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\(+\)
\(-\)

Adding & Subtracting
Rational Numbers

Master the rules for combining positive and negative fractions, decimals, and integers using number lines and proven strategies.

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

Visualizing with the Number Line

A number line lets us see what adding and subtracting rational numbers really means. Adding a positive number moves you right. Adding a negative number moves you left.

\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(+2\)
 start end

Subtracting Means Moving Left

Subtracting a positive number is the same as adding its opposite. On the number line, this means moving left.

\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(-3\)
 start end

The Core Rules for Addition

1
Same signs: Add the absolute values and keep the shared sign. \((-4) + (-3) = -(4+3) = -7\)
2
Different signs: Subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value. \((-7) + 4 = -(7-4) = -3\)

The Subtraction Rule

Every subtraction problem can be converted to an addition problem. This is the single most important rule to remember:

Example 1
\(5 - 8 = 5 + (-8) = -3\)
Example 2
\(-3 - (-7) = -3 + 7 = 4\)

Same Signs vs. Different Signs

Same Signs Different Signs
Operation Add absolute values Subtract absolute values
Sign of Result Keep the shared sign Sign of larger absolute value
Example \((-3)+(-5)=-8\) \((-3)+5=+2\)
Number Line Both arrows same direction Arrows in opposite directions

How to Apply the Rules

1
Rewrite subtraction as addition: Change \(a - b\) to \(a + (-b)\).
2
Find absolute values: Determine \(|a|\) and \(|b|\).
3
Same or different signs? If same, add the absolute values. If different, subtract the smaller from the larger.
4
Determine the sign: Same signs: keep it. Different signs: use the sign of the number with the larger absolute value.
5
Verify on a number line if you are unsure.

Example 1: Adding Integers

Evaluate \((-6) + (+4)\).

\(-7\)
\(-6\)
\(-5\)
\(-4\)
\(-3\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(+4\)
start: \(-6\)
end: \(-2\)

Example 2: Adding Fractions

Evaluate \(\dfrac{3}{4} + \left(-\dfrac{1}{2}\right)\).

1
Find a common denominator: LCD of \(4\) and \(2\) is \(4\). Rewrite: \(\frac{3}{4} + \left(-\frac{2}{4}\right)\).
2
Different signs: Subtract absolute values of the numerators: \(|3| - |-2| = 3 - 2 = 1\).
3
Determine sign: \(\frac{3}{4}\) has the larger absolute value and is positive, so the result is positive.

Example 3: Subtracting Negatives

Evaluate \(-2.5 - (-4.3)\).

1
Rewrite as addition: \(-2.5 - (-4.3) = -2.5 + 4.3\).
2
Different signs: \(|4.3| - |-2.5| = 4.3 - 2.5 = 1.8\).
3
Sign: \(4.3\) has the larger absolute value and is positive.

Example 4: Mixed Numbers

Evaluate \(-1\frac{1}{3} + \left(-2\frac{2}{3}\right)\).

1
Convert to improper fractions: \(-\frac{4}{3} + \left(-\frac{8}{3}\right)\).
2
Same signs (both negative): Add absolute values: \(\frac{4}{3} + \frac{8}{3} = \frac{12}{3} = 4\).
3
Keep the shared sign: Both are negative, so the result is \(-4\).

Number Line with Fractions

Show \(-\frac{1}{2} + \frac{3}{4}\) on a number line.

\(-1\)
\(-\frac{1}{2}\)
\(0\)
\(\frac{1}{2}\)
\(1\)
\(+\frac{3}{4}\)
start: \(-\frac{1}{2}\)
end: \(\frac{1}{4}\)

Common Mistakes to Avoid

✕ Dropping the negative
Treating \(-3 + 5\) as \(3 + 5 = 8\) instead of \(5 - 3 = 2\). Always track signs carefully.
✕ Double negative confusion
Writing \(4 - (-2) = 2\) instead of \(4 + 2 = 6\). Subtracting a negative adds.
✕ Forgetting the LCD
Adding \(\frac{1}{3} + \frac{1}{4}\) as \(\frac{2}{7}\). You need a common denominator: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).

Quick Reference Summary

Situation Rule Example
Positive + Positive Add, result positive \(3 + 5 = 8\)
Negative + Negative Add absolute values, result negative \((-3)+(-5)=-8\)
Positive + Negative Subtract, sign of larger \(7+(-4)=3\)
Subtract a Positive Add its opposite (negative) \(3-7=3+(-7)=-4\)
Subtract a Negative Add its opposite (positive) \(3-(-2)=3+2=5\)

Question 1 of 5

What is \((-8) + 3\)?

Question 2 of 5

What is \(6 - (-4)\)?

Question 3 of 5

What is \(\dfrac{2}{5} + \left(-\dfrac{4}{5}\right)\)?

Question 4 of 5

A diver is at \(-12.5\) meters. She ascends \(7.8\) meters. What is her new depth?

Question 5 of 5

Which number line correctly shows \(2 + (-5)\)?

\(-4\)
\(-3\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(-5\)
start: \(2\)
end: \(-3\)