
 * Add 6 + 3

It's not long before you have memorised the results of adding smaller numbers so you no longer have to count, (on your fingers or otherwise!), to find the answer to simple additions.

This pattern repeats itself again and again. When learning a new arithmetic operation, you initially learn how to perform it, then after practising you tend to memorise the results of the operation on the most common (often smaller) numbers so you don't actually have to perform the operation to get to the answer. This is perhaps the first and most basic arithmetic technique, i.e. memorise and you don't have to calculate.

Now it is obviously impossible to memorise the results of all possible calculations, but it is clear that a certain amount of memorisation is essential, and that some additional memorisation above the minimum requirement will vastly improve the efficiency of many calculations. Where to draw the line is difficult to say, but the following would be good basic set of results to memorise, (most of us will have already memorised these results long ago).

$$   \begin{align} 2+6&=? \\     5+4&=? \\      3+4&=? \\      6+4&=? \\      etc.    \end{align} $$ $$    \begin{align} 1+?&=10 \\     2+?&=10 \\      3+?&=10 \\      4+?&=10 \\      etc.    \end{align} $$
 * All sums up to 10.
 * All compositions of 10

Just memorising the above lets you calculate any sum of two single digit numbers by partitioning and reorganising (see later), but it saves a lot of time to also memorise the sum of all pairs of single digit numbers and not just those sums up to 10. e.g. $$ \begin{align} 2+9&=? \\     8+8&=? \\      4+7&=? \\      7+6&=? \\      9+7&=? \\      etc.    \end{align} $$ Note that it is perhaps surprising how few people have memorised the above sums. You may think you have them memorised, but consider carefully; are you actually recovering the results from memory or are you working them out by partitioning and reorganisation. (e.g. Do you know that 7+6=13, or do you think "it's one less than 7+7", or "it's one more than 6+6". Do you actually know that 9+7=16, or are you thinking "9+7 is the same as 10+6".
 * All single digit sums

Addition is Commutative
One of the first things you learn about addition is that it is commutative. That is, it doesn't matter whether you add 5 to 6 or add 6 to 5, the answer is the same no matter what order the terms are in. e.g.: $$   \begin{align} 2+9  &= 9+2   &= 11 \\      4+15  &= 15+4  &=19\\      5+31  &= 31+5  &=36\\      7+53  &= 53+7  &=60\\      23+61 &= 61+23 &=84 \\      etc. \\ generally: \\ a+b &= b+a \end{align} $$

This idea is implicitly understood by everyone, (e.g. if asked to do 3+54 you would almost certainly think of 54+3 before coming up with 57), but it is important to explicitly state that addition is commutative as this rule is important in many addition algorithms. (It should also be noted that other mathematical operators are not commutative, e.g. 6-5 is not the same as 5-6.)

Addition is Associative
Another of the most basic addition laws is that addition is associative. That is, if you have a number of additions to perform it doesn't matter which order you do them in. e.g.:

$$   \begin{align} (2+9)+5 &= 2+(9+5)  &= 16 \\      (4+15)+7 &= 4+(15+7) &=26\\      etc. \\ generally: \\ (a+b)+c &= a+(b+c) \end{align} $$

The First Technique: Extended Counting
Soon after you have memorised how to count you then quickly learn how to add one to any number, e.g. you know that 8999+1=9000 even though you have almost certainly never counted from 1 to 9000! Note that you do this calculation without formally adding the digits and carrying the result. e.g. you don't do this:

$$   \begin{matrix} 8 & 9 & 9 & 9 & \\       &   & _1& 1 & +1 \\      \hline & & & 0 \end{matrix} \quad \Rightarrow \quad \begin{matrix} 8 & 9 & 9 & 9 & \\       & _1&   & 1 & +1 \\      \hline & & 0 & 0 \end{matrix} \quad \Rightarrow \quad \begin{matrix} 8 & 9 & 9 & 9 & \\     _1&   &   & 1 & +1 \\      \hline & 0 & 0 & 0 \end{matrix} \quad \Rightarrow \quad \begin{matrix} 8 & 9 & 9 & 9 & \\       &   &   & 1 & +1 \\      \hline 9 & 0 & 0 & 0 \end{matrix} $$

You instead instinctively know that when adding one to any number there are only two possible outcomes.


 * 1) If the last digit is not a nine then the answer is just the number with the last digit increased by one
 * 2) If all the consecutive last digits are nines then you replace the nines with zeroes and add 1 to the first digit (working from right to left) that isn't a nine.

Subtracting 9
Subtracting 9 is a very easy way. For example : 10-1=9 100-1=99  Here, we observe that while subtracting 1 from 0's we net the answer is only as 9's

Subtraction from a power of 10
You will see in the multiplication section the Vedic Mathematics Sutra Vertically and Crosswise is used to multiply numbers near a power of 10, (e.g. 10, 100, 1000, etc.). The first step in this technique is to subtract the numbers you are working with from the nearest power of 10. Luckily another sutra can help with this initial subtraction.

All From 9 And The Last From 10 tells us how to subtract a number from the next highest power of 10; we simply subtract each digit of the number in question from 9 apart from the last one which we subtract from 10. It's as simple as that. For example:

$$ \begin{matrix} 9 & 9 & 9 & 10 \\   8 & 6 & 7 & 5 & - \\    \downarrow & \downarrow & \downarrow & \downarrow \\ 1 & 3 & 2 & 5 \end{matrix} $$ 10000-8675=1325
 * Subtract 8675 from 10000

The number you are subtracting must have the same number of digits as the number of zeroes in the power of 10 you are subtracting from. If your number has less digits than this you must pad the number with leading zeroes. e.g.

$$ \begin{matrix} 9 & 9 & 9 & 9 & 10 \\   0 & 0 & 8 & 7 & 5 & - \\    \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 9 & 9 & 1 & 2 & 5 \end{matrix} $$ 100000-875=99125
 * Subtract 875 from 100000