The Number System

Adding & Subtracting Rational Numbers

Adding and subtracting positive and negative fractions and decimals.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Combining Positive and Negative Amounts

A hiker starts at an elevation of −20 feet (below a reference point) and climbs 35 feet. Before reading on, predict the hiker's new elevation — and notice that you're really adding a negative number and a positive number together.

Picture this on a number line: starting at −20 and moving 35 units in the positive direction lands at 15. That number-line movement is the real meaning behind adding signed numbers — moving right for a positive addend, left for a negative one — and it explains both sign rules you'll use constantly with rational numbers.

When two numbers share the same sign, moving further in that same direction means their distances from zero add together, and the sum keeps that shared sign. When two numbers have opposite signs, they pull in opposite directions, so their distances partly cancel — the sum's size is the difference of their distances, and its sign matches whichever number was farther from zero.

Worked Example — Adding Two Negative Numbers

Find −4.5 + (−3.2). Both numbers move in the negative direction, so their distances from zero add: 4.5 + 3.2 = 7.7, keeping the negative sign: −7.7.

Worked Example — Adding Numbers with Opposite Signs

Find −20 + 35, the hiker's elevation from the opening question. The distances from zero are 20 and 35; their difference is 35 − 20 = 15. Since 35 is farther from zero and positive, the sum takes a positive sign: 15.

Worked Example — Subtracting a Negative Number

Find 8 − (−5). Subtracting a negative number is the same as adding its opposite: 8 − (−5) = 8 + 5 = 13.

a - b = a + (-b)

Tip

Rewriting every subtraction as 'add the opposite' turns every problem into an addition problem, so only one set of sign rules — for addition — needs to be remembered at all.

Common Mistakes

Subtracting the distances from zero when adding two numbers with the same sign, instead of adding them.
Same-sign numbers move further in the same direction on the number line — their distances from zero add together, not subtract.
Forgetting to flip the sign when subtracting a negative number, treating 8 − (−5) the same as 8 − 5.
Subtracting a negative number is the same as adding its positive opposite — 8 − (−5) becomes 8 + 5, not 8 − 5.

Key Takeaways

Adding rational numbers can be pictured as movement on a number line — right for positive, left for negative.
Same-sign numbers add their distances from zero and keep the shared sign; opposite-sign numbers subtract their distances, keeping the sign of the larger one.
Subtracting a number is always the same as adding its opposite.

Summary

Number-line movement explains why the sign rules for adding and subtracting rational numbers work. The next lesson extends rational-number reasoning to multiplication and division.

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