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# References for Sūtras and Upasūtras

## Introducing commonly used terms in Vedic Mathematics.

This section is a reference to commonly used terms, with explanations and examples - with respect to Vedic Mathematics, and its usage.

Beejank
Also known as Digit-Sum, it is the sum of all the digits of a number, and the process of continued till a single digit is reached. That is, if the sum of digits has two or more digits, then it is again summed till we get a single-digit value.
 For example, the Beejank of 16 = 1 + 6 = 7Also, the Beejank of 1152 is 1 + 1 + 5 + 2 = 9And, the Beejank of 871 = 8 + 7 + 1 = 16 = 1 + 6 = 7Finally, Beejank of 1567 is 1 + 5 + 6 + 7 = 19 = 1 + 9 = 10 = 1 + 0 = 1
It should be noted that the digit 9 passes no value to the Beejank:
 If we take two numbers, 2798 and 278 Beejank of 2798 is 2 + 7 + 9 + 8 = 26 = 2 + 6 = 8 And, Beejank of 278 is 2 + 7 + 8 = 17 = 1 + 7 = 8 We get the same Beejanks.  Similarly, for 13 and 22 (that is, 13 + 9) Beejank of 13 is 1 + 3 = 4 And, Beejank of 22 is 2 + 2 = 4 We get the same Beejanks.  As such, when 9 is added (or subtracted) from a number - the Beejank remains the same.
For this reason, while calculating Beejanks, the digit 9, and digits that sum up to 9 may be ommitted:
 Beejank of 139 is 1 + 3 = 4 (Because 9 may be omitted)  Similarly, Beejank of 13761 is 1 + 7 + 1 = 9 (Because 3 and 6 may be omitted, as 3 + 6 = 9)  If we get a negative Beejank, which is possible in calculations, it is converted into a positive Beejank by adding 9. For example, If Beejank is -7, then it is -7 + 9 = 2

It is the one more of a number, or the next number.
 For example, the Ekadhika of 17 = 17 + 1 = 18And, the Ekadhika of 99 = 99 + 1 = 100

Ekayuna (ekanyūna)
It is the one less of a number, or the previous number.
 For example, the Ekayuna of 13 = 13 - 1 = 12And, the Ekayuna of 1000 = 1000 - 1 = 999

Purak (pūrak)
It is the complemantary number with reference to 10 (usually, unless explicitly specified otherwise).
 For example, the Purak of 2 = 10 - 2 = 8And, the Purak of 9 = 10 - 9 = 1

Rekhank
A number with bar (at the top) is used to denote a negative number.
 For example, 3 = -3, 45 = -45 And, 1429 = -1429

Shuddhikaran (śūddhikaran)
It is important to note that the numbers are formatted with a ',' (coma) in conventional method. This means that 1,000 simply means 1000. In Vedic Mathematics, since the operations are primarily on digits, it is conventional that a format be used to separate the digits.
 For example: [1,4,2] simply means 142, in a Base10 Number Format - the digits being separated by a coma, for clarity.  Many 'schools of thoughts' use a Divider, like 1/4/2 or 1|4|2 - although, it means the same, we will be using a coma throughout this website.
During computations, it is possible that a number is derived as [1,24,3]. Such numbers are considered aśūddha or impure - because only one digit should take the position of a placeholder. Such numbers are processed using śūddhikaran by retaining the last digit, and carrying forward the additional part.
 For example: So, [1,24,3] = [1+2,4,3] = 343  The simplest way to visualize this would be, to note that [1,24,3] implies 1 in Hundred's Place, 24 is Ten's Place and 3 in One's Place. Which is, (1 × 100) + (24 × 10) + (3 × 1) = 100 + 240 + 3 = 343 It is often denoted as [1,24,3] to mark the carry-over part.  It should be noted here that the coma separates the digits, that is the Unit's place from the Ten's place, the Ten's place from the Thousand's place, and so on. And, the number is put into a bracket to imply that the number is aśūddha.
In the event that a negative digit is derived, it is know as a mishrank or a vinculum number, discussed below.
 For example: So, [1,16,-9] = [1,16,9]  Such numbers are 'purified' with its complement from (m × 10) and m is carried forward. As such, [1,16,9] = [1,16,11] = [1,16-1,1] = [1,15,1] = 251 But, [1,16,9] = [1,24,9] = [1-2,4,9] = [1,4,9] = -51  Again, the simplest way to visualize this would be, to note that [1,16,9] implies 1 in Hundred's Place, 16 in Ten's Place and -9 in One's Place. Which is, (1 × 100) + (16 × 10) + (-9 × 1) = 100 + 160 - 9 = 251 Similarly, [1,16,9] = (1 × 100) + (-16 × 10) + (9 × 1) = 100 - 160 + 9 = -51

Mishrank, or Vinculum Numbers
Vinculum numbers are representative numbers to denote a number in a manner that it contains both positive and negative digits. The vinculum numbers are denoted by Rekhank, and uses the concept of Purak. This is the reason that Rekhank is associated with Purak.
 For example, 8 = 10 - 2 = 12 Also, 97 = 100 - 3 = 103 And, 289 = 300 - 11 = 311  Note, that the negative digits have a bar above them (Rekhank), but the conventional positional notation is to be maintained. This means, that there cannot be more than two distinct positive and negative parts: For example, 289 = 304 - 15 has to be converted to 300 + 4 - 15= 300 - 11= 311
Now, there are more than one ways to represent a number. This essentially means that there are more than one Vinculum number to represent the same number:
 For example, 86 = 90 - 4 = 94 Again, 86 = 100 - 14 = 114 Thus, 86 = 94 = 114 = 1914 = 19914 = ...
Also, negative numbers can be represented using the Vinculum representation:
 For example, 7 = -7 = -10 + 3 = 13  Also, 36 = -36 = -100 + 64 = 164  And, 978 = -978 = -1000 + 22 = 1022  Note, that by mere observing the vinculum number one can conclude if the number (in its general form) is negative or postive. A bar to the left means representation of a negative number, and a bar to the right means a representation of a positive number.
Vinculum numbers can be as easily converted back to its general (normalized) form, as easily it was derived - by merely following the positional notation:
 For example, 12 = 1 x 10 + (-2) x 1 = 10 - 2 = 8  And, 311 = 3 x 100 + (-1) x 10 + (-1) x 1 = 300 - 10 - 1 = 300 - 11 = 289  For negative numbers, 1022 = (-1) x 1000 + 0 x 100 + 2 x 10 + 2 x 1 = -1000 + 0 + 20 + 2 = -1000 + 22 = -978
Please find below examples of operations using Vinculum numbers by applying Vedic Mathematics, primarily for academic reasons:
 Conversion of general (normalized) number to Vinculum: Using Sūtra 7: saṅkalana vyavakalanābhyāṃ (By addition, and by subtraction): Steps: 1. Add the number to itself; 2. Use a digit-wise subtraction.  For example, Vinculum number for 8 = (8 + 8) - 8 (digit-wise) = 16 - 8 (digit-wise)   16 -  8 -----   12 =[(1 - 0),(6 - 8)] = [1,-2] = 12  Again, Vinculum number for 315 = (315 + 315) - 315 (digit-wise) = 630 - 315 (digit-wise)   630 - 315 ------   325 =[(6 - 3),(3 - 1),(0 - 5)] = [3,2,-5] = 325  Vinculum number of a negative number -978 = -[(978 + 978) - 978 (digit-wise)] = -[1956 - 978 (digit-wise)]   1956 -  978 -------   1022 = -[(1 - 0),(9 - 9),(5 - 7),(6 - 8)] = -[1,0,-2,-2] = -1,0,2,2 (Since, -0 = 0) = 1022  Addition/Subtraction of Vinculum numbers: Let us take two numbers: 7 = 13, and 6 = 14   13 + 14 -----   27 =[(1 + 1),(-3 + -4)] = [2,-7] = 27 = 20 - 7 = 13  Subtraction can similarly done, by adding a negative number.

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