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Prealgebra 2e

Key Concepts

Prealgebra 2eKey Concepts

Key Concepts

7.1 Rational and Irrational Numbers

  • Real numbers
    • The image shows a large rectangle labeled “Real Numbers”. The rectangle is split in half vertically. The right half is labeled “Irrational Numbers”. The left half is labeled “Rational Numbers” and contains three concentric rectangles. The outer most rectangle is labeled “Integers”, the next rectangle is “Whole Numbers” and the inner most rectangle is “Natural Numbers”.

7.2 Commutative and Associative Properties

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

7.3 Distributive Property

  • Distributive Property:
    • If a,b,ca,b,c are real numbers then
      • a(b+c)=ab+aca(b+c)=ab+ac
      • (b+c)a=ba+ca(b+c)a=ba+ca
      • a(b-c)=ab-aca(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=a0+a=aa+0=a0+a=a 0 is the additive identity
    • Identity Property of Multiplication: For any real number a: a⋅1=a1⋅a=aa⋅1=a1⋅a=a 1 is the multiplicative identity
  • Inverse Properties
    • Inverse Property of Addition: For any real number a: a+(-a)=0-aa+(-a)=0-a is the additive inverse of a
    • Inverse Property of Multiplication: For any real number a: (a≠0)a⋅1a=11a(a≠0)a⋅1a=11a is the multiplicative inverse of a
  • Properties of Zero
    • Multiplication by Zero: For any real number a, a⋅0=00⋅a=0The product of any number and 0 is 0. a⋅0=00⋅a=0The product of any number and 0 is 0.
    • Division of Zero: For any real number a, 0a=0Zero divided by any real number, except itself, is zero. 0a=0Zero divided by any real number, except itself, is zero.
    • Division by Zero: For any real number a, a0a0 is undefined and a÷0a÷0 is undefined. Division by zero is undefined.
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