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\(<\)
\(=\)
\(>\)

Ordering
Rational Numbers

Master comparing and ordering integers, fractions, and decimals on the number line: the foundation of algebraic reasoning.

Pre-Algebra

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). This includes integers, fractions, terminating decimals, and repeating decimals.

Integers
\(-5,\; 0,\; 7\) can be written as \(\frac{-5}{1},\; \frac{0}{1},\; \frac{7}{1}\)
Fractions
\(\frac{3}{4},\; -\frac{2}{5},\; \frac{11}{3}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\)

The Number Line: Your Best Tool

Every rational number has an exact position on the number line. Numbers increase as you move right and decrease as you move left.

\(-3\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(-1.5\)
\(0.5\)
\(1.5\)
 ← Smaller Larger →

Understanding Negative Numbers

Negative numbers sit to the left of zero. A critical idea: the more negative a number is, the smaller it is, even though the digit looks bigger.

\(-5\)
\(-3\)
\(0\)
\(2\)
\(4\)
\(-5\)
\(-3\)
\(-5 < -3\)

The Comparison Rule

To compare any two rational numbers \(p\) and \(q\):

1
Convert to a common form: either both to decimals, or both to fractions with a common denominator.
2
Compare the values. If decimals, compare digit by digit from left to right. If fractions with the same denominator, compare numerators.
3
Write the result using \(<\), \(>\), or \(=\).

Cross-Multiplication Shortcut

To compare \(\frac{a}{b}\) and \(\frac{c}{d}\) without finding a common denominator, cross-multiply and compare the products.

$$\frac{3}{4}$$
\(?\)
$$\frac{5}{7}$$
\(3 \times 7 = 21\)
\(5 \times 4 = 20\)
\(21 > 20, \text{ so } \frac{3}{4} > \frac{5}{7}\)

Comparing vs. Ordering

Comparing
Deciding the relationship between two numbers: is \(a < b\), \(a > b\), or \(a = b\)?
Ordering
Arranging three or more numbers from least to greatest (or greatest to least).

Comparing uses symbols: \(<\), \(>\), \(=\). Ordering uses a sorted list, sometimes written with \(<\) between values: \(-2 < -\frac{1}{2} < 0.3 < \frac{4}{5}\).

Step-by-Step: How to Order

1
Convert everything to decimals (or find a common denominator). Decimals are usually easier to compare.
2
Separate negatives from positives. All negatives are less than all positives.
3
Order each group. For negatives: the number closest to zero is greatest. For positives: the number furthest from zero is greatest.
4
Combine: negatives (smallest first), then zero (if present), then positives.
5
Write using original forms and inequality symbols \(<\) between them.

Example 1: Simple Ordering

Order from least to greatest: \(\frac{1}{2},\; -3,\; 0.75,\; -0.5\)

1
Convert to decimals: \(\frac{1}{2} = 0.5\), \(-3 = -3.0\), \(0.75 = 0.75\), \(-0.5 = -0.5\)
2
Negatives: \(-3.0\) and \(-0.5\). Positives: \(0.5\) and \(0.75\)
3
Order negatives: \(-3.0 < -0.5\). Order positives: \(0.5 < 0.75\)
\(-3\)
\(0\)
\(-3\)
\(-0.5\)
\(\frac{1}{2}\)
\(0.75\)

Example 2: Fraction Challenge

Order from least to greatest: \(\frac{2}{3},\; \frac{3}{5},\; \frac{7}{10}\)

1
Find common denominator: LCD of \(3, 5, 10\) is \(30\).
2
Convert: \(\frac{2}{3} = \frac{20}{30}\), \(\frac{3}{5} = \frac{18}{30}\), \(\frac{7}{10} = \frac{21}{30}\)
3
Compare numerators: \(18 < 20 < 21\)

Example 3: Mixed Types

Order from greatest to least: \(-\frac{3}{4},\; 1.2,\; -0.8,\; \frac{5}{6},\; 0\)

1
Convert to decimals: \(-\frac{3}{4} = -0.75\), \(1.2\), \(-0.8\), \(\frac{5}{6} \approx 0.833\), \(0\)
2
Sort descending: \(1.2 > 0.833 > 0 > -0.75 > -0.8\)
\(0\)
\(-0.8\)
\(-\frac{3}{4}\)
\(\frac{5}{6}\)
\(1.2\)

The Absolute Value Trap

Absolute value \(|x|\) measures the distance from zero: it is always non-negative. But distance from zero is not the same as the number's value.

\(0\)
\(-4\)
\(3\)
\(|-4| = 4\)
\(|3| = 3\)

Special Cases to Know

Zero is Neutral
\(0\) is greater than every negative number and less than every positive number.
Equivalent Forms
\(\frac{1}{2} = 0.5 = \frac{2}{4} = \frac{50}{100}\). Always simplify or convert before comparing.
Opposite Signs
Any positive number is always greater than any negative number: \(0.001 > -1000\).

Common Mistakes to Avoid

✕ Ignoring the Sign
Thinking \(-7 > -2\) because \(7 > 2\). Remember: more negative means smaller.
✕ Comparing Unlike Forms
Comparing \(\frac{3}{8}\) and \(0.4\) without converting. Always use the same form first.
✕ Wrong Cross-Multiply
Mixing up which product goes with which fraction. Each product stays with the numerator it started from.

Quick Reference Table

Method When to Use Example
Convert to Decimals Mixed types (fractions + decimals) \(\frac{3}{8}\) vs. \(0.4\) → \(0.375\) vs. \(0.4\)
Common Denominator Two or more fractions \(\frac{2}{3}\) vs. \(\frac{5}{8}\) → \(\frac{16}{24}\) vs. \(\frac{15}{24}\)
Cross-Multiply Quick comparison of two fractions \(\frac{2}{3}\) vs. \(\frac{5}{8}\): \(16\) vs. \(15\)
Number Line Visual check, especially with negatives Plot and read left to right

Question 1 of 5

Which symbol correctly completes the statement: \(-\frac{5}{6}\) ___ \(-\frac{2}{3}\)?

Question 2 of 5

Which list is ordered from least to greatest?

Question 3 of 5

Using cross-multiplication, which fraction is greater: \(\frac{4}{9}\) or \(\frac{5}{11}\)?

Question 4 of 5

A number line shows point \(P\) at \(-2.5\) and point \(Q\) at \(-1.75\). Which statement is true?

Question 5 of 5

Order from greatest to least: \(\frac{7}{8},\; 0.9,\; -\frac{1}{2},\; \frac{3}{4}\)