1.9 Properties of Real Numbers

Real numbers have unique properties that make algebra work smoothly. These properties let us rearrange terms, simplify expressions, and solve equations with confidence. They're the building blocks of algebraic operations.

Understanding these properties is crucial for mastering algebra. From the commutative property to the distributive property, each rule plays a vital role in manipulating equations and expressions. They're the tools you'll use to tackle more complex math problems.

Properties of Real Numbers

Rearranging algebraic terms

• Commutative property allows swapping the order of terms in addition and multiplication without altering the result
  • Addition: $a + b = b + a$ (3 + 5 = 5 + 3)
  • Multiplication: $a \times b = b \times a$ (2 × 7 = 7 × 2)
• Associative property permits regrouping terms in addition and multiplication without changing the outcome
  • Addition: $(a + b) + c = a + (b + c)$ $((2 + 3) + 4 = 2 + (3 + 4))$
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$ $((2 × 3) × 4 = 2 × (3 × 4))$
• Combining commutative and associative properties enables flexible rearrangement and regrouping of terms in algebraic expressions
  • $3x + 2y + 5x = (3x + 5x) + 2y = 8x + 2y$ (rearranging and regrouping like terms)
• The closure property ensures that the result of addition or multiplication of real numbers is always a real number

Identity and inverse properties

• Identity property leaves the result unchanged when adding 0 or multiplying by 1
  • Addition: $a + 0 = a$ (7 + 0 = 7)
  • Multiplication: $a \times 1 = a$ (4 × 1 = 4)
• Inverse property results in the identity element when adding the additive inverse (negative) or multiplying by the multiplicative inverse (reciprocal)
  • Addition: $a + (-a) = 0$ (5 + (-5) = 0)
  • Multiplication: $a \times \frac{1}{a} = 1$ (for $a \neq 0$) $(6 \times \frac{1}{6} = 1)$
• Solving equations using identity and inverse properties
  1. Addition: $x + 5 = 10 \rightarrow x + 5 + (-5) = 10 + (-5) \rightarrow x = 5$
  2. Multiplication: $3x = 12 \rightarrow 3x \times \frac{1}{3} = 12 \times \frac{1}{3} \rightarrow x = 4$

Zero's unique mathematical properties

• Addition with zero results in the original number
  • $a + 0 = a$ (10 + 0 = 10)
• Multiplication by zero always yields zero
  • $a \times 0 = 0$ (-4 × 0 = 0)
• Division by zero is undefined and not allowed
  • Attempting to divide by zero leads to an undefined result or an error ($\frac{5}{0}$ is undefined)
• Zero as an exponent equals one for any non-zero base
  • $a^0 = 1$ (for $a \neq 0$) $(2^0 = 1)$

Distributive property in algebra

• Distributive property states that multiplying a factor by a sum is equivalent to multiplying the factor by each term in the sum and then adding the results
  • $a(b + c) = ab + ac$ $(2(3 + 4) = 2 \times 3 + 2 \times 4)$
• Expanding expressions involves multiplying the factor outside the parentheses by each term inside
  • $2(3x + 4) = 2 \times 3x + 2 \times 4 = 6x + 8$
• Simplifying expressions requires combining like terms after expanding
  • $3(2x + 1) + 4(x - 3) = 6x + 3 + 4x - 12 = 10x - 9$
• Factoring expressions reverses the distributive property by identifying the common factor and factoring it out
  • $5x + 10 = 5(x + 2)$ (factoring out the common factor of 5)

Additional Properties of Real Numbers

• The field axioms encompass the commutative, associative, distributive, and identity properties, forming the foundation of real number operations
• The density property states that between any two real numbers, there is always another real number
• The completeness property ensures that every non-empty set of real numbers with an upper bound has a least upper bound
• The trichotomy property states that for any two real numbers a and b, exactly one of the following is true: a < b, a = b, or a > b