Update: This article was last updated on 6th February 2025 to reflect the accuracy and up-to-date information on the page.
Introducing our new series on Vedic Mathematics: “Unveiling the 16 Sutras“. This ancient system of mathematics, developed in ancient India, provides several techniques and sutras (formulas) for doing calculations fast and efficiently.
Recommended Reading: Vedic Math Sutras: Sutra 2-Nikhilam Navatashcaramam Dashatah
In this educational journey, we are going to uncover the ancient techniques, and we are starting with the mighty sutra called Ekadhikena Purvena, a shortcut to square.
Translated as “by one more than the previous,” this sutra is the foundation for mental calculations and is the key to unlocking mathematical prowess.
Ekadhikena Purvena‘s power is the acceleration of computation, making it highly useful to astronomers and engineers in ancient India. The concept was very handy in astronomical calculations, where exact and rapid calculations were required for tracking planetary positions, predicting eclipses, and determining celestial alignments.
Ekadhikena Purvena in Engineering and Modern Applications
Indian mathematicians such as Brahmagupta and Bhaskara II applied the same methodology as Ekadhikena Purvena to solve algebra and geometric problems. The sutra has been effective enough to remain valuable in the present too in various disciplines like:
Computer algorithms – Quick computation in logic of programming Astronomical modeling – Rapid estimation techniques for tracking celestial bodies Engineering calculations – Quick multiplication techniques of the laws of physics and building formulas
Despite being an ancient technique, Ekadhikena Purvena remains highly applicable in modern mathematics and competitive exams. Its efficiency proves how timeless Vedic Math techniques are, making them valuable for students as well as professionals.
This is a versatile technique, with easy multiplication when the two numbers have a unit digit sum of 10.
Steps
Here is the step-by-step process of applying the Ekadhikena Purvena sutra:
“Ekadhikena Purvena” is a sutra (formula) in Vedic mathematics, which means “By one more than the previous one.” This sutra is used to square numbers close to a power of 10, typically when the number’s last digit(s) is(are) small.
But 45 is far from a power of 10. If we apply the sutra directly, the calculations are pretty complicated.
However, if you still want to use the Ekadhikena Purvena method, you can use the nearest base of 50, which is more convenient.
1. Let’s apply the Ekadhikena Purvena sutra to find the square of 45.
First, split the number into two parts: the part that doesn’t include the 5 (4 in this case) and the 5.
A: 4
B: 5
2. Now, take the part of the number that doesn’t include the 5 (4) and add 1, making it 5.
C: 4+1=5
3. Multiply the original number that doesn’t include the 5 (4) by the number you just calculated (5). This gives you 20.
AXC
4×5= 20
4. Then append 25 (the square of 5) to the result from step 3.
So, the result is 2025
The next one will be Nikhilam Navatashcaraman Dashatah.
Common Mistakes & Tips for Using Ekadhikena Purvena
While Ekadhikena Purvena is a powerful mental math technique, beginners often make mistakes when applying the sutra. Here are some common errors and tips to avoid them:
Common Mistakes
1. Forgetting to add one – The core idea of Ekadhikena Purvena is to add one more than the previous number. Missing this step leads to incorrect calculations.
2. Misplacing digits while appending 25 – The second part of the answer (25) should always be placed at the end. Some students mistakenly place it in the middle.
3. Applying it to incorrect numbers – This sutra works best for numbers ending in 5. Applying it to other numbers directly can cause confusion.
4. Skipping mental verification – Although Ekadhikena Purvena accelerates the calculations, it is always advisable to quickly check the results in order to avoid mistakes.
Quick Tips for Rapid and Accurate Calculations
Practice with simple numbers first – Before trying the larger numbers, try Ekadhikena Purvena with smaller values such as 15², 25², and 35². Understand the logic behind the sutra – Knowing that Ekadhikena Purvena means “one more than the previous” helps in grasping the concept better. Write intermediate steps if needed – Until you master the sutra, jot down calculations to reinforce accuracy. Use visualization techniques – Picture the steps in your mind for faster recall.
Mastering Ekadhikena Purvena not only develops the ability to mentally calculate rapidly but also instills confidence in mental handling of complex math problems. Continue practicing, and soon squaring numbers mentally will become second nature!
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