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Mechanic's Rule also known as Babylonian method is a Cauchy sequence to calculate the square root of a number.

${\displaystyle x_{0}>0,a>0,\ \ x_{\text{n+1}}={\frac {1}{2}}\left(x_{n}+{\frac {a}{x_{n}}}\right)}$

This equation can be used to generate an infinite sequence of approximations ${\displaystyle x_{0},\ x_{1},\ x_{2},\ x_{3}...\ }$ that gets closer and closer to ${\displaystyle {\sqrt {a}}}$

In general, ${\displaystyle x_{\text{n+1}}={\frac {1}{k}}\left((k-1)x_{n}+{\frac {a}{x_{n}^{k-1}}}\right)}$ will converge to ${\displaystyle {\sqrt[{k}]{a}}}$

Let,

${\displaystyle x_{\text{n+1}}={\frac {1}{2}}\left(x_{n}+{\frac {a}{x_{n}}}\right)}$

Therefore,

${\displaystyle {\begin{aligned}x_{\text{n+1}}-{\sqrt {a}}&={\frac {1}{2}}\left(x_{n}+{\frac {a}{x_{n}}}\right)-{\sqrt {a}}\\&={\frac {x_{n}^{2}-2x_{n}{\sqrt {a}}+a}{2x_{n}}}\end{aligned}}}$

${\displaystyle x_{\text{n+1}}-{\sqrt {a}}={\frac {(x_{n}-{\sqrt {a}})^{2}}{2x_{n}}}}$

(1)

Similarlx,

${\displaystyle x_{\text{n+1}}+{\sqrt {a}}={\frac {(x_{n}+{\sqrt {a}})^{2}}{2x_{n}}}}$

(2)

Dividing Eq. 1 bx Eq. 2,

${\displaystyle {\begin{aligned}{\frac {x_{\text{n+1}}-{\sqrt {a}}}{x_{\text{n+1}}+{\sqrt {a}}}}&={\frac {\frac {(x_{n}-{\sqrt {a}})^{2}}{2x_{n}}}{\frac {(x_{n}+{\sqrt {a}})^{2}}{2x_{n}}}}\\&={\frac {(x_{n}-{\sqrt {a}})^{2}}{(x_{n}+{\sqrt {a}})^{2}}}\\\end{aligned}}}$

Therefore,

${\displaystyle {\begin{aligned}{\frac {x_{\text{n+1}}-{\sqrt {a}}}{x_{\text{n+1}}+{\sqrt {a}}}}&={\frac {(x_{n}-{\sqrt {a}})^{2}}{(x_{n}+{\sqrt {a}})^{2}}}\\&={\frac {(x_{\text{n-1}}-{\sqrt {a}})^{2^{2}}}{(x_{\text{n-1}}+{\sqrt {a}})^{2^{2}}}}\\&={\frac {(x_{\text{n-2}}-{\sqrt {a}})^{2^{3}}}{(x_{\text{n-2}}+{\sqrt {a}})^{2^{3}}}}\end{aligned}}}$

${\displaystyle {\frac {x_{\text{n+1}}-{\sqrt {a}}}{x_{\text{n+1}}+{\sqrt {a}}}}={\frac {(x_{0}-{\sqrt {a}})^{2^{n+1}}}{(x_{0}+{\sqrt {a}})^{2^{n+1}}}}}$

For ${\displaystyle x_{\text{0}}>0\,\!}$ and ${\displaystyle {\sqrt {a}}>0}$, we have ${\displaystyle \left|x_{0}-{\sqrt {a}}\right|<x_{0}+{\sqrt {a}}}$

${\displaystyle \lim _{n\to \infty }{\frac {x_{\text{n+1}}-{\sqrt {a}}}{x_{\text{n+1}}+{\sqrt {a}}}}=\lim _{n\to \infty }{\frac {(x_{0}-{\sqrt {a}})^{2^{n+1}}}{(x_{0}+{\sqrt {a}})^{2^{n+1}}}}=0}$

Therefore,

${\displaystyle \lim _{n\to \infty }x_{n}={\sqrt {a}}}$

Indian cuisine encompasses a delectable variety of sub-cuisines and cooking styles that vary very widely, reflecting the diversity of the Indian subcontinent, even though there is a certain centrality to the general ingredients used. While there are a lot of popular cookbooks and websites featuring recipes from these varied Indian sub-cuisines , the terms used in the recipes sometimes tend to be multi-lingual and region-specific, mostly based on the author's specific sub-ethnicity, the popularity of a given vegetable/spice in a given sub-cuisine within India, etc.

Indian cuisine is overwhelmingly vegetarian friendly and employs a variety of different fruits, vegetables, grains, and spices which vary in name from region to region within the country. Most Indian restaurants serve predominantly Punjabi/North Indian cuisine, while a limited few serve a very limited choice of some South Indian dishes like Dosa. But for the connoisseurs, India offers a complex and eclectic array of sub-cuisines to explore, which are equally vegetarian friendly and a delight to the taste buds.

Even for Indians, this wide variety of vegetables, fruits, grains and spices used in various Indian sub-cuisines can be mind boggling because of the variety of region-specific names used for identifying the food items. Indian vegetable markets and grocery stores get their wholesale supplies from suppliers belonging to various regions/ethnicities from all over India and elsewhere, and the food suppliers/packagers mostly use sub-ethnic, region-specific item/ingredient names on the respective signs/labels used to identify specific vegetables, fruits, grains and spices based on their respective regions of origin. This further aggravates the confusion in identifying specific items/ingredients, especially for international consumers/expatriates looking to procure vegetables, fruits, grains and spices specific to Indian sub-cuisines.

This article attempts to centralize, compile and tabulate the various vegetables, fruits, grains and spices that are commonly employed in various Indian sub-cuisines to help reduce this confusion in identifying and procuring various Indian food ingredients, especially in the cross-regional, international markets/contexts.

The following is a list of common Indian ingredients, as well as their names in the various local languages spoken.

{{DEFAULTSORT:Multilingual List Of Indian Vegetables, Spices & Grains}} Category:Indian cuisine Category:Languages of India Category:India-related lists Category:Indian ingredients

'Indian animals and birds are vary in name from region to region within the country. The following is a list of common wild and domestic, Indian animals, birds and fish and their names in the various local languages spoken.

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