{
    "api": "YAML JSON TOON Database",
    "version": "1.0.0",
    "format": "json",
    "dataset": {
        "slug": "common-mathematical-constants",
        "title": "Common Mathematical Constants",
        "description": "Widely used mathematical constants and their approximate values.",
        "summary": "The mathematical constants that recur across fields, each with its symbol, a decimal value to ten places and what it measures: pi, e, the golden ratio, square roots, the imaginary unit, and further constants such as tau, the Euler-Mascheroni constant, Catalan's constant, Apery's constant and the Feigenbaum constant.",
        "use_when": [
            "You need a constant's value or its symbol and meaning.",
            "You want to cite a constant with an explicit number of digits."
        ],
        "caveats": [
            "Values are rounded to ten decimal places and are approximations; most constants are irrational.",
            "For some constants (the Euler-Mascheroni constant, Catalan's constant) it is not known whether they are irrational."
        ],
        "related": [
            "geometric-formulas",
            "trigonometric-identities",
            "statistics-basics"
        ],
        "category": "Mathematics",
        "category_slug": "mathematics",
        "tags": "math,constants,reference",
        "view_count": 0,
        "entry_count": 17,
        "fields": [
            {
                "section": "data",
                "field": "name",
                "type": "string",
                "example": "Pi"
            },
            {
                "section": "data",
                "field": "symbol",
                "type": "string",
                "example": "pi"
            },
            {
                "section": "data",
                "field": "value",
                "type": "number|string",
                "example": "3.1415926535"
            },
            {
                "section": "data",
                "field": "description",
                "type": "string",
                "example": "Ratio of a circle circumference to its diameter."
            }
        ]
    },
    "data": [
        {
            "name": "Pi",
            "symbol": "pi",
            "value": 3.1415926535,
            "description": "Ratio of a circle circumference to its diameter."
        },
        {
            "name": "Euler's Number",
            "symbol": "e",
            "value": 2.7182818284,
            "description": "Base of the natural logarithm."
        },
        {
            "name": "Golden Ratio",
            "symbol": "phi",
            "value": 1.6180339887,
            "description": "Ratio where the sum of two quantities divided by the larger equals the larger divided by the smaller."
        },
        {
            "name": "Square Root of 2",
            "symbol": "sqrt(2)",
            "value": 1.4142135623,
            "description": "Positive algebraic number whose square is 2."
        },
        {
            "name": "Square Root of 3",
            "symbol": "sqrt(3)",
            "value": 1.7320508075,
            "description": "Positive algebraic number whose square is 3."
        },
        {
            "name": "Natural Logarithm of 2",
            "symbol": "ln(2)",
            "value": 0.6931471805,
            "description": "Natural logarithm of 2."
        },
        {
            "name": "Imaginary Unit",
            "symbol": "i",
            "value": "sqrt(-1)",
            "description": "Defined by the property that its square is -1."
        },
        {
            "name": "Tau",
            "symbol": "tau",
            "value": 6.2831853072,
            "description": "Ratio of a circle circumference to its radius; equal to 2 pi."
        },
        {
            "name": "Euler-Mascheroni Constant",
            "symbol": "gamma",
            "value": 0.5772156649,
            "description": "Limiting difference between the harmonic series and the natural logarithm."
        },
        {
            "name": "Square Root of 5",
            "symbol": "sqrt(5)",
            "value": 2.2360679775,
            "description": "Appears in the golden ratio, which equals (1 + sqrt(5)) / 2."
        },
        {
            "name": "Natural Logarithm of 10",
            "symbol": "ln(10)",
            "value": 2.302585093,
            "description": "Converts between natural and base-10 logarithms."
        },
        {
            "name": "Catalan's Constant",
            "symbol": "G",
            "value": 0.9159655942,
            "description": "Alternating sum of the reciprocals of odd squares; appears in combinatorics and number theory."
        },
        {
            "name": "Apery's Constant",
            "symbol": "zeta(3)",
            "value": 1.2020569032,
            "description": "Value of the Riemann zeta function at 3; Apery proved it is irrational."
        },
        {
            "name": "Silver Ratio",
            "symbol": "delta_S",
            "value": 2.4142135624,
            "description": "Equal to 1 + sqrt(2); the continued fraction [2; 2, 2, 2, ...]."
        },
        {
            "name": "Plastic Number",
            "symbol": "rho",
            "value": 1.3247179572,
            "description": "The real root of x^3 = x + 1, the smallest Pisot number."
        },
        {
            "name": "Khinchin's Constant",
            "symbol": "K",
            "value": 2.6854520011,
            "description": "Geometric mean of the continued-fraction coefficients of almost every real number."
        },
        {
            "name": "Feigenbaum Constant delta",
            "symbol": "delta",
            "value": 4.6692016091,
            "description": "Limiting ratio of successive period-doubling bifurcation intervals in many chaotic systems."
        }
    ]
}