The opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.
Opposites are mirror images across 0: the same distance from 0 on opposite sides. That is why every number plus its opposite equals 0.
Formal definition and notation
The opposite of a numbera is the number −a for which a + (−a) = 0. On the number line, a and −a lie the same distance, |a|, from 0 on opposite sides. Taking the opposite twice gives back the number: −(−a) = a. The only number equal to its own opposite is 0. Mathematicians also call the opposite the additive inverse, because adding it undoes adding a.
Symbol
Read it as
Meaning
−a
the opposite of a
the number that adds to a to make 0; it is positive when a is negative
−(−3)
the opposite of negative three
equals 3
−5
negative five
the opposite of 5; read a minus sign in front of a number as negative
Key facts
A number and its opposite have the same absolute value: |−9| = |9| = 9.
To find the opposite, change the sign: 12 becomes −12, −3.5 becomes 3.5, 23 becomes −23.
Absolute value is the distance from 0, so it is never negative: |−8| = 8 and |8| = 8. The opposite flips the side: the opposite of 8 is −8.
Oppositevs.Reciprocal
The reciprocal flips a fraction so the product is 1: the reciprocal of 5 is 15. The opposite changes the sign so the sum is 0: the opposite of 5 is −5.
Not a different idea: additive inverse is the formal name for the opposite, used in algebra when solving equations.
Quick check: 3 questions
Question 1 of 3
What is the opposite of −6?
Show answer and explanation
Answer: A.−6 is 6 units left of 0, so its opposite is 6 units right: 6. Choice B keeps the same number, and C and D are reciprocals, which multiply to 1 instead of adding to 0.
Question 2 of 3
Which pair of numbers are opposites?
Show answer and explanation
Answer: D.−4 + 4 = 0, so they are opposites. 3 and 13 are reciprocals, −2 and −2 are equal, and 5 and −0.5 are not the same distance from 0.
Question 3 of 3
If −x = 9, what is x?
Show answer and explanation
Answer: B.−x is the opposite of x. If the opposite of x is 9, then x = −9. Check: −(−9) = 9. Choice A forgets that −x and x have opposite signs.
Opposite: frequently asked questions
Is the opposite of a number always negative?
No. The opposite of a positive number is negative, but the opposite of a negative number is positive: the opposite of −12 is 12. The opposite of 0 is 0, which is neither. A good test: a number and its opposite always add to 0, so they must be on different sides of 0 unless the number is 0 itself.
Is 0 its own opposite?
Yes. 0 is zero units from 0, so the number the same distance away on the other side is 0 again, and 0 + 0 = 0. It is the only number that equals its own opposite. That is why 0 is neither positive nor negative: it sits exactly at the middle of the number line.
Are opposite and additive inverse the same thing?
Yes. Additive inverse is the more formal name, used in algebra. It describes what the opposite does: adding it to the number gives 0, the additive identity. In equations you use it to undo addition, as in x + 7 = 10, where adding −7 to both sides gives x = 3.
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