Nikhilam Sutra

The Nikhilam Sutra: An Important Mathematical Rule

The Nikhilam Sutra is a mathematical rule. that mathematicians use to solve complex mathematical equations. Mathematicians use the rule to find the value of a particular mathematical expression. It is considered one of the most important mathematical rules in ancient India.

Origin

Scholars in ancient India used the Nikhilam Sutra to solve complex mathematical equations during the Vedic period. Later, ancient Indian mathematicians such as Aryabhata and Brahmagupta refined the rule, making significant contributions to the field.

The concept of “Nikhilam” meaning “all” or “complete,” forms the basis of the Sutra. The Sutra states “All from 9 and the last from 10” meaning that mathematicians use all numbers from 9 and the last number from 10 in solving equations.

Uses

Mathematicians use the Nikhilam Sutra to solve a variety of mathematical equations, including those in algebra, geometry, and trigonometry. The method is particularly useful for solving equations with large numbers and complex calculations. The method of “completing the square” involves rearranging the terms of an equation for easier solving.

Advantages

The Nikhilam Sutra is a simple and straightforward method. It requires only basic mathematical operations such as addition, subtraction, multiplication, and division. Despite its simplicity, it is a powerful tool for solving a wide range of mathematical equations. Mathematicians have used the equation for centuries in fields such as engineering, physics, and finance.

Conclusion

In conclusion, the Nikhilam Sutra has its origin in ancient India and is considered one of the most important mathematical rules in ancient India. Mathematicians refined and developed the rule during the Vedic period. 

The rule, based on the concept of “Nikhilam,” helps mathematicians solve a variety of mathematical equations, including those in algebra, geometry, and trigonometry. The simple and straightforward method, relying on basic mathematical operations, is a powerful tool for solving a wide range of mathematical equations. Today, mathematicians continue to use the Nikhilam Sutra in fields such as engineering, physics, and finance, making its significance timeless.

Examples of Nikhilam Sutra in Action

Algebraic Equations:

To solve the equation x^2 + 6x = 35 using the Nikhilam Sutra, mathematicians first rearrange the terms to x^2 + 6x – 35 = 0. Then, they complete the square by adding and subtracting (6/2)^2 = 9 to both sides of the equation, resulting in x^2 + 6x + 9 – 9 – 35 = 0. Next, mathematicians factor the left side of the equation to (x+3)^2 = -26. Since the square of a real number cannot be negative, mathematicians can conclude that there is no solution to this equation.

Geometric Problems:

To find the length of the hypotenuse of a right triangle with sides of length 3, 4, and 5, mathematicians first square the lengths of the legs and add them (3^2 + 4^2 = 9 + 16 = 25). Then, they take the square root of the sum, which gives the length of the hypotenuse (sqrt(25) = 5).

Trigonometric problems:

To find the value of sin(30°) using Nikhilam Sutra, use the trigonometric identity sin²(x) + cos²(x) = 1. Since you know that cos(30°) = √3/2, substitute this value into the identity and solve for sin(30°) by subtracting cos²(30°) from both sides. This gives you sin²(30°) = 1 – (√3/2)² = 1 – 3/4 = 1/4 and taking the square root of both sides gives you sin(30°) = √(1/4) = 1/2.