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Mathematics LibreTexts

7.S: The Properties of Real Numbers (Summary)

  • Page ID
    6903
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    Key Terms

    Additive Identity The additive identity is 0. When zero is added to any number, it does not change the value.
    Additive Inverse The opposite of a number is its additive inverse. The additive inverse of a is −a .
    Irrational number A number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
    Multiplicative Identity The multiplicative identity is 1. When one multiplies any number, it does not change the value.
    Multiplicative Inverse The reciprocal of a number is its multiplicative inverse. The multiplicative inverse of a is \(\frac{1}{a}\).
    Rational number A number that can be written in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0. Its decimal form stops or repeats.
    Real number A number that is either rational or irrational.

    Key Concepts

    7.1 - Rational and Irrational Numbers

    • Real numbers

    7.2 - Commutative and Associative Properties

    • Commutative Properties
      • Commutative Property of Addition: If a, b are real numbers, then a + b = b + a
      • Commutative Property of Multiplication: If a, b are real numbers, then a • b = b • a
    • Associative Properties
      • Associative Property of Addition: If a, b, c are real numbers then (a + b) + c = a + (b + c)
      • Associative Property of Multiplication: If a, b, c are real numbers then (a • b) • c = a • (b • c)

    7.3 - Distributive Property

    • Distributive Property:
      • If a, b, c are real numbers then 
        • a(b + c) = ab + ac
        • (b + c)a = ba + ca
        • a(b • c) = ab • ac

    7.4 - Properties of Identity, Inverses, and Zero

    • Identity Properties
      • Identity Property of Addition: For any real number a: a + 0 = a, 0 + a = a
        • 0 is the additive identity
      • Identity Property of Multiplication: For any real number a: a • 1 = a, 1 • a = a
        • 1 is the multiplicative identity
    • Inverse Properties
      • Inverse Property of Addition: For any real number a: a + (- a) = 0
        • - a is the additive inverse of a
      • Inverse Property of Multiplication: For any real number a: (a ≠ 0) a • \(\frac{1}{a}\) = 1
        • \(\frac{1}{a}\) is the multiplicative inverse of a
    • Properties of Zero
      • Multiplication by Zero: For any real number a, a • 0 = 0, 0 • a = 0
        • The product of any number and 0 is 0.
      • Division of Zero: For any real number a, 0 a = 0, 0 + a = 0
        • Zero divided by any real number, except itself, is zero.
      • Division by Zero: For any real number a, \(\frac{0}{a}\) is undefined and a ÷ 0 is undefined.
        • Division by zero is undefined.

    Contributors