Basic Rules and Properties of Algebra

Addition

\[ a + b = b + a \]

Examples:

• Real numbers: \( 2 + 3 = 3 + 2 \)
• Algebraic expressions: \( x^2 + x = x + x^2 \)

Multiplication

\[ a \times b = b \times a \]

Examples:

• Real numbers: \( 5 \times 7 = 7 \times 5 \)
• Algebraic expressions: \( (x^3 - 2) \times x = x \times (x^3 - 2) \)

Addition

\[ (a + b) + c = a + (b + c) \]

Examples:

• Real numbers: \( (2 + 3) + 6 = 2 + (3 + 6) \)
• Algebraic expressions: \( (x^3 + 2x) + x = x^3 + (2x + x) \)

Multiplication

\[ (a \times b) \times c = a \times (b \times c) \]

Examples:

• Real numbers: \( (7 \times 3) \times 10 = 7 \times (3 \times 10) \)
• Algebraic expressions: \( (x^2 \times 5x) \times x = x^2 \times (5x \times x) \)

Addition and Multiplication

\[ a \times (b + c) = a \times b + a \times c \]

and

\[ (a + b) \times c = a \times c + b \times c \]

Examples:

• Real numbers: \( 2 \times (2 + 8) = 2 \times 2 + 2 \times 8 \)
• Real numbers: \( (2 + 8) \times 10 = 2 \times 10 + 8 \times 10 \)
• Algebraic expressions: \( x \times (x^4 + x) = x \times x^4 + x \times x \)
• Algebraic expressions: \( (x^4 + x) \times x^2 = x^4 \times x^2 + x \times x^2 \)

The Additive Inverse

The additive inverse of a number \( a \) is \( -a \). When you add a number to its inverse, the result is zero.

\[ a + (-a) = 0 \]

Example:

• The additive inverse of \( -6 \) is \( -(-6) = 6 \), and \( -6 + 6 = 0 \)

The Reciprocal (Multiplicative Inverse)

The reciprocal of a non-zero real number \( a \) is \( \frac{1}{a} \). Multiplying a number by its reciprocal yields one.

\[ a \times \left(\frac{1}{a}\right) = 1 \]

Example:

• The reciprocal of \( 5 \) is \( \frac{1}{5} \), and \( 5 \times \left(\frac{1}{5}\right) = 1 \)

The Additive Identity

The additive identity is \( 0 \). Adding zero to any number leaves it unchanged.

\[ a + 0 = 0 + a = a \]

The Multiplicative Identity

The multiplicative identity is \( 1 \). Multiplying any number by one leaves it unchanged.

\[ a \times 1 = 1 \times a = a \]