भारतीय गणित

ह्या लेखाचा/विभागाचा इंग्रजी किंवा अमराठी भाषेतून मराठी भाषेत भाषांतर करावयाचे बाकी आहे. अनुवाद करण्यास आपलाही सहयोग हवा आहे. ऑनलाईन शब्दकोश आणि इतर सहाय्या करिता भाषांतर प्रकल्पास भेट द्या.

भारतीय उपखंडात गणिताची सुरुवात वैदिक पूर्व ते उत्तर वैदिक काळातील (सुमारे इ. स. पू. ७००० - इ. स. २००) यजुर्वेद, शतपथ ब्राह्मण व शुल्ब सूत्रे इत्यादी ग्रंथांमध्ये प्राथमिक स्वरूपांत झाली. यजुर्वेद संहितेमधून आज प्रचलित असलेल्या दशमान पद्धतीचा उगम झाला. गणिताच्या दृष्टीने शुल्ब सूत्रांस विशेष महत्त्व देता येईल. शुल्ब सूत्रे हे बौधायन, मानव, आपस्तंब व कात्यायन यांनी इ. स. पू. ८०० ते इ. स. पू. ३०० च्या सुमारास लिहिली आहेत. या सूत्रांमध्ये भूमिती, बौधायनचा सिद्धान्त (पायथागोसचा सिद्धान्त), बौधायनची त्रिके (पायथागोसची त्रिके), पाय (π)ची ३.१४ ही किंमत आणि संख्यांचे वर्गमूळ या विषयांवर चर्चा केलेली आढळते. पाणिनी या संस्कृत व्याकरणकाराने त्यांच्या ग्रंथात बुलियन लॉजिक आणि कंटेक्स्ट फ्री ग्रामर यांचा प्रथमत: उपयोग केलेला आढळतो. त्यांना बॅकस-नौर फॉर्म (BNF)चा अग्रदूत मानले जाते. छंदशास्त्रावरच्या लेखनात मेरु प्रस्तार, द्विपद प्रमेय, पिङ्गल संख्या आणि ${\displaystyle 2^{n}}$ची कॉंबिनेटरियल आयडेंटिटी याबद्दल पिङ्गल या संगीत तज्ज्ञाला बरेचसे ज्ञात होते असे आढळून येते.

भारतीय गणिताच्या दृष्टीने मध्ययुगीन कालखंड (इ.स. ४०० ते इ. स. १६००) हा विशेष म्हणता येईल. आज उपयोगात येणारी दशमान पद्धत मध्ययुगीन भारतात शोधली गेली. भारतीय गणित तज्ज्ञांनी शून्याचा एक संख्या म्हणून प्रथमत: अभ्यास केला. मध्ययुगीन कालखंडात पहिला आर्यभट, आचार्य वराहमिहिर, आचार्य ब्रह्मगुप्त, पहिला भास्कराचार्य, आचार्य वीरसेन, महावीर आचार्य, श्रीधराचार्य, मंजुल, दुसरा आर्यभट, आचार्य श्रीपती, नेमीचंद्र सिद्धान्त चक्रवर्ती, दुसरा भास्कराचार्य, संगमग्रामचे माधवाचार्य, ज्येष्ठदेव व नीलकंठ सोमयाजी अशा अनेक शास्त्रज्ञांचे गणित व खगोलशास्त्रात मोठे योगदान आहे.

ह्या शास्त्रज्ञांनी पाय (π)ची किंमत, दशमान पद्धत, शून्य, ऋण संख्या, अनंत, अंकगणित, बीजगणित, क्षेत्रमिती, भूमिती, त्रिकोणमिती, त्रिकोणमितीय चलने व त्यांचे तक्ते, क्षेत्रमिती, वर्ग व घन समीकरणे, डायोफॅंटाईन समीकरणे, कंटिन्यूड फ्रॅक्शन्स, घातांक शृंखला, अनंतांश कॅलक्युस आणि विभेदीय व समाकलनीय कॅलक्युलसची प्रारंभिक संकल्पना ह्या गणितीय विषयांवर; तर सौर मंडळाची सूर्य केन्द्रीयता, पृथ्वीचा परिवलन व परिभ्रमण काल, सूर्योदयाचे समीकरण, चंद्राचे वर्धमान, सूर्यग्रहण, चंद्रग्रहण, ग्रहांचे परागमन, फिरते ग्रहगोलीय अक्षांश, ग्रहांचे परस्पर सबंध, ग्रहांचे ताऱ्यांशी सबंध आणि गुरुत्वाकर्षणाची प्रारंभिक संकल्पना ह्या खगोलशास्त्रीय विषयांवर संशोधन केले."प्रात्यक्षिक शिस्त" म्हणून गणिताचा अभ्यास BC व्या शतकापासून पायथागोरियन लोकांपासून सुरू होतो, ज्यांनी "ग्रीक" या शब्दाची रचना प्राचीन ग्रीक ancient (गणिता) पासून केली होती, ज्याचा अर्थ "शिक्षणाचा विषय" होता. []] ग्रीक गणिताने या पद्धती मोठ्या प्रमाणात परिष्कृत केल्या (विशेषतः निष्ठावान तर्क आणि पुरावांमध्ये गणिताची कडकपणा यांच्या माध्यमातून) आणि गणिताच्या विषयाचा विस्तार केला. []] त्यांनी सैद्धांतिक गणितामध्ये अक्षरशः कोणतेही योगदान दिले नसले तरी, प्राचीन रोमी लोक सर्वेक्षण, स्ट्रक्चरल अभियांत्रिकी, यांत्रिकी अभियांत्रिकी, बुककीपिंग, चंद्र आणि सौर दिनदर्शिका तयार करणे, तसेच कला व हस्तकला यांमध्ये गणितांचा उपयोग करीत असत. चिनी गणिताने प्लेस व्हॅल्यू सिस्टम आणि नकारात्मक क्रमांकाचा प्रथम वापर यासह प्रारंभिक योगदान दिले. []] []] हिंदु-अरेबिक अंक प्रणाली आणि त्याच्या कारवायांचा उपयोग करण्याचे नियम, जगभरात आज वापरल्या जाणाऱ्या, पहिल्या शतकातील एडीच्या काळात विकसित झाले आणि मुहम्मद इब्न मसा यांच्या कार्याद्वारे इस्लामिक गणिताद्वारे पाश्चात्य जगात प्रसारित केले गेले. अल-ख्वरीझ्मा. []] []] इस्लामिक गणिताने यामधून या संस्कृतींना ज्ञात गणिताचा विस्तार व विस्तार केला. [१०] मेक्सिको आणि मध्य अमेरिकेच्या माया सभ्यतेने विकसित केलेले गणित ही या परंपरांपेक्षा स्वतंत्र परंतु स्वतंत्र आहे, जिथे शून्या ही संकल्पनेला माया अंकांमध्ये प्रमाणित प्रतीक दिले गेले.

साचा:पहा । वेद

साचा:पहा । शुल्ब सूत्रे

${\displaystyle {\sqrt {2}}=1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}\approx 1.4142156\ldots }$

${\displaystyle {\sqrt {2}}=1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297.}$

An important landmark of the Vedic period was the work of Sanskrit grammarian, (c. 520–460 BCE). His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages).टिपोनोमेट्रिक फंक्शन्सची पहिली टेबल्स हिप्परकस (सी .१ 90 ० - सी .१२० बीसीई) आणि मेनेलास (सी. –०-११40० सीई) यांनी बनविली होती, परंतु त्या गमावल्या गेल्या. टॉलेमीच्या अस्तित्वातील टेबलसह (इ. सी. - ० - सी .१6868 CE सीई), ते सर्व जीवांचे सारण्या होते, अर्ध्या जीवांचे नव्हे, म्हणजे साइन फंक्शनचे. [१] भारतीय गणितज्ञ आर्यभान (सा.यु. 47 47–-–50०) यांनी तयार केलेले टेबल आतापर्यंतचे पहिले साईन टेबल मानले जाते. [१] अर्याभासाचे टेबल प्राचीन भारतातील मानक साईन टेबल राहिले. या टेबलाची अचूकता सुधारण्यासाठी सतत प्रयत्न होत राहिले. संगमाग्रमाच्या माधव (सी .१5050० - सी .१25२)) आणि माधव यांनी साईन टेबलाचे टॅब्युलेशन साईन आणि कोसाइन फंक्शन्सच्या पॉवर सिरीजच्या विस्ताराचा शोध लावला

• Sine (Jya).
• Cosine (Kojya).
• Inverse sine (Otkram jya).

• Tangent.
• Secant.

त्रिकोणमिती

(हे सुद्धा पहा : Aryabhata's sine table)

• Introduced the trigonometric functions.
• Defined the sine (jya) as the modern relationship between half an angle and half a chord.
• Defined the cosine (kojya).
• Defined the versine (utkrama-jya).
• Defined the inverse sine (otkram jya).
• Gave methods of calculating their approximate numerical values.
• Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
• Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.
• Spherical trigonometry.

अंकगणित

• Continued fractions.

बीजगणित

• Solutions of simultaneous quadratic equations.
• Whole number solutions of linear equations by a method equivalent to the modern method.
• General solution of the indeterminate linear equation .

गणितीय खगोलशास्त्र

• Accurate calculations for astronomical constants, such as the:
  • सूर्यग्रहण.
  • चंद्रग्रहण.
  • The formula for the sum of the cubes, which was an important step in the development of integral calculus.^[१]

• ${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$
• ${\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)}$
• ${\displaystyle {\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)}$

Brahmagupta's theorem

Brahmagupta's formula

${\displaystyle A={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\,}$

Brahmagupta's Theorem on rational triangles

${\displaystyle \ ax^{2}+bx=c}$

Brahmagupta's Identity

${\displaystyle \ (x^{2}-Ny^{2})(x'^{2}-Ny'^{2})=(xx'+Nyy')^{2}-N(xy'+x'y)^{2}}$

Lemma (Brahmagupta)

If ${\displaystyle x=x_{1},\ \ y=y_{1}\ \ }$ is a solution of ${\displaystyle \ \ x^{2}-Ny^{2}=k_{1},}$ and,

${\displaystyle x=x_{2},\ \ y=y_{2}\ \ }$ is a solution of ${\displaystyle \ \ x^{2}-Ny^{2}=k_{2},}$, then:

${\displaystyle x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\ \ }$ is a solution of ${\displaystyle \ x^{2}-Ny^{2}=k_{1}k_{2}}$

Theorem (Brahmagupta)

${\displaystyle \ x^{2}-Ny^{2}=k}$ has an integer solution for any one of ${\displaystyle \ k=\pm 4,\pm 2,-1}$ then Pell's equation:

${\displaystyle \ x^{2}-Ny^{2}=1}$

Example (Brahmagupta)

Find integers ${\displaystyle \ x,\ y\ }$ such that:

${\displaystyle \ x^{2}-92y^{2}=1}$

${\displaystyle \ x=1151,\ y=120}$

• Zero
• Squares
• Cubes
• square roots, cube roots, and the series extending beyond these
• Plane geometry
• Solid geometry
• Problems relating to the casting of shadows
• Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle.

Mahavira also:

• Asserted that the square root of a negative number did not exist
• Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.
• Solved cubic equations.
• Solved quartic equations.
• Solved some quintic equations and higher-order polynomials.
• Gave the general solutions of the higher order polynomial equations:
  • ${\displaystyle \ ax^{n}=q}$
  • ${\displaystyle a{\frac {x^{n}-1}{x-1}}=p}$
• Solved indeterminate quadratic equations.
• Solved indeterminate cubic equations.
• Solved indeterminate higher order equations.

• A good rule for finding the volume of a sphere.
• The formula for solving quadratic equations.

• Elementary operations
• Extracting square and cube roots.
• Fractions.
• Eight rules given for operations involving zero.
• Methods of summation of different arithmetic and geometric series, which were to become standard references in later works.

=== दुसरा आर्यभट (इ. स. ९२० - इ. स. १०००)=== दुसरा आर्यभट्ट यानी सिद्धत्शिरोमनी हा गनितवरिल ग्रंथ लिहिला.त्यातील लिलवती हे प्रकरण प्रसिद्ध आहे. गुरुत्वाकर्षण हे ही आर्यभट्ट यांना न्युटनच्या आधी माहित होते.

• General solution of the simultaneous indeterminate linear equation.
• सूर्यग्रहण
• चंद्रग्रहण

अंकगणित

• Interest computation
• Arithmetical and geometrical progressions
• Plane geometry
• Solid geometry
• The shadow of the gnomon
• Solutions of combinations
• Gave a proof for division by zero being infinity.

बीजगणित

• The recognition of a positive number having two square roots.
• Surds.
• Operations with products of several unknowns.
• The solutions of:
  • Quadratic equations.
  • Cubic equations.
  • Quartic equations.
  • Equations with more than one unknown.
  • Quadratic equations with more than one unknown.
  • The general form of Pell's equation using the chakravala method.
  • The general indeterminate quadratic equation using the chakravala method.
  • Indeterminate cubic equations.
  • Indeterminate quartic equations.
  • Indeterminate higher-order polynomial equations.

भूमिती

• Gave a proof of the Pythagorean theorem.

Calculus

• Conceived of differential calculus.
• Discovered the derivative.
• Discovered the differential coefficient.
• Developed differentiation.
• Stated Rolle's theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis).
• Derived the differential of the sine function.
• Computed π, correct to five decimal places.
• Calculated the length of the Earth's revolution around the Sun to 9 decimal places.

त्रिकोणमिती

• Developments of spherical trigonometry
• The trigonometric formulas:
  • ${\displaystyle \ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)}$
  • ${\displaystyle \ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)}$

हेसुद्धा पाहा: केरळ गणित व खगोलशास्त्र अभ्यास केंद्र

${\displaystyle r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}y/x\leq 1.}$

${\displaystyle \sin x=x-x{\frac {x^{2}}{(2^{2}+2)r^{2}}}+x{\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots }$

${\displaystyle r-\cos x=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}{\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,}$

where, for r = 1, the series reduces to the standard power series for these trigonometric functions, for example:

• • ${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$

and

• • ${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }$
• Use of rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle, was not used.)^[२]
• Use of series expansion of ${\displaystyle \arctan x}$ to obtain an infinite series expression (later known as Gregory series) for ${\displaystyle \pi }$:^[२]

${\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }$

• A rational approximation of error for the finite sum of their series of interest. For example, the error, ${\displaystyle f_{i}(n+1)}$, (for n odd, and i = 1, 2, 3) for the series:

${\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots +(-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)}$

${\displaystyle {\text{where }}f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.}$

• Manipulation of error term to derive a faster converging series for ${\displaystyle \pi }$:^[२]

${\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots }$

• Using the improved series to derive a rational expression,^[२] 104348/33215 for π correct up to nine decimal places, i.e. 3.141592653.
• Use of an intuitive notion of limit to compute these results.^[२]
• A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.^[३] However, they did not formulate the notion of a function, or have knowledge of the exponential or logarithmic functi

${\displaystyle {\begin{aligned}&x+y=a,\ x-y=b,\ xy=c,x^{2}+y^{2}=d,\\[8pt]&x^{2}-y^{2}=e,\ x^{3}+y^{3}=f,\ x^{3}-y^{3}=g\end{aligned}}}$

• शुल्ब सूत्रे
• केरळ गणित व खगोलशास्त्र अभ्यास केंद्र
• सूर्य सिद्धांत
• आचार्य ब्रह्मगुप्त
• बख्शाली पर्णलेख
• भारतीय गणित तज्ञांची यादी
• भारतीय विज्ञान आणि तंत्रज्ञान
• भारतीय तर्कशास्त्र
• भारतीय खगोलशास्त्र
• गणिताचा इतिहास
• हिंदू धर्मग्रंथांमद्धे संख्यांची यादी
• भारती कृष्ण तीर्थ कृत गणित लेखन

86. ^ Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. 46. ISBN 3-540-64767-8.

87. ^ Britannica Concise Encyclopedia (2007), entry algebra

• Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages, ISBN 3-7643-7291-5.
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• Vedic Maths Archived 2015-02-07 at the Wayback Machine.
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साचा:India topics साचा:Indian mathematics