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

$$x_0>0,a>0,\ \ x_{\text{n+1}}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)$$

This equation can be used to generate an infinite sequence of approximations $$x_0,\ x_1,\ x_2,\ x_3...\ $$ that gets closer and closer to $$\sqrt{a}$$

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

Proof
Let,
 * $$x_{\text{n+1}}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)$$

Therefore,

\begin{align} 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{align} $$

Similarlx,

Dividing $$ bx $$,

\begin{align} \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{align} $$

Therefore,

\begin{align} \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{align} $$



\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 $$x_{\text{0}} > 0\,\!$$ and $$\sqrt{a} > 0$$, we have $$\left | x_0-\sqrt{a} \right | < x_0+\sqrt{a}$$



\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,

\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.

Fruits
Category:Indian cuisine Category:Languages of India Category:India-related lists Category:Indian ingredients

Multilingual list of Indian animals and birds
'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.