MSC #5 – Vertically and Crosswise: Developing Short-Cuts
The Vertically and Crosswise sutra is the basis of the general multiplication method in Vedic Mathematics whether using numbers or polynomials. It is also used in division, squaring, and extracting square roots and computations involving fractions and mixed numbers. It is also useful in analytic geometry and in trigonometry when using triples. This type of Multiplication can be done from either from left to right or right to left. In a 2 x 2 left to right multiplication, there are three steps: first is a vertical multiplication of the tens’ digits, then a middle cross-multiplication step and finally another vertical multiplication of the units’ digits. In mental calculations, it is often easier to start with the middle crosswise step because it is the most complicated. Doing so also gives us a deeper understanding of how many multiplication shortcuts are developed. Our example for today shows how to quickly square numbers near 50. As shown in the figure, the middle crosswise step will give [(5 x 2) + (5 x 2)] = (5 + 5) x 2 = 20. Since 5 + 5 = 10, the result of this crosswise step will always be the units digit followed by a zero. The first and the last steps in this squaring follows the Sutra, the First by the First and the Last by the Last: 5 x 5 = 25 and 2^2 which is written as two digits, 04. The quick method of squaring numbers near fifty is: the first part of the answer is 25 + excess over or deficiency from fifty while the second part is the square of the excess or deficiency. Note that the second part of the answer must occupy two places. Adding a leading zero or “carrying” may be done as needed. Thus 54^2 = 25 + 4 | 4^2 = 29|16 = 2916 61^2 = 25 + 11| 11^2 = 36| 121 = 36+1|21 = 3721 For numbers below 50, we can use bar notation which is explained in MSC #4 – Using Bar Numbers (https://www.math-inic.com/blog/msc-4-using-bar-numbers/) 49^2 = 5(1)^2 = 25 + (1) | (1)^2 = 24|01 = 2401 44^2 = 5(6)^2 = 25 + (6) | (6)^2 = 19|36 = 1936 We can also perform this squaring short cut even without using bar numbers. 38^2 = (50 – 12)^2 = 25 – 12 | 12^4 = 13 | 144 = 13 + 1|44 = 1444 More short cuts like squaring numbers near 500, 5000, multiplying numbers like 36 x 76 where the last digits are the same while he initial digits are complementary and 77 x 82 will be discussed in our e-book, Master Strategies in Computing #5 – Vetrically and Crosswise.