7.7.2: Key Concepts

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

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 $a \cdot b = b \cdot a$
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 $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

Distributive Property:
- If a,b,ca,b,c are real numbers then
  - $a (b + c) = a b + a c$
  - $(b + c) a = b a + c a$
  - $a (b - c) = a b - a c$

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: $a \cdot 1 = a 1 \cdot a = a$ 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: $(a \neq 0) a \cdot \frac{1}{a} = 1 \frac{1}{a}$ is the multiplicative inverse of a
Properties of Zero
- Multiplication by Zero: For any real number a, $\begin{matrix} a \cdot 0 = 0 & 0 \cdot a = & 0 & The & product of any number and 0 is 0. \end{matrix}$
- Division of Zero: For any real number a, $\begin{matrix} \frac{0}{a} = & 0 & Zero & divided by any real number, except itself, is zero. \end{matrix}$
- Division by Zero: For any real number a, $\frac{a}{0}$ is undefined and $a \div 0$ is undefined. Division by zero is undefined.

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