### Brahmagupta – The Genius of Mathematics

### Born

**Brahmagupta**, son of an astrologer named **Jisnugupta**, was born in **598 A.D** perhaps in **bhinmal (Old name was Bhillamala).** However, there is no conclusive evidence available relating to the birth place of

**Brahmagupta.**Now

**Bhinmal**is a town in the

**Jalore District**of

**Rajasthan**,

**India**but at that time it was the apparent capital of the

**Gurjaradesa**, the second largest kingdom of Western India, comprising the

**southern Rajasthan**and

**northern Gujarat**in modern-day India. It was also a center of learning for

**mathematics**and

**astronomy**. He was a

**Shaivite**by religion.

**Shaivism**is one of the major branches of

**Hinduism**and it is devoted to worship of the god

**Shiva**.

**Education & Career**

**Brahmagupta** became an astronomer of the ** Brahmapaksha school** (one of the four major schools of

**Indian astronomy**during that period). He studied the five traditional

**on Indian astronomy as well as the work of other astronomers including**

*siddhanthas***Aryabhata I**,

**Latadeva**,

**Pradyumna**,

**Varahamihira**,

**Simha**,

**Srisena**,

**Vijayanandin**and

**Vishnuchandra**.

At the age of **30**, he composed **Brahma-sphuta-siddhanta** (Correctly Established Doctrine of Brahma) in **628 AD** having **25 chapters** while living in **Bhinmal**. The first **ten chapters** cover the topics of **mean longitudes** & **true longitudes** of the **planets**, the three problems of **diurnal rotation,** **lunar eclipses & solar eclipses,** **risings** and **settings**; the **moon’s crescent**; the **moon’s shadow,** **conjunctions** of the **planets** with each other and **conjunctions** of the **planets** with the **fixed stars**.

The remaining **fifteen chapters** contain key chapters on **mathematics** including **algebra**, **geometry**,** trigonometry** and **algorithmic**. Τhe text is notable for its **mathematical content** as it contains rules regarding **zero**, **negative** and** positive numbers**, rules for computing **square roots**, methods of solving **linear** and **quadratic equations**, rules for **summing series**, **Brahmagupta’s identity**, **Brahmagupta’s formula, Brahmagupta’s matrix** and **Brahmagupta’s theorem**.

Later, **Brahmagupta** moved to **Ujjain **as it** **was also a major centre for **astronomy **in** India**. At the mature age of **67**, he composed his next well known work** ****Khanda-khadyaka. **The other books he wrote** “Durkeamynarda” **and** “Cadamakela”.**

**Known For**

**Brahmagupta** was known mostly through his works, which cover **mathematical and astronomical** topics and significantly combine the two. He did tremendous work in several streams of **mathematics** and **astronomy**. Here is the list of his most famous works;

**Arithmetic**

Indian arithmetic was well known in Medieval Europe as **“Modus Indoram”** meaning method of the Indians. In **Brahma-sphuta-siddhanta**, Multiplication was named **Gomutrika**. In the beginning of **chapter 12** of **Brahma-sphuta-siddhanta**, entitled **Calculation**, **Brahmagupta** explained how to find the **cube** and** cube-root** of an integer and gave rules facilitating the computation of **squares** and **square roots**. He also gave rules for dealing with five types of combinations of **fractions**:

*1) a*/*c*+ *b*/*c*; 2) *a*/*c* × *b*/*d*; 3) *a*/1 + *b*/*d*; 4) *a*/*c* + *b*/*d* × *a*/*c* = *a*(*d* + *b*)/*cd*; and 5) *a*/*c* − *b*/*d* × *a*/*c* = *a*(*d* − *b*)/*cd*.

#### Series

**Brahmagupta** gave the **sum of the squares** of the first ** n natural numbers** as

**n(n+1)(2n+1)/6**and the

**sum of the cubes**of the first

**n**natural numbers as

**(n(n+1)/2)**. However, no proofs has been given about how did he discover these formulae.

^{2}**Number System**

**Brahmagupta** is the first who gave concrete rules of using **positive numbers**, **negative numbers**, and **zero**. The **Brahma-sphuta-siddhanta** is one the earliest known text to treat zero as a number. In **Chapter 18** of his **Brahma-sphuta-siddhanta***, *he established the basic mathematical rules which are-

- The sum of two positive quantities is positive
**[A + B =C]**. - The sum of two negative quantities is negative
**[-A + (-B) = – C]**. - A negative number minus zero is a negative number
**[-A – 0 = -A]**. - A positive number minus zero is a positive number
**[A – 0 = A]**. - Zero minus zero is a zero
**[0 – 0 = 0]**. - The sum of zero and zero is zero
**[0 + 0 = 0]**. - The sum of a positive and a negative is their difference
**[A + (-B) = A -B]**or if they are equal, zero**[A + (-A) = 0]**. - A negative number subtracted from zero is a positive number
**[0 – (-A) = A]**. - A positive number subtracted from zero is a negative number
**[0 – (A) = -A]**. - The product of zero by a negative number or positive number is zero
**[0 x –A = 0] or [0 x A = 0]** - The product of zero by zero is zero
**[0 x 0 = 0]**. - The product of two positive numbers is a positive number
**[A x B = C]**. - The product of two negative numbers is a positive number
**[-A x -B = C]**. - The product of a negative number and a positive number is a negative number
**[-A x B = -C].** - The product of a positive number and a negative number is a negative number
**[A x -B = -C]** - Positive number divided by positive or negative by negative is positive
**[ A/B = C Or -A/-B = C ]** - Positive divided by negative is negative. Negative divided by positive is negative
**[ A/-B = -C Or -A/B = -C ]** - A positive or negative number when divided by zero is a fraction with the zero as denominator
- Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator
- Zero divided by zero is zero
**[ 0/0 = 0 ]**

Here we can see that the **Brahmagupta’s **postulates relating to **negative and positive numbers** and **zero** are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

#### Algebra

In **chapter 18** of **Brahma-sphuta-siddhanta***, *he* *developed the solution of the general **linear equation** of form **bx + c = dx + e***,*

equivalent to **x = e − c/b − d**, where *c* and *e are constants*.

He further gave two equivalent solutions to the general quadratic equation of form **ax ^{2} + bx = **

**c**

*,*equivalent to

and,

He went on to solve systems of **simultaneous indeterminate equations** stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable’s coefficient. In particular, he recommended using **“the pulverizer”** to solve equations with **multiple unknowns**.

**Brahmagupta’s formula**

In **Chapter 12** of **Brahma-sphuta-siddhanta** , he gave his **remarkable** and **most famous** result in geometry is his formula for **cyclic quadrilaterals** (one that can be inscribed in a circle). It gives the area **K** of a **cyclic quadrilateral** whose sides have lengths ** a, b, c, d** as

where ** S**, the semi-perimeter and is given as,

**Brahmagupta’s theorem**

In **Chapter 12** of **Brahma-sphuta-siddhanta***, *his theorem states that if a **cyclic quadrilateral** is orthodiagonal (that is, has perpendicular diagonals), then the **perpendicular** to a side from the point of intersection of the diagonals always **bisects** the opposite side.

More specifically, let** A, B, C** and

**be four points on a**

*D***circle**such that the

**lines**and

*AC***BD**are

**perpendiculars**. The intersection of

**AC**and

**BD**is

**M**. Now, Drop the

**perpendicular**from

**M**to the line

**BC**, calling the intersection

**E**. Let

**F**be the

**intersection**of the line

**EM**and the edge

**AD**. Then, the

**theorem**states that

**F**is the

**midpoint of**

**AD**

*or*

**AF=FD.**

#### Brahmagupta–Fibonacci identity

In **algebra**, the **Brahmagupta–Fibonacci identity** or simply **Brahmagupta identity** says that the product of **two sums** each of **two squares** is itself a **sum of two squares**. In other words, the set of all sums of two squares is closed under multiplication. Specifically:

The **identity** was actually first found in **Diophantus’ Arithmetica** (III, 19), of the t

**hird century A.D**. It was rediscovered by

**Brahmagupta**, who generalized it to the

**Brahmagupta identity**and used it in his study of what is now called

**Pell’s equation, namely**

**X**^{2}− NY^{2}= 1#### Brahmagupta matrix

To solve the **indeterminate equation, Brahmagupta** gave an iterative method of deriving new solutions from the known ones by his **samasa-bhavana**, the principle of composition. He gave the following matrix ,

where **B** is called as the **Brahmagupta matrix**. To know more about** Brahmagupta matrix** and **Brahmagupta polynomials** follow the link; THE BRAHMAGUPTA POLYNOMIALS.

#### Astronomy

In **7th Chapter** of **Brahma-sphuta-siddhanta** entitled **Lunar Crescent**, Some of the important contributions made by **Brahmagupta **are: methods for calculating the position of **heavenly bodies** over time (ephemerides), positions of the **planets**, their **rising** and **setting**, **planetary conjunctions**, and the calculation of **solar** and **lunar eclipses**.

In his second work **Khanda-khadyaka, **he uses the interpolation formula to compute** values of sines.**

**Died**

He is believed to have died in Ujjain between **665** to **670 A.D**. He was one of the greatest mathematicians in Indian history and his contributions to mathematics and science have made significant. A writer of his own time, **Bhaksara II**, called him **“ Ganita Chakra Chudamani”**, which means

**“the gem in the circle of mathematicians.”**

** **

##### References

*THE BRAHMAGUPTA POLYNOMIALS by E. R. Suryanarayan, Department of Mathematics, University of Rhode Island, Kingston, RI02881 (Submitted May 1994)*