Artículos sobre Desarrollo Web, Tecnología y Matemáticas

Teorema fundamental de las fracciones

Publicado: 04/05/2019 15:14:38

Volver Teorema fundamental de las fracciones
Teorema: Si a, b, c y d son números reales cualesquiera, pero a y b no nulos, entonces:

    1. (a / b) * (c / d) = (a * c) / (b * d)

    2. (a / b) / (c / d) = (a / b) * (d / c), c != 0

    3. a / b + c / b = (a + c) / b

    4. a / b + c / d = [(a * d) + (b * c)] / (b * d)

    5. (a * c) / (b * c) = a / b, c != 0

    6. a / b = c / d => a * d = b * c

    7. -a / b = a / -b = -(a / b)

Demostración:

    1. (a / b) * (c / d) = (a * b ^ -1) * (c * d ^ -1) =
        (a * c) * (b ^ -1 * d ^ -1) = (a * c) * (b * d) ^ -1 =
        (a * c) / (b * d). []

    2. (a / b) / (c / d) = (a / b) * (c * d ^ -1) ^ -1 =
        (a / b) * (c ^ -1 * d) =
        (a / b) * (d / c). []

    3. a / b + c / b = a * 1 / b + c * 1 / b =
        (a + c) * 1 / b = (a + c) / b. []

    4. a / b + c / d = 1 * (a / b) + 1 * (c / d) =
        (d / d) * (a / b) + (b / b) * (c / d) =
        (a * d) / (b * d) + (b * c) / (b * d) =
        (a * d + b * c) / (b * d). []

    5. (a * c) / (b * c) = (a / b) * (c / c) =
        (a / b) * 1 = a / b. []

    6. a / b = c / d => (a / b) * (b * d) = (c / d) * (b * d) =>
        (a * d) * (b / b) = (b * c) * (d / d) =>
        a * d * 1 = b * c * 1 =>
        a * d = b * c. []

    7. -a / b =
        -a * b ^ -1 = -1 * a * b ^ -1 =
        a * (-1 * b ^ -1) = a * ((-1) ^ -1 * b ^ -1) =
        a * (-1 * b) ^ -1 = a * (-b) ^ -1 =
        a / -b =
        -1 * (a * b ^ -1) = -1 * a / b =
        -(a / b). []

Volver