Mathematics and Its Importance
[edit] Mathematics
[edit] Pi as irrational
Aryabhata worked on the approximation for Pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes: caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."[12]
This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.
After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[3]
[edit] Mensuration and trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as tribhujasya phalashariram samadalakoti bhujardhasamvargah that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."[14]
Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus, which means "cove" or "bay". And after that, the sinus became sine in English.[15]
[edit] Indeterminate equations
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya: Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7
That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as
[edit] Pi as irrational
Aryabhata worked on the approximation for Pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes: caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."[12]
This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.
After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[3]
[edit] Mensuration and trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as tribhujasya phalashariram samadalakoti bhujardhasamvargah that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."[14]
Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus, which means "cove" or "bay". And after that, the sinus became sine in English.[15]
[edit] Indeterminate equations
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya: Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7
That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as