Comparison Formula

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

The Formula

a−b>0  ⟹  a>ba - b > 0 \implies a > b; a−b<0  ⟹  a<ba - b < 0 \implies a < b; a−b=0  ⟹  a=ba - 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>35 > 3 (five is greater than three), 2.5<32.5 < 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∈Ra, b \in \mathbb{R}: a>b  ⟺  a−b>0a > b \iff a - b > 0; a<b  ⟺  b−a>0a < b \iff b - a > 0; a=b  ⟺  a−b=0a = b \iff a - b = 0. The relation ≤\leq is a total order (reflexive, antisymmetric, transitive, total).

Worked Examples

Example 1

easy
Compare 58\dfrac{5}{8} and 35\dfrac{3}{5}. Which is greater?

Answer

58>35\dfrac{5}{8} > \dfrac{3}{5}

First step

1
Find a common denominator: lcm(8,5)=40\text{lcm}(8,5) = 40.

Full solution

  1. 2
    Convert: 58=2540\dfrac{5}{8} = \dfrac{25}{40} and 35=2440\dfrac{3}{5} = \dfrac{24}{40}.
  2. 3
    Since 25>2425 > 24, we have 2540>2440\dfrac{25}{40} > \dfrac{24}{40}, so 58>35\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 −73-\dfrac{7}{3}, −2.4-2.4, −52-\dfrac{5}{2}, and −2-2 in order from least to greatest.

Example 3

easy
You have 3 cookies. Your friend has 5 cookies. Who has more?

Common Mistakes

  • Pointing the symbol the wrong way - the open end always faces the larger number (5<85 < 8).
  • Comparing different forms without converting - put fractions, decimals, percents in one form first.
  • Using > or < when the values are equal - if they match exactly, the symbol is =.

Why This Formula Matters

Comparison turns the informal 'more/less' idea into the symbolic <<, >>, == language that all of inequalities and ordering depend on. Getting the symbol direction right (open end faces the bigger number) is a small habit that prevents endless inequality errors. Recognizing it by "Am I relating two values with one symbol, with the open end facing the larger?" — rather than by familiar numbers — is what lets a student tell it apart from ordering numbers and more and less and inequalities in a mixed problem set.

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?

Comparison turns the informal 'more/less' idea into the symbolic <<, >>, == language that all of inequalities and ordering depend on. Getting the symbol direction right (open end faces the bigger number) is a small habit that prevents endless inequality errors. Recognizing it by "Am I relating two values with one symbol, with the open end facing the larger?" — rather than by familiar numbers — is what lets a student tell it apart from ordering numbers and more and less and inequalities in a mixed problem set.

What do students get wrong about Comparison?

The procedure for comparison is the easy part; the trap is pointing the symbol the wrong way. Asking "Am I relating two values with one symbol, with the open end facing the larger?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Comparison formula?

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

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