Comparison Formula

The Formula

a - b > 0 \implies a > b; a - b < 0 \implies a < b; a - b = 0 \implies a = b

When to use: Which is bigger? Which is smaller? Are they the same? Comparison answers these questions with precision.

Quick Example

5 > 3 (five is greater than three), 2.5 < 3 (2.5 is less than 3)

Notation

< (less than), > (greater than), = (equal to), \leq (less than or equal), \geq (greater than or equal)

What This Formula Means

Determining how two quantities relate in terms of size or value, using the symbols <, >, or =.

Which is bigger? Which is smaller? Are they the same? Comparison answers these questions with precision.

Formal View

For a, b \in \mathbb{R}: a > b \iff a - b > 0; a < b \iff b - a > 0; a = b \iff a - b = 0. The relation \leq is a total order (reflexive, antisymmetric, transitive, total).

Worked Examples

Example 1

easy
Compare \dfrac{5}{8} and \dfrac{3}{5}. Which is greater?

Solution

  1. 1
    Find a common denominator: \text{lcm}(8,5) = 40.
  2. 2
    Convert: \dfrac{5}{8} = \dfrac{25}{40} and \dfrac{3}{5} = \dfrac{24}{40}.
  3. 3
    Since 25 > 24, we have \dfrac{25}{40} > \dfrac{24}{40}, so \dfrac{5}{8} > \dfrac{3}{5}.

Answer

\dfrac{5}{8} > \dfrac{3}{5}
To compare fractions with different denominators, rewrite them with a common denominator, then compare numerators directly. The fraction with the larger numerator is the greater fraction.

Example 2

medium
Place -\dfrac{7}{3}, -2.4, -\dfrac{5}{2}, and -2 in order from least to greatest.

Common Mistakes

  • Comparing fractions by looking only at numerators — \frac{1}{3} is not greater than \frac{1}{5} just because '3 is less'; actually \frac{1}{3} > \frac{1}{5}, but you need common denominators to see why
  • Thinking -1 is greater than -10 'because 1 is smaller' for the wrong reason — -1 > -10 is true, but the reasoning must be about position on the number line
  • Mixing up the direction of < and > — writing 5 < 3 when meaning 5 > 3

Why This Formula Matters

Foundation for inequalities, optimization, and decision-making.

Frequently Asked Questions

What is the Comparison formula?

Determining how two quantities relate in terms of size or value, using the symbols <, >, or =.

How do you use the Comparison formula?

Which is bigger? Which is smaller? Are they the same? Comparison answers these questions with precision.

What do the symbols mean in the Comparison formula?

< (less than), > (greater than), = (equal to), \leq (less than or equal), \geq (greater than or equal)

Why is the Comparison formula important in Math?

Foundation for inequalities, optimization, and decision-making.

What do students get wrong about Comparison?

Comparing negative numbers or fractions with different denominators.

What should I learn before the Comparison formula?

Before studying the Comparison formula, you should understand: more less.

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