Binomial Theorem

In mathematics, the binomial theorem is the most widely used theorem. A binomial statement raised to any finite power can be expanded using the binomial theorem. It’s used to answer issues in combinatory, algebra, calculus, and probability, among other subjects. 

It is used to compare two huge numbers, to discover the remainder when a number raised to a large exponent is divided by another number, and in probability to determine whether an experiment will succeed or fail. The binomial theorem is also utilized in weather forecasting, forecasting the national economy in the coming years, and IP address distribution. Let’s take a closer look at the Binomial Theorem.

Binomial Expression

The Binomial Expression is a mathematical expression made up of two terms that include addition and subtraction operations. The distributive property must be utilized to multiply the binomials, and the equal terms must be joined to add the binomials. Binomial expressions such as (1+x), (x+y), (x²+xy), and (2a+3b) are a few examples.

(a+b)⁰ = 1 

(a+b)¹ = 1a+1b 

(a+b)² = 1a²+2ab+1b² 

(a+b)³ = 1a³+3a²b+3ab²+1b³ 

(a+b)⁴ = 1a⁴+4a³b+6a²b²+4ab³+1b⁴ 

Binomial Expansion

The binomial expansion for (a+b)ⁿ can be stated using the Pascal triangle. The expansion of (a+b)⁴ can be written starting from the fifth row. And (a+b)⁵ can be written from the sixth-row expansion.

As a result, we can write (a+b)⁵  = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵. The binomial expansion is made up of several terms, including:

• General Term is given by T_r+1 = ⁿC_r a_n – rbr

• Middle Term

• The total number of terms in an expansion of (a+b)ⁿ is n+1.

• The sum of the powers of a and b equals n.

History of Binomial Theorem

Since at least the 4th century BCE, when the Greek mathematician Euclid described the particular case of the binomial theorem with exponent 2, specific examples of the binomial theorem have been known. The cube binomial theorem was known in India by the 6th century AD, according to evidence.

Ancient Indian mathematicians were interested in binomial coefficients, which are combinatorial values that describe the number of ways to choose k objects out of n without replacement. The Chandashastra of Indian lyricist Pingala (200 BCE), which gives a way for its solution, is the first recorded reference to this contentious dilemma. Halayudh, a commentator from the 10th century AD, discusses this strategy using what is now known as Pascal’s triangle.

Indian mathematicians apparently learned how to formulate it as a quotient [ n! / ((nk)!k!)]  by the 6th century AD. The 12th-century work Lilavati by Bhaskara contains a precise exposition of this rule. To our knowledge, al-book, Qaraji’s which is quoted by al-Samawal in his “Al-Bahir,” contains the first formulation of the binomial theorem and the table of binomial coefficients.

Al-Qaraji used an early version of mathematical induction to describe the triangular pattern of binomial coefficients and to produce mathematical proofs of both the binomial theorem and Pascal’s triangle. Although many of his mathematical works have vanished, Persian poet and mathematician Omar Khayyam was likely familiar with the high order formula.

In the 13th-century mathematical works of Yang Hui and Chu Shih-Chih, the binomial expansion of smaller degrees was known. Although those works are still missing, Yang Hui attributes this strategy to an older 11th-century treatise by Jia Jian. Michael Stifel coined the term “binomial coefficients” in 1544 and demonstrated how to use them to express (1+a)ⁿ (1+a)^n-1 using “Pascal’s Triangle.” Blaise Pascal studied symmetry triangles extensively in his “Trate des Triangles” Arithmetic. The pattern of numbers, however, was already known to late Renaissance European mathematicians such as Stifel, Niccol Fontana Tartaglia, and Simon Stevin.

The generalized binomial theorem, which holds for any rational exponent, is attributed to Isaac Newton.

Use of Binomial Theorem 

• The binomial theorem is mentioned in the comedic opera The Pirates of Penzance’s song Major-Gen

• Professor Moriarty is said to be preparing a treatise on the binomial theorem, according to Sherlock Holmes

• “Newton’s binomial is as beautiful as Venus de Milo,” wrote Fernando Pessoa, who went by the unusual name of lvaro de Campos. The truth is that few individuals are aware of it.”

• Alan Turing mentions Isaac Newton’s work on the binomial theorem during his first meeting with Commander Denniston at Bletchley Park in the 2014 film The Imitation Game

Geometric Interpretation of Binomial Theorem

With the binomial theorem n = 2, it is mathematically evident that a square of side a + b can be sliced into sides of a square a, sides of a square b, and with sides of a square c. A and B are two rectangles. A cube with side a + b can be chopped into a cube with side a, a cube with side b, three a×a×b rectangular boxes, and three a×b×b rectangular boxes with n = 3.                          

Conclusion

We learned about the Binomial theorem, its features, and instances in this article, as well as how important it is in the mathematical world. It is used to answer issues in combinatory, algebra, calculus, probability, statistics, and data science, among other areas of mathematics. 

Ancient Indian mathematicians were interested in binomial coefficients, which are combinatorial values that describe the number of ways to choose k objects out of n without replacement.