{
    "name": "Complex Number Modulus Calculator",
    "url": "https://calculators.geekmidst.com/calculators/complex-number-modulus-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/complex-number-modulus-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the modulus (absolute value) of a complex number as an exact simplified surd and a decimal, plus its conjugate, unit vector and argument.",
    "summary": "The modulus |a + bi| = √(a² + b²) is the distance of the number from 0 on the complex plane. It is given here exactly, such as 3√5, and as a decimal.",
    "also_called": [
        "Complex absolute value calculator",
        "Complex magnitude calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "a",
            "label": "Real part a",
            "type": "number",
            "default": "3"
        },
        {
            "name": "b",
            "label": "Imaginary part b",
            "type": "number",
            "default": "6"
        }
    ],
    "outputs": [
        {
            "key": "modulus",
            "label": "Modulus |z|",
            "format": "number"
        },
        {
            "key": "exact",
            "label": "Exact value",
            "format": "text"
        },
        {
            "key": "modulus_sq",
            "label": "|z|² = z·z̄",
            "format": "number"
        },
        {
            "key": "conjugate",
            "label": "Conjugate z̄",
            "format": "text"
        },
        {
            "key": "unit",
            "label": "Unit complex number z ÷ |z|",
            "format": "text"
        },
        {
            "key": "arg_deg",
            "label": "Argument",
            "format": "number"
        }
    ],
    "formula": {
        "text": "|a + bi| = √(a² + b²);  z̄ = a − bi;  z·z̄ = a² + b²",
        "variables": {
            "a": "the real part",
            "b": "the imaginary part"
        }
    },
    "worked_example": {
        "inputs": {
            "a": "3",
            "b": "6"
        },
        "steps": [
            "a² + b² = 9 + 36 = 45.",
            "√45 = √(9 × 5) = 3√5 ≈ 6.7082039325.",
            "The conjugate flips the sign of the imaginary part: 3 − 6i."
        ],
        "result": "|3 + 6i| = 3√5 ≈ 6.7082."
    },
    "interpretation": "The modulus is always zero or positive and equals 0 only for z = 0. Dividing z by its modulus gives the unit complex number with the same direction.",
    "assumptions": [
        "The exact form is given when a and b are integers or terminating decimals; otherwise the decimal is shown."
    ],
    "limitations": [
        "The exact surd is simplified for a² + b² (after clearing decimals) up to 10¹³."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How is the modulus related to the Pythagorean theorem?",
            "answer": "a and b are the legs of a right triangle on the complex plane and |z| is its hypotenuse."
        },
        {
            "question": "Why is z times its conjugate real?",
            "answer": "(a + bi)(a − bi) = a² − (bi)² = a² + b², which is |z|²."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Complex Number Polar Form Converter",
            "url": "https://calculators.geekmidst.com/calculators/complex-number-polar-conversion-calculator"
        },
        {
            "name": "Complex Number Addition Calculator",
            "url": "https://calculators.geekmidst.com/calculators/complex-number-addition-calculator"
        },
        {
            "name": "Pythagorean Theorem Calculator",
            "url": "https://calculators.geekmidst.com/calculators/pythagorean-theorem-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}