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परिकर्माष्टके वर्गमूलम् – Finding Square-root
December 8, 2015 Square-root
परिकर्माष्टके वर्गमूलम्
अथ वर्ग-मूलम् | वर्ग-मूले करण-सूत्रम्
त्यक्त्वान्त्याद्विषमात् कृतिं द्विगुणयेन्मूलं समे तद्धृते |
त्यक्त्वा लब्धकृतिंस्तदाद्यविषमाल्लब्धं द्विनिघ्नं न्यसेत् |
पङ्क्त्यां पङ्क्तिहृते समेऽन्यविषमात्त्यक्त्वाप्तवर्गं फलम् |
पङ्क्त्यां तद्द्विगुणं न्यसेदिति मुहुः पङ्क्तेर्दलं स्यात्पदम् ||२२||
पदच्छेदैः –
त्यक्त्वा अन्त्यात् विषमात् कृतिम् द्वि-गुणयेत् मूलम् समे तद्-हृते |
त्यक्त्वा लब्ध-कृतिम् तद्-आद्य-विषमात् लब्धम् द्वि-निघ्नम् न्यसेत् |
पङ्क्त्याम् पङ्क्ति-हृते समे अन्य-विषमात् त्यक्त्वा आप्त-वर्गम् फलम् |
पङ्क्त्याम् तत् द्वि-गुणम् न्यसेत् इति मुहुः पङ्क्तेः दलम् स्यात् पदम् ||२२||
From the point of view of appreciation of poetic elements,
• Because the procedure is somewhat longish, भास्कराचार्य has chosenशार्दूलविक्रीडितवृत्तम्, which provides for 19 letters in each line.
• One may note that starting word in lines 1 and 2 is त्यक्त्वा and starting word of lines 3 and 4 is पङ्क्त्याम्. That is some help in committing the verse to memory.
अन्वयाः –
1. अन्त्यात् विषमात् कृतिम् त्यक्त्वा
2. मूलम् द्वि-गुणयेत्
3. तद् समे हृते
4. लब्ध-कृतिम् आद्य-विषमात् त्यक्त्वा
5. तद् लब्धम् द्वि-निघ्नम् पङ्क्त्याम् न्यसेत्
6. पङ्क्ति-हृते समे
7. आप्त-वर्गम् फलम् अन्य-विषमात् त्यक्त्वा
8. तत् द्वि-गुणम् पङ्क्त्याम् न्यसेत्
9. इति मुहुः
10. पङ्क्तेः दलम् पदम् स्यात्
Some observations –
• Although from the अन्वयाः, one may look at it as a 10-step procedure, the mention इति मुहुः (like this again) suggests iteration. Number of iterations will depend upon the number of digits in the given number.
• There are many words, which are used repetitively. They are mathematical terminologies.
o The word त्यक्त्वा appears 3 times in steps 1, 4 and 7, In the first step, it means 'separating'. In steps 4 and 7, it means 'subtracting'.
o The word कृति in steps 1 and 4 implies specific operation.
o The phrases द्वि-गुणयेत् in step 2, द्वि-निघ्नम् in step 5 and द्वि-गुणम् in step 8 all mean multiplication by two.
o Correspondingly the word दलम् in step 10 means half.
o The phrase पङ्क्त्याम् न्यसेत् (= put it into पङ्क्ति) is both at step 5 and also in step 8. This word पङ्क्ति (= a row, a line) is also there in Steps 6 and 10. The word पङ्क्ति has a specific meaning. Here in the context of the procedure for finding square-root it is "divider-line". It is not one particular line. The divider keeps developing afresh step by step.
o There is also the word विषमात् in steps 1, 4 and 7. When in the given number, pairs are identified leftwards, taking digits of unit's place and tenth's place as one pair and so on, each pair has a विषम (= odd) and सम (= even), One on the right is विषम (= odd) and one on the left is सम (= even), because one is always reading leftward.
The phrase अन्त्यात् विषमात् in step 1 refers to the digit or pair at the left-most end after pairing all other digits. If it is only one digit, it is विषम
The phrase आद्य-विषमात् in step 4 refers to the same अन्त्य-विषम referred to in step 1.
The phrase अन्य-विषमात् in step 7 refers to the next pair.
Keeping these observations in mind, one can attempt interpretation of the अन्वयाः and the श्लोक –
1. अन्त्यात् विषमात् कृतिम् त्यक्त्वा –
1. त्यक्त्वा = separating
2. अन्त्यात् विषमात् = from the leftmost odd digit (or pair)
3. कृतिम् = largest deductible square. Note, that this is interpretation and not just the literal meaning of कृति
2. मूलम् द्वि-गुणयेत् –
1. मूलम् = square-root of the largest deductible square
2. द्वि-गुणयेत् = be doubled
3. तद् समे हृते –
1. तद् = square-root of the largest deductible square
2. समे हृते = multiplied by equal number, i.e. squared
4. लब्ध-कृतिम् आद्य-विषमात् त्यक्त्वा –
1. त्यक्त्वा = subtracting
2. लब्ध-कृतिम् = result of the operation (the largest deductible square)
1. आद्य-विषमात् = from first odd (pair or number)
1. तद् लब्धम् द्वि-निघ्नम् पङ्क्त्याम् न्यसेत् –
1. तद् लब्धम् = the square-root
2. द्वि-निघ्नम् = doubled
3. पङ्क्त्याम् न्यसेत् = be placed in a divider line
2. पङ्क्ति-हृते समे
3. आप्त-वर्गम् फलम् अन्य-विषमात् त्यक्त्वा
4. तत् द्वि-गुणम् पङ्क्त्याम् न्यसेत् –
1. तत् द्वि-गुणम् = that doubled
2. पङ्क्त्याम् न्यसेत् = be placed in a divider line
5. इति मुहुः = like this, again (till the end)
6. पङ्क्तेः दलम् पदम् स्यात् –
1. पङ्क्तेः दलम् = half of divider line
2. पदम् स्यात् = is the answer
Let us take an example problem to find square root of 18769
Step 1 = त्यक्त्वा to separate अन्त्य विषम last odd, we have left-ward विषम-सम pairs from unit's place as विषम 9 सम 6 विषम 7 सम 8 and are left with left-most विषम 1
Step 2 The largest deductible square for left-most विषम 1 is 1 only.
Step 3 The मूलम् is 1. Its square is 1.
Step 4. त्यक्त्वा by deducting from आद्यविषमात् i.e. from 1, this कृतिम् operation, i.e. square of 1 is 1,
Step 5 Remainder लब्धकृति (1 – 1 =) 0
Step 6 तद् लब्धम् the square-root i.e. 1 द्वि-निघ्नम् doubled, i.e. 2 पङ्क्त्याम् न्यसेत् be put in a divider line.
Step 7 Now we have to proceed with अन्य विषम the next pair i.e. 87 ahead of the previous remainder 0. So we have the dividend 087. And on the divider line पङ्क्त्याम् we have 2. We have to add a digit ahead of this 2 on the divider line, such that the product of the digit with the new number on the divider line will be less than the dividend 087. One can judge that 24 x 4 = 96, which is more than 087. But 23 x 3 = 69 is less than 087. On subtraction we have the remainder 18.
Step 8 We add 3 to 23, which gets us 26 on yet another divider line. This 26 is same as putting double of 3 to the first number 2 on the first divider line.
Step 9. We have to now again इति मुहुः get the remaining अन्य विषम the next pair i.e. 69 ahead of the remainder 18. So we have the dividend 1869. And we have 26 on the divider line. We can judge that a digit 268 x 8 = 2144 is much larger than 1869. But 267 x 7 = 1869.
Step 10 The remainder now is 0. We have to add 7 to 267 on the previous divider line. Result is 274. Half दलम् of 274 is 137, which becomes स्यात् the answer पदम्.
May be, that the following will be easier to understand –
Number of which square-root is required → 18769
Separating by pairing 1 87 69
अन्त्य विषम = 1
Largest deductible square = 1
Remainder = 0
मूलम् of Largest deductible square = 1
पङ्क्ति (2×1 =) 2_ Dividend 0 87
By judgement 23 Subtract (23×3=) 69
Remainder 18
Next पङ्क्ति (23+3=) 26_ Dividend 18 69
By judgement 267 Subtract (267×7=) 1869
Remainder = 0
Next पङ्क्ति (267+7=) 274
Answer = 274 / 2 = 137
Let us take yet another example, 915836
Here अन्त्य विषम.is a two-digit number 91.
Largest deductible square is 81.
Its मूलम् is 9.
Remainder = 10
First पङ्क्ति double of 9, hence 18 _
Ahead of the remainder 10, we put अन्य विषम 58. The dividend is 1058.
We can judge that 186 x 6 = 1116, while 185 x 5 = 925. So we work with 925.
Remainder is 1058 – 925 = 133.
New पङ्क्ति 185 + 5 = 190 _
Ahead of the remainder 133, we put अन्य विषम 36. So dividend is 13336.
We have पङ्क्ति 190_.
We can judge that 1907 x 7 = 13349. We work with 1906 x 6 = 11436.
Remainder is 13336 – 11436 = 1900.
New पङ्क्ति 1906 + 6 = 1912_
We have no more अन्य विषम left in the given number but remainder is also not 0, So square-root of 915836 is somewhat more than half of 1912 (1912/2 = 956). We can check that square of 957 (957^2 = 915849).
Actually we can continue इति मुहुः by taking अन्य विषम as 00. With '00' added to previous remainder 1900, next dividend becomes 190000. Next divider is 1912_.
We can judge that 19129 x 9 = 172161. It may be noted that we cannot put in the divider-line a digit larger than 9.
Remainder (190000 – 172161 =) 17839. New divider line is 19129 + 9 = 19138. Half of this is 956.9. Note, that because the square-root of 915836 is >956, but <957, we have to put a decimal point after 956.
We can continue इति मुहुः to whatever level of accuracy we desire. Since 915836 is not a perfect square, the remainder may not become zero to any place after the decimal point. Presently we know that square-root of 915836 is > 956.9.
There is one उद्देशक-श्लोक.
अत्र उद्देशकः in उपजातिवृत्तम् –
मूलञ्चतुर्णाञ्च तथा नवानां पूर्वं कृतानाञ्च सखे कृतीनाम् |
पृथक् पृथक् वर्ग-पदानि विद्धि बुद्धेर्विवृद्धिर्यदि तेऽत्र जाता ||२३||
पदच्छेदैः
मूलम् चतुर्णाम् च तथा नवानाम् पूर्वम् कृतानाम् च सखे कृतीनाम् |
पृथक् पृथक् वर्ग-पदानि विद्धि बुद्धेः विवृद्धिः यदि ते अत्र जाता ||२३||
अन्वयेन –
सखे यदि ते अत्र बुद्धेः विवृद्धिः जाता चतुर्णाम् तथा नवानाम् च मूलम् पूर्वम् कृतानाम् कृतीनाम् च वर्ग-पदानि पृथक् पृथक् विद्धि |
• सखे = Dear
• यदि ते अत्र बुद्धेः विवृद्धिः जाता = If your intellect is enhanced
• चतुर्णाम् (मूलम्) तथा = know square-roots of 4
• नवानाम् च मूलम् = square-root of 9
• पूर्वम् वर्ग-पदानि कृतानाम् कृतीनाम् च (मूलम्) = square-roots of numbers which were made earlier as squares
• पृथक् पृथक् विद्धि = know them separately.
न्यासः ४/ ९/ ८१/ १९६/ ८८२०९/ १००१०००२५/ लब्धानि क्रमेण मूलानि २/ ३/ ९/ १४/ २९७/ १०००५//
This उद्देशक only presents questions/problems, those too, as converse of the उद्देशक of finding squares.
इति वर्ग-मूलम्//
Here ends (discussion of) square-root.
शुभमस्तु !
परिकर्माष्टके वर्गमूलम् – Finding Square-root
December 8, 2015 Square-root
परिकर्माष्टके वर्गमूलम्
अथ वर्ग-मूलम् | वर्ग-मूले करण-सूत्रम्
त्यक्त्वान्त्याद्विषमात् कृतिं द्विगुणयेन्मूलं समे तद्धृते |
त्यक्त्वा लब्धकृतिंस्तदाद्यविषमाल्लब्धं द्विनिघ्नं न्यसेत् |
पङ्क्त्यां पङ्क्तिहृते समेऽन्यविषमात्त्यक्त्वाप्तवर्गं फलम् |
पङ्क्त्यां तद्द्विगुणं न्यसेदिति मुहुः पङ्क्तेर्दलं स्यात्पदम् ||२२||
पदच्छेदैः –
त्यक्त्वा अन्त्यात् विषमात् कृतिम् द्वि-गुणयेत् मूलम् समे तद्-हृते |
त्यक्त्वा लब्ध-कृतिम् तद्-आद्य-विषमात् लब्धम् द्वि-निघ्नम् न्यसेत् |
पङ्क्त्याम् पङ्क्ति-हृते समे अन्य-विषमात् त्यक्त्वा आप्त-वर्गम् फलम् |
पङ्क्त्याम् तत् द्वि-गुणम् न्यसेत् इति मुहुः पङ्क्तेः दलम् स्यात् पदम् ||२२||
From the point of view of appreciation of poetic elements,
• Because the procedure is somewhat longish, भास्कराचार्य has chosenशार्दूलविक्रीडितवृत्तम्, which provides for 19 letters in each line.
• One may note that starting word in lines 1 and 2 is त्यक्त्वा and starting word of lines 3 and 4 is पङ्क्त्याम्. That is some help in committing the verse to memory.
अन्वयाः –
1. अन्त्यात् विषमात् कृतिम् त्यक्त्वा
2. मूलम् द्वि-गुणयेत्
3. तद् समे हृते
4. लब्ध-कृतिम् आद्य-विषमात् त्यक्त्वा
5. तद् लब्धम् द्वि-निघ्नम् पङ्क्त्याम् न्यसेत्
6. पङ्क्ति-हृते समे
7. आप्त-वर्गम् फलम् अन्य-विषमात् त्यक्त्वा
8. तत् द्वि-गुणम् पङ्क्त्याम् न्यसेत्
9. इति मुहुः
10. पङ्क्तेः दलम् पदम् स्यात्
Some observations –
• Although from the अन्वयाः, one may look at it as a 10-step procedure, the mention इति मुहुः (like this again) suggests iteration. Number of iterations will depend upon the number of digits in the given number.
• There are many words, which are used repetitively. They are mathematical terminologies.
o The word त्यक्त्वा appears 3 times in steps 1, 4 and 7, In the first step, it means 'separating'. In steps 4 and 7, it means 'subtracting'.
o The word कृति in steps 1 and 4 implies specific operation.
o The phrases द्वि-गुणयेत् in step 2, द्वि-निघ्नम् in step 5 and द्वि-गुणम् in step 8 all mean multiplication by two.
o Correspondingly the word दलम् in step 10 means half.
o The phrase पङ्क्त्याम् न्यसेत् (= put it into पङ्क्ति) is both at step 5 and also in step 8. This word पङ्क्ति (= a row, a line) is also there in Steps 6 and 10. The word पङ्क्ति has a specific meaning. Here in the context of the procedure for finding square-root it is "divider-line". It is not one particular line. The divider keeps developing afresh step by step.
o There is also the word विषमात् in steps 1, 4 and 7. When in the given number, pairs are identified leftwards, taking digits of unit's place and tenth's place as one pair and so on, each pair has a विषम (= odd) and सम (= even), One on the right is विषम (= odd) and one on the left is सम (= even), because one is always reading leftward.
The phrase अन्त्यात् विषमात् in step 1 refers to the digit or pair at the left-most end after pairing all other digits. If it is only one digit, it is विषम
The phrase आद्य-विषमात् in step 4 refers to the same अन्त्य-विषम referred to in step 1.
The phrase अन्य-विषमात् in step 7 refers to the next pair.
Keeping these observations in mind, one can attempt interpretation of the अन्वयाः and the श्लोक –
1. अन्त्यात् विषमात् कृतिम् त्यक्त्वा –
1. त्यक्त्वा = separating
2. अन्त्यात् विषमात् = from the leftmost odd digit (or pair)
3. कृतिम् = largest deductible square. Note, that this is interpretation and not just the literal meaning of कृति
2. मूलम् द्वि-गुणयेत् –
1. मूलम् = square-root of the largest deductible square
2. द्वि-गुणयेत् = be doubled
3. तद् समे हृते –
1. तद् = square-root of the largest deductible square
2. समे हृते = multiplied by equal number, i.e. squared
4. लब्ध-कृतिम् आद्य-विषमात् त्यक्त्वा –
1. त्यक्त्वा = subtracting
2. लब्ध-कृतिम् = result of the operation (the largest deductible square)
1. आद्य-विषमात् = from first odd (pair or number)
1. तद् लब्धम् द्वि-निघ्नम् पङ्क्त्याम् न्यसेत् –
1. तद् लब्धम् = the square-root
2. द्वि-निघ्नम् = doubled
3. पङ्क्त्याम् न्यसेत् = be placed in a divider line
2. पङ्क्ति-हृते समे
3. आप्त-वर्गम् फलम् अन्य-विषमात् त्यक्त्वा
4. तत् द्वि-गुणम् पङ्क्त्याम् न्यसेत् –
1. तत् द्वि-गुणम् = that doubled
2. पङ्क्त्याम् न्यसेत् = be placed in a divider line
5. इति मुहुः = like this, again (till the end)
6. पङ्क्तेः दलम् पदम् स्यात् –
1. पङ्क्तेः दलम् = half of divider line
2. पदम् स्यात् = is the answer
Let us take an example problem to find square root of 18769
Step 1 = त्यक्त्वा to separate अन्त्य विषम last odd, we have left-ward विषम-सम pairs from unit's place as विषम 9 सम 6 विषम 7 सम 8 and are left with left-most विषम 1
Step 2 The largest deductible square for left-most विषम 1 is 1 only.
Step 3 The मूलम् is 1. Its square is 1.
Step 4. त्यक्त्वा by deducting from आद्यविषमात् i.e. from 1, this कृतिम् operation, i.e. square of 1 is 1,
Step 5 Remainder लब्धकृति (1 – 1 =) 0
Step 6 तद् लब्धम् the square-root i.e. 1 द्वि-निघ्नम् doubled, i.e. 2 पङ्क्त्याम् न्यसेत् be put in a divider line.
Step 7 Now we have to proceed with अन्य विषम the next pair i.e. 87 ahead of the previous remainder 0. So we have the dividend 087. And on the divider line पङ्क्त्याम् we have 2. We have to add a digit ahead of this 2 on the divider line, such that the product of the digit with the new number on the divider line will be less than the dividend 087. One can judge that 24 x 4 = 96, which is more than 087. But 23 x 3 = 69 is less than 087. On subtraction we have the remainder 18.
Step 8 We add 3 to 23, which gets us 26 on yet another divider line. This 26 is same as putting double of 3 to the first number 2 on the first divider line.
Step 9. We have to now again इति मुहुः get the remaining अन्य विषम the next pair i.e. 69 ahead of the remainder 18. So we have the dividend 1869. And we have 26 on the divider line. We can judge that a digit 268 x 8 = 2144 is much larger than 1869. But 267 x 7 = 1869.
Step 10 The remainder now is 0. We have to add 7 to 267 on the previous divider line. Result is 274. Half दलम् of 274 is 137, which becomes स्यात् the answer पदम्.
May be, that the following will be easier to understand –
Number of which square-root is required → 18769
Separating by pairing 1 87 69
अन्त्य विषम = 1
Largest deductible square = 1
Remainder = 0
मूलम् of Largest deductible square = 1
पङ्क्ति (2×1 =) 2_ Dividend 0 87
By judgement 23 Subtract (23×3=) 69
Remainder 18
Next पङ्क्ति (23+3=) 26_ Dividend 18 69
By judgement 267 Subtract (267×7=) 1869
Remainder = 0
Next पङ्क्ति (267+7=) 274
Answer = 274 / 2 = 137
Let us take yet another example, 915836
Here अन्त्य विषम.is a two-digit number 91.
Largest deductible square is 81.
Its मूलम् is 9.
Remainder = 10
First पङ्क्ति double of 9, hence 18 _
Ahead of the remainder 10, we put अन्य विषम 58. The dividend is 1058.
We can judge that 186 x 6 = 1116, while 185 x 5 = 925. So we work with 925.
Remainder is 1058 – 925 = 133.
New पङ्क्ति 185 + 5 = 190 _
Ahead of the remainder 133, we put अन्य विषम 36. So dividend is 13336.
We have पङ्क्ति 190_.
We can judge that 1907 x 7 = 13349. We work with 1906 x 6 = 11436.
Remainder is 13336 – 11436 = 1900.
New पङ्क्ति 1906 + 6 = 1912_
We have no more अन्य विषम left in the given number but remainder is also not 0, So square-root of 915836 is somewhat more than half of 1912 (1912/2 = 956). We can check that square of 957 (957^2 = 915849).
Actually we can continue इति मुहुः by taking अन्य विषम as 00. With '00' added to previous remainder 1900, next dividend becomes 190000. Next divider is 1912_.
We can judge that 19129 x 9 = 172161. It may be noted that we cannot put in the divider-line a digit larger than 9.
Remainder (190000 – 172161 =) 17839. New divider line is 19129 + 9 = 19138. Half of this is 956.9. Note, that because the square-root of 915836 is >956, but <957, we have to put a decimal point after 956.
We can continue इति मुहुः to whatever level of accuracy we desire. Since 915836 is not a perfect square, the remainder may not become zero to any place after the decimal point. Presently we know that square-root of 915836 is > 956.9.
There is one उद्देशक-श्लोक.
अत्र उद्देशकः in उपजातिवृत्तम् –
मूलञ्चतुर्णाञ्च तथा नवानां पूर्वं कृतानाञ्च सखे कृतीनाम् |
पृथक् पृथक् वर्ग-पदानि विद्धि बुद्धेर्विवृद्धिर्यदि तेऽत्र जाता ||२३||
पदच्छेदैः
मूलम् चतुर्णाम् च तथा नवानाम् पूर्वम् कृतानाम् च सखे कृतीनाम् |
पृथक् पृथक् वर्ग-पदानि विद्धि बुद्धेः विवृद्धिः यदि ते अत्र जाता ||२३||
अन्वयेन –
सखे यदि ते अत्र बुद्धेः विवृद्धिः जाता चतुर्णाम् तथा नवानाम् च मूलम् पूर्वम् कृतानाम् कृतीनाम् च वर्ग-पदानि पृथक् पृथक् विद्धि |
• सखे = Dear
• यदि ते अत्र बुद्धेः विवृद्धिः जाता = If your intellect is enhanced
• चतुर्णाम् (मूलम्) तथा = know square-roots of 4
• नवानाम् च मूलम् = square-root of 9
• पूर्वम् वर्ग-पदानि कृतानाम् कृतीनाम् च (मूलम्) = square-roots of numbers which were made earlier as squares
• पृथक् पृथक् विद्धि = know them separately.
न्यासः ४/ ९/ ८१/ १९६/ ८८२०९/ १००१०००२५/ लब्धानि क्रमेण मूलानि २/ ३/ ९/ १४/ २९७/ १०००५//
This उद्देशक only presents questions/problems, those too, as converse of the उद्देशक of finding squares.
इति वर्ग-मूलम्//
Here ends (discussion of) square-root.
शुभमस्तु !