"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { em: "em", h3: "h3", li: "li", ol: "ol", p: "p", path: "path", span: "span", strong: "strong", svg: "svg", table: "table", tbody: "tbody", td: "td", th: "th", thead: "thead", tr: "tr", ul: "ul", ..._provideComponents(), ...props.components }, {ComparisonTable, Definition, English, Explanation, Hinglish, ImportantQA, MCQ, SubTopicHeader, Summary} = _components; if (!ComparisonTable) _missingMdxReference("ComparisonTable", true); if (!Definition) _missingMdxReference("Definition", true); if (!English) _missingMdxReference("English", true); if (!Explanation) _missingMdxReference("Explanation", true); if (!Hinglish) _missingMdxReference("Hinglish", true); if (!ImportantQA) _missingMdxReference("ImportantQA", true); if (!MCQ) _missingMdxReference("MCQ", true); if (!SubTopicHeader) _missingMdxReference("SubTopicHeader", true); if (!Summary) _missingMdxReference("Summary", true); return _jsxs(_Fragment, { children: [_jsx(SubTopicHeader, { id: "4.3", title: "Bakhshali Iterative Method (IKS)" }), "\n", _jsxs(Definition, { term: "Bakhshali Method", children: [_jsx(English, { children: _jsx(_components.p, { children: "The Bakhshali Method is an ancient Indian mathematical technique found in the Bakhshali Manuscript (written on birch bark, dated around the 3rd-4th century CE) for calculating the approximate square root of a non-square number. It is extremely efficient, yielding accuracy equivalent to two iterations of the Newton-Raphson method in a single compound step." }) }), _jsx(Hinglish, { children: _jsx(_components.p, { children: "Bakhshali Method ek ancient Indian mathematical formula hai jo Bakhshali Manuscript (birch bark yani bhojpatra par likhi gayi 3rd-4th century CE ki book) se mila hai. Iska use non-square numbers (jaise 10, 29) ka approximate square root nikalne ke liye kiya jata hai. Yeh method itna efficient hai ki ek hi step mein Newton-Raphson ke do steps jitna accurate answer de deta hai." }) })] }), "\n", _jsxs(Explanation, { children: [_jsxs(English, { children: [_jsx(_components.h3, { children: "The Bakhshali Square Root Formula" }), _jsxs(_components.p, { children: ["To find the square root of a non-square number ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) }) }), ":"] }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Step 1: Identify the nearest perfect square:" }), " Let ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) }) }), " be the perfect square closest to ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) }) }), ". ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) }) }), " is its square root."] }), "\n", _jsxs(_components.li, { children: [_jsxs(_components.strong, { children: ["Step 2: Find the difference (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) }) }), "):"] }), " Calculate the difference between ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) }) }), " and ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) }) }), ":\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", 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[_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] }) }), "\n", _jsxs(_components.em, { children: ["(Note: ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) }) }), " can be positive or negative)."] })] }), "\n", _jsxs(_components.li, { children: [_jsxs(_components.strong, { children: ["Step 3: Calculate the first adjustment (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" })] }) }) }), "):"] }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2251em", 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}), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "b" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: [_jsxs(_components.strong, { children: ["Step 4: Calculate the first approximation (", _jsx(_components.span, { className: "katex", children: 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})] })] }) }) }), "):"] }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7519em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord 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}) })] }), _jsxs(_components.span, { style: { top: "-3.2255em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line mtight", style: { borderBottomWidth: "0.049em" } })] }), _jsxs(_components.span, { style: { top: "-3.384em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "b" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.344em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter sizing reset-size3 size6" })] }), _jsx(_components.span, { className: "mclose sizing reset-size3 size6 mtight delimcenter", style: { top: "0.075em" }, children: _jsx(_components.span, { className: "mtight", children: ")" }) })] }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.1039em" }, children: _jsxs(_components.span, { style: { top: "-3.1436em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.81em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) })] }), "\n"] }), _jsx(_components.h3, { children: "Solved Example: Find the square root of 10" }), _jsxs(_components.p, { children: ["Let ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "10" })] })] }) }), ":"] }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The nearest perfect square is ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "9" })] }) }) }), ", so ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6684em", verticalAlign: "-0.024em" } }), _jsx(_components.span, { className: "mord", children: "9" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "⟹" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "3" })] })] }) }), "."] }), "\n", _jsxs(_components.li, { children: ["The difference ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "10" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "9" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] }) }), "."] }), "\n", _jsxs(_components.li, { children: ["Calculate ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2251em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: 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"0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "721" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "3.16228" })] })] }) })] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "Comparison:" }), " Actual ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.04em", verticalAlign: "-0.1328em" } }), _jsx(_components.span, { className: "mord sqrt", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9072em" }, children: [_jsxs(_components.span, { className: "svg-align", style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", style: { paddingLeft: "0.833em" }, children: _jsx(_components.span, { className: "mord", children: "10" }) })] }), _jsxs(_components.span, { style: { top: "-2.8672em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "hide-tail", style: { minWidth: "0.853em", height: "1.08em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "400em", height: "1.08em", viewBox: "0 0 400000 1080", preserveAspectRatio: "xMinYMin slice", children: _jsx(_components.path, { d: "M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1328em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "3.162277" })] })] }) }), ". The Bakhshali approximation is accurate up to 4 decimal places in a single step!"] }), "\n"] })] }), _jsxs(Hinglish, { children: [_jsx(_components.h3, { children: "Bakhshali Square Root Formula Ke Steps" }), _jsxs(_components.p, { children: ["Kisi non-square number ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) }) }), " ka square root nikalne ke liye:"] }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Step 1: Sabse paas ka perfect square check karein:" }), " Let ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) }) }), " sabse paas ka perfect square hai aur ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) }) }), " uska root hai."] }), "\n", _jsxs(_components.li, { children: [_jsxs(_components.strong, { children: ["Step 2: Difference (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) }) }), ") nikalein:"] }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] }) }), "\n", _jsxs(_components.em, { children: ["(Note: ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) }) }), " positive ya negative dono ho sakta hai)."] })] }), "\n", _jsxs(_components.li, { children: [_jsxs(_components.strong, { children: ["Step 3: First correction parameter (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" })] }) }) }), ") nikalein:"] }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2251em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8801em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "A" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "b" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: [_jsxs(_components.strong, { children: ["Step 4: First approximation (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7519em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] })] }) }) }), ") nikalein:"] }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7519em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2251em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8801em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "A" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "b" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Step 5: Final Bakhshali approximation value:" }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.04em", verticalAlign: "-0.2106em" } }), _jsx(_components.span, { className: "mord sqrt", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8294em" }, children: [_jsxs(_components.span, { className: "svg-align", style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", style: { paddingLeft: "0.833em" }, children: _jsx(_components.span, { className: "mord mathnormal", children: "Q" }) })] }), _jsxs(_components.span, { style: { top: "-2.7894em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "hide-tail", style: { minWidth: "0.853em", height: "1.08em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "400em", height: "1.08em", viewBox: "0 0 400000 1080", preserveAspectRatio: "xMinYMin slice", children: _jsx(_components.path, { d: "M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2106em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8352em", verticalAlign: "-0.0833em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.3629em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.0179em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6828em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) })] }), "\n"] }), _jsxs(_components.h3, { children: ["Solved Example: ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.04em", verticalAlign: "-0.1328em" } }), _jsx(_components.span, { className: "mord sqrt", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9072em" }, children: [_jsxs(_components.span, { className: "svg-align", style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", style: { paddingLeft: "0.833em" }, children: _jsx(_components.span, { className: "mord", children: "10" }) })] }), _jsxs(_components.span, { style: { top: "-2.8672em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "hide-tail", style: { minWidth: "0.853em", height: "1.08em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "400em", height: "1.08em", viewBox: "0 0 400000 1080", preserveAspectRatio: "xMinYMin slice", children: _jsx(_components.path, { d: "M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1328em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }) }), " calculate karein"] }), _jsxs(_components.p, { children: ["Let ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "10" })] })] }) }), ":"] }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Nearest perfect square ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", 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}), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1901em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8451em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: 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_jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "3.16228" })] })] }) })] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "Comparison:" }), " Real value ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.04em", verticalAlign: "-0.1328em" } }), _jsx(_components.span, { className: "mord sqrt", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: 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"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1328em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } 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Bakhshali method se single step mein hi 4 decimal places tak correct answer mil gaya!"] }), "\n"] })] }), _jsx(ComparisonTable, { children: _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsxs(_components.th, { style: { textAlign: "left" }, children: ["Number ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) }) })] }), _jsxs(_components.th, { style: { textAlign: "left" }, children: ["Nearest Square ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) }) })] }), _jsx(_components.th, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) }) }) }), _jsx(_components.th, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { 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children: _jsx(_components.span, { className: "mord mtight", children: "18" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "0.0556" })] })] }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "8.94427" })] }) }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "8.944271" })] }) }) }) })] })] })] }) })] }), "\n", _jsxs(Summary, { children: [_jsx(English, { children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Bakhshali method is an ancient Indian numerical root-finding algorithm." }), "\n", _jsx(_components.li, { children: "It is quadratically convergent and equivalent to two steps of Newton-Raphson." }), "\n", _jsxs(_components.li, { children: ["Works for both positive differences (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] }) }), ") and negative differences (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] }) }), ")."] }), "\n"] }) }), _jsx(Hinglish, { children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Bakhshali method ancient Indian mathematics (IKS) ka ek highly accurate square root solver hai." }), "\n", _jsx(_components.li, { children: "Yeh equivalent to double-iteration Newton-Raphson hai aur tezi se precise answer deta hai." }), "\n", _jsxs(_components.li, { children: ["Yeh method positive difference (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] }) }), ") aur negative difference (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] }) }), ") dono ke liye equal apply hota hai."] }), "\n"] }) })] }), "\n", _jsx(ImportantQA, { questions: [{ "q": "State the Bakhshali formula for finding the approximate square root of a non-square number.", "marks": 2, "a": { "components": [{ "type": "Explanation", "content": "The Bakhshali formula to approximate the square root of a non-square number $Q$ (where $A^2$ is the nearest perfect square and $b = Q - A^2$) is:\n$$\\sqrt{Q} \\approx A' - \\frac{P^2}{2A'}$$\nwhere:\n- $P = \\frac{b}{2A}$\n- $A' = A + P$Bakhshali formula kisi non-square number $Q$ ka approximate square root nikalne ke liye (jahan $A^2$ nearest perfect square hai aur $b = Q - A^2$):\n$$\\sqrt{Q} \\approx A' - \\frac{P^2}{2A'}$$\nwhere:\n- $P = \\frac{b}{2A}$\n- $A' = A + P$" }] } }, { "q": "Calculate the approximate square root of 29 using the Bakhshali method.", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Given: $Q = 29$.\n1. Nearest perfect square is $25$. So $A^2 = 25 \\implies A = 5$.\n2. Difference $b = Q - A^2 = 29 - 25 = 4$.\n3. Calculate $P = \\frac{b}{2A} = \\frac{4}{2(5)} = 0.4$.\n4. Calculate $A' = A + P = 5 + 0.4 = 5.4$.\n5. Apply Bakhshali formula:\n $$\\sqrt{29} \\approx A' - \\frac{P^2}{2A'}$$\n $$\\sqrt{29} \\approx 5.4 - \\frac{(0.4)^2}{2(5.4)}$$\n $$= 5.4 - \\frac{0.16}{10.8} = 5.4 - 0.01481 = 5.38519$$.\n(Actual root of 29 is $\\approx 5.38516$).Diya hai: $Q = 29$.\n1. Nearest perfect square $25$ hai, toh $A^2 = 25 \\implies A = 5$.\n2. Difference $b = 29 - 25 = 4$.\n3. $P = \\frac{b}{2A} = \\frac{4}{10} = 0.4$.\n4. $A' = A + P = 5 + 0.4 = 5.4$.\n5. Bakhshali formula lagane par:\n $$\\sqrt{29} \\approx 5.4 - \\frac{(0.4)^2}{2(5.4)} = 5.4 - 0.01481 = 5.38519$$.\n(Real root of 29 is $\\approx 5.38516$)." }] } }, { "q": "Find the approximate square root of 80 using the Bakhshali method.", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Given: $Q = 80$.\n1. Nearest perfect square is $81$. So $A^2 = 81 \\implies A = 9$.\n2. Difference $b = Q - A^2 = 80 - 81 = -1$ (negative difference).\n3. Calculate $P = \\frac{b}{2A} = \\frac{-1}{2(9)} = -\\frac{1}{18} \\approx -0.05556$.\n4. Calculate $A' = A + P = 9 - 0.05556 = 8.94444$.\n5. Apply Bakhshali formula:\n $$\\sqrt{80} \\approx A' - \\frac{P^2}{2A'}$$\n $$\\sqrt{80} \\approx 8.94444 - \\frac{(-0.05556)^2}{2(8.94444)}$$\n $$= 8.94444 - \\frac{0.003086}{17.88889} = 8.94444 - 0.00017 = 8.94427$$.\n(Actual root of 80 is $\\approx 8.944271$).Diya hai: $Q = 80$.\n1. Nearest perfect square $81$ hai, toh $A^2 = 81 \\implies A = 9$.\n2. Difference $b = 80 - 81 = -1$ (Negative difference).\n3. $P = \\frac{-1}{2(9)} = -\\frac{1}{18} \\approx -0.05556$.\n4. $A' = A + P = 9 - 0.05556 = 8.94444$.\n5. Bakhshali formula lagane par:\n $$\\sqrt{80} \\approx 8.94444 - \\frac{(-0.05556)^2}{2(8.94444)} = 8.94427$$.\n(Real root of 80 is $\\approx 8.944271$)." }] } }] }), "\n", _jsx(MCQ, { questions: [{ "id": "2044017e-79f8-464d-803b-4eb654fed42f", "q": "The Bakhshali method is an ancient technique from which country?", "options": ["Greece", "India", "Egypt", "China"], "answer": 1, "hint": "It is part of the Indian Knowledge Systems (IKS) in mathematics.", "explanation": "The Bakhshali method originated in ancient India, documented in the Bakhshali Manuscript (written on birch bark)." }, { "id": "8a8b1ed9-f8f9-4c97-9cc6-6ab4a4c8003f", "q": "What is the Bakhshali method primarily used to calculate?", "options": ["Approximate roots of polynomials", "Approximate square roots of non-square numbers", "Exact values of pi", "Solutions of simultaneous equations"], "answer": 1, "hint": "It calculates numbers like sqrt(10) or sqrt(29).", "explanation": "The Bakhshali method is specifically designed for approximating the square root of a positive real number that is not a perfect square." }, { "id": "8eddcd0a-5187-498b-873d-aaba7a9d8bae", "q": "If we want to find the square root of 27, what is the best value for A^2?", "options": ["16", "36", "25", "49"], "answer": 2, "hint": "Find the perfect square closest to 27.", "explanation": "The perfect square closest to 27 is 25 (since 27 - 25 = 2, which is closer than 36 - 27 = 9). So A^2 = 25." }, { "id": "ed62c695-c90a-409a-8466-960fdb4e8863", "q": "What is the relation between the Bakhshali method and the Newton-Raphson method?", "options": ["Bakhshali is equivalent to one Newton-Raphson iteration", "Bakhshali is equivalent to two Newton-Raphson iterations", "They have completely different convergence rates", "Newton-Raphson is slower than Bakhshali"], "answer": 1, "hint": "Bakhshali is equivalent to double-iteration of the Babylonian (Newton) method.", "explanation": "Mathematically, one single compound step of the Bakhshali method provides the exact same accuracy as two full successive iterations of the Newton-Raphson method." }, { "id": "2ed94926-8e51-4375-8bf5-6d433dc35897", "q": "If Q = 10 and A = 3, what is the value of the difference parameter 'b'?", "options": ["1", "-1", "2", "0.5"], "answer": 0, "hint": "b = Q - A^2.", "explanation": "b = Q - A^2 = 10 - 3^2 = 10 - 9 = 1." }, { "id": "a1907430-3ccc-46cf-aea7-94c790580f5a", "q": "In Bakhshali method, if P = b/(2A) and A' = A + P, what is the final formula for the square root of Q?", "options": ["A' + P^2 / (2A')", "A' - P^2 / (2A')", "A' - P / (2A')", "A' + P / (2A')"], "answer": 1, "hint": "It subtracts a second-order adjustment from the first approximation A'.", "explanation": "The Bakhshali formula states that the square root is approximately equal to A' - P^2 / (2A')." }, { "id": "0655beb5-1c3b-489a-8b41-abd8b8c6f319", "q": "For Q = 15, if we choose the nearest perfect square as 16, what is the value of 'b'?", "options": ["1", "-1", "2", "-2"], "answer": 1, "hint": "b = Q - A^2 where Q = 15 and A^2 = 16.", "explanation": "b = 15 - 16 = -1. In Bakhshali method, 'b' can be negative." }, { "id": "45bf59f5-bfc8-429b-8a1f-e8dd3eceb18b", "q": "Find the value of P if we want to calculate the square root of 26 using Bakhshali method.", "options": ["1/10", "1/5", "2/5", "1/26"], "answer": 0, "hint": "Q = 26, nearest square is 25 (A = 5). b = 26 - 25 = 1. Calculate b/(2A).", "explanation": "P = b/(2A) = 1/(2*5) = 1/10." }, { "id": "ac616b41-7e09-4035-befa-a5949f5906a1", "q": "What is the first approximation A' for finding the square root of 17 using Bakhshali method?", "options": ["4.125", "4.25", "4.0625", "4.5"], "answer": 0, "hint": "Q = 17, A = 4, b = 1. A' = A + b/(2A).", "explanation": "A' = 4 + 1/(2*4) = 4 + 1/8 = 4 + 0.125 = 4.125." }, { "id": "8304f3b8-1279-4d25-b813-2c1c5a69d883", "q": "Calculate the second adjustment term P^2/(2A') for Q = 10 (where A=3, P=1/6, A'=19/6).", "options": ["1/228", "1/72", "1/12", "1/19"], "answer": 0, "hint": "Calculate (1/36) / (2 * 19/6) = (1/36) / (19/3) = (1/36) * (3/19).", "explanation": "P^2 / (2A') = (1/6)^2 / (2 * 19/6) = (1/36) / (19/3) = 1/36 * 3/19 = 1/(12 * 19) = 1/228." }], topicId: "4.3" })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent }; function _missingMdxReference(id, component) { throw new Error("Expected " + (component ? "component" : "object") + " `" + id + "` to be defined: you likely forgot to import, pass, or provide it."); }