03-07-2012, 12:32 PM
Vedic mathematics
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INTRODUCTION
Vedic mathematics - a gift given to this world by the ancient sages of India. A system which is far simpler and more enjoyable than modern mathematics. The simplicity of Vedic Mathematics means that calculations can be carried out mentally though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods; they are not limited to one method. This leads to more creative, interested and intelligent pupils. Vedic Mathematics refers to the technique of Calculations based on a set of 16 Sutras, or aphorisms, as algorithms and their upa-sutras or corollaries derived from these Sutras. Any mathematical problems (algebra, arithmetic, geometry or trigonometry) solved mentally with these sutras. Vedic Mathematics is more coherent than modern mathematics.
Vedic Mathematics offers a fresh and highly efficient approach to mathematics covering a wide range - starts with elementary multiplication and concludes with a relatively advanced topic, the solution of non-linear partial differential equations. But the Vedic scheme is not simply a collection of rapid methods; it is a system, a unified approach. Vedic Mathematics extensively exploits the properties of numbers in every practical application.
VEDIC SUTRAS
The word ‘Vedic’ is derived from the word “veda’ which means the store-house of all knowledge. Vedic mathematics is mainly based on 16 Sutras (or aphorisms) dealing with various branches of mathematics like arithmetic, algebra, geometry etc. These Sutras along with their brief meanings are enlisted below alphabetically
1) (Anurupye) Shunyamanyat – If one is in ratio, the other is zero
2) Chalana-Kalanabyham – Differences and Similarities.
3) Ekadhikina Purvena – By one more than the previous one
4) Ekanyunena Purvena – By one less than the previous one
5) Gunakasamuchyah – The factors of the sum is equal to the sum of the factors
6) Gunitasamuchyah – The product of the sum is equal to the sum of the product
7) Nikhilam Navatashcaramam
Dashatah – All from 9 and the last from 10
8) Paraavartya Yojayet – Transpose and adjust.
9) Puranapuranabyham – By the completion or noncompletion
10) Sankalana-vyavakalanabhyam – By addition and by subtraction
11) Shesanyankena Charamena – The remainders by the last digit
12) Shunyam Saamyasamuccaye – When the sum is the same that sum is zero
13) Sopaantyadvayamantyam – The ultimate and twice the penultimate
14) Urdhva-tiryakbyham – Vertically and crosswise
METHODS AND PERFORMANCES – EXISTING SYSTEMS
There are number of techniques that to perform binary multiplication. In general, the choice is based upon factors such as latency, throughput, area, and design complexity. More efficient parallel approach uses some sort of array or tree of full adders to sum partial products. Array multiplier, Booth Multiplier and Wallace Tree multipliers are some of the standard approaches to have hardware implementation of binary multiplier which are suitable for VLSI implementation at CMOS level.
ARRAY MULTIPLIER
Array multiplier is an efficient layout of a combinational multiplier. Multiplication of two binary number can be obtained with one micro-operation by using a combinational circuit that forms the product bit all at once thus making it a fast way of multiplying two numbers since only delay is the time for the signals to propagate through the gates that forms the multiplication array. In array multiplier, consider two binary numbers A and B, of m and n bits. There are mn summands that are produced in parallel by a set of mn AND gates. n x n multiplier requires n (n-2) full adders, n half-adders and n2 AND gates. Also, in array multiplier worst case delay would be (2n+1) td. Array Multiplier gives more power consumption as well as optimum number of components required, but delay for this multiplier is larger. It also requires larger number of gates because of which area is also increased; due to this array multiplier is less economical Thus, it is a fast multiplier but hardware complexity is high.