Key takeaway: The additive inverse of a number is the number that, when added to it, makes a sum of zero; it has the same distance from zero but lies on the other side.
Grade 77.NS.A.1aNumbers and Operations
Definition of Additive Inverse
The additive inverse of a number is its opposite. Add the two together and they cancel completely, leaving 0, the additive identity.
On a number line, a number and its additive inverse sit the same distance from zero, one to the right and one to the left. Every integer has exactly one, and zero is its own: \(0 + 0 = 0\). The idea is what makes subtraction possible, since taking away a number is the same as adding its inverse.
Formula and notation
The additive inverse of \(a\) is written \(-a\), and the key property is:
\[a + (-a) = 0\]
For example, \(-(-6) = 6\) because the inverse of \(-6\) is \(6\). Writing the opposite of a negative number turns it positive.
Key facts
- A number and its opposite have the same absolute value.
- Zero is its own opposite.
- Subtracting \(b\) gives the same result as adding the opposite of \(b\).
- Positive numbers have negative inverses, and negative numbers have positive ones.
Additive Inverse example
Find the additive inverse of \(-27\) and of \(19\). Then check each one by adding.
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The inverse of \(-27\) is its opposite, \(27\). Check: \(-27 + 27 = 0\).
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The inverse of \(19\) is \(-19\). Check: \(19 + (-19) = 0\).
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Use it in a problem: a diver at \(-27\) feet needs to rise \(27\) feet to return to sea level, which is 0.
The additive inverses are 27 and \(-19\).
Common mistake
Watch out
Lena says the additive inverse of \(-9\) is \(-9\), because “the inverse of a negative stays negative.”
Why it is wrong
Check by adding: \(-9 + (-9) = -18\), not 0. The sum is zero only when the signs are opposite, so the inverse of \(-9\) is \(9\), and \(-9 + 9 = 0\). Do not confuse it with the reciprocal \(-\frac{1}{9}\), which is the multiplicative inverse and is found by flipping a fraction, not by changing a sign. Adding the pair is always the quickest test: if the total is not zero, the number is not the inverse.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Numbers and Operations