Comparing Numbers: A Formal Algebraic Approach
🎯 Learning Objectives
- Understand the formal definition of comparing numbers using differences
- Apply comparison rules to integers, fractions, decimals, and algebraic expressions
- Use inequality symbols correctly and interpret them on a number line
- Solve real-world problems involving number comparison
📐 Formal Definition: Comparing Two Numbers
The comparison of two numbers is based entirely on the sign of their difference. For any two real numbers $a$ and $b$:
$a > b$
$$a - b > 0$$
The number $a$ is greater than $b$ if and only if the difference $a - b$ is positive.
$a < b$
$$a - b < 0$$
The number $a$ is less than $b$ if and only if the difference $a - b$ is negative.
$a = b$
$$a - b = 0$$
The number $a$ is equal to $b$ if and only if the difference $a - b$ equals zero.