Opposite Integers
Integers and Their Opposites: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Decide directionPositive numbers move right; negative numbers move left.
- Combine distancesAdd distances when the signs match and subtract distances when the signs differ.
- Give the signUse the direction with the larger distance to decide the final sign.
Worked examples
Different signs
- Start at -9.
- Adding 14 moves right 14 spaces.
- You pass zero and land at 5.
Subtract a negative
- Subtracting a negative means add the opposite.
- Rewrite as 6 + 8.
- Add the distances.
Try one before moving on
Integers and Their Opposites: pop-up practice
Opposite integers are a foundational concept for the GED Mathematical Reasoning test. Every integer on the number line has a mirror image the same distance from zero, but on the opposite side. Understanding opposite integers builds the groundwork for all signed-number arithmetic, absolute value, and algebraic reasoning you will encounter on test day.
What Are Opposite Integers?
Two integers are opposites if they are the same distance from zero on the number line but on different sides. The opposite of a positive integer is negative, and the opposite of a negative integer is positive. The one special case: zero is its own opposite because it sits exactly at the center.
Notation: the opposite of any number \(\color{blue}{n}\) is written \(\color{blue}{-n}\). So the opposite of \(\color{blue}{7}\) is \(\color{blue}{-7}\), and the opposite of \(\color{blue}{-7}\) is \(\color{blue}{-(-7) = 7}\).
Rules for Finding Opposite Integers
1. Change the sign
To find the opposite of any integer, simply change its sign from positive to negative (or from negative to positive).
- Opposite of \(\color{blue}{5}\) → \(\color{blue}{-5}\)
- Opposite of \(\color{blue}{-8}\) → \(\color{blue}{8}\)
- Opposite of \(\color{blue}{-12}\) → \(\color{blue}{12}\)
2. Opposites sum to zero
A number and its opposite always add up to zero. This is called the Additive Inverse Property.
- \(\color{blue}{7 + (-7) = 0}\)
- \(\color{blue}{(-15) + 15 = 0}\)
- \(\color{blue}{0 + 0 = 0}\)
3. Double negative equals positive
The opposite of an opposite returns to the original number: \(\color{blue}{-(-n) = n}\).
- \(\color{blue}{-(-3) = 3}\)
- \(\color{blue}{-(-10) = 10}\)
Step-by-Step Summary
- Identify the integer you are given.
- If it is positive, place a minus sign in front of it.
- If it is negative, remove the minus sign (or apply the double-negative rule).
- If it is zero, the opposite is still zero.
- Check: the original number plus its opposite must equal \(\color{blue}{0}\).
Watch: Opposite of a Number (Video Lesson)
This Khan Academy lesson explains opposite integers and how to find them on the number line:
Opposite Integers – Worked Examples
Example 1: What is the opposite of \(\color{blue}{9}\)?
Change the sign: the opposite of \(\color{blue}{9}\) is \(\color{blue}{-9}\).
Check: \(\color{blue}{9 + (-9) = 0}\) ✓
Example 2: What is the opposite of \(\color{blue}{-15}\)?
Remove the negative sign: the opposite of \(\color{blue}{-15}\) is \(\color{blue}{15}\).
Check: \(\color{blue}{-15 + 15 = 0}\) ✓
Example 3: Simplify \(\color{blue}{-(-7)}\).
The opposite of \(\color{blue}{-7}\) is \(\color{blue}{7}\). So \(\color{blue}{-(-7) = 7}\).
Example 4: A scuba diver is at an elevation of \(\color{blue}{-30}\) feet (30 feet below sea level). What integer represents the same distance above sea level?
The opposite of \(\color{blue}{-30}\) is \(\color{blue}{30}\). The diver would need to be at \(\color{blue}{30}\) feet above sea level.
More Practice: Negative Numbers (Video Lesson)
Math Antics explains negative numbers and opposites with clear visual examples:
Exercises for Opposite Integers
Find the opposite of each integer.
- \(\color{blue}{12}\)
- \(\color{blue}{-7}\)
- \(\color{blue}{-25}\)
- \(\color{blue}{0}\)
- \(\color{blue}{-(-4)}\)
- A temperature of \(\color{blue}{-18°F}\). What is the opposite temperature?
Answers
- \(\color{blue}{-12}\)
- \(\color{blue}{7}\)
- \(\color{blue}{25}\)
- \(\color{blue}{0}\)
- \(\color{blue}{4}\)
- \(\color{blue}{18°F}\)
Frequently Asked Questions
What is the opposite of zero?
Zero is its own opposite. Because \(\color{blue}{0 + 0 = 0}\), zero satisfies the definition of being the additive inverse of itself.
Is the opposite of an integer the same as its absolute value?
No. The absolute value of \(\color{blue}{-7}\) is \(\color{blue}{7}\) (always non-negative), while the opposite of \(\color{blue}{-7}\) is also \(\color{blue}{7}\). However, the absolute value of \(\color{blue}{7}\) is \(\color{blue}{7}\), while the opposite of \(\color{blue}{7}\) is \(\color{blue}{-7}\). They differ for positive integers.
How do opposite integers appear on the GED test?
GED questions may ask you to identify an integer’s opposite, apply the additive inverse property, or interpret real-world contexts such as elevation, temperature, or debt where opposites represent equal-but-reversed quantities.
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