For example, the Beejank of 16 = 1 + 6 = 7 Also, the Beejank of 1152 is 1 + 1 + 5 + 2 = 9 And, the Beejank of 871 = 8 + 7 + 1 = 16 = 1 + 6 = 7 Finally, Beejank of 1567 is 1 + 5 + 6 + 7 = 19 = 1 + 9 = 10 = 1 + 0 = 1 |
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. |
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 |
For example, the Ekadhika of 17 = 17 + 1 = 18 And, the Ekadhika of 99 = 99 + 1 = 100 |
For example, the Ekayuna of 13 = 13 - 1 = 12 And, the Ekayuna of 1000 = 1000 - 1 = 999 |
For example, the Purak of 2 = 10 - 2 = 8 And, the Purak of 9 = 10 - 9 = 1 |
For example, 3 = -3, 45 = -45 And, 1429 = -1429 |
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. |
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,^{2}4,3] 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. |
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,^{1}1] = [1,16-1,1] = [1,15,1] = 251 But, [1,16,9] = [1,^{2}4,9] = [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 |
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 |
For example, 86 = 90 - 4 = 94 Again, 86 = 100 - 14 = 114 Thus, 86 = 94 = 114 = 1914 = 19914 = ... |
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. |
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 |
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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