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Additive Identity — Definition, Formula & Examples

The additive identity is the number 0, because adding 0 to any number leaves that number unchanged. For example, 7+0=77 + 0 = 7 and −3+0=−3-3 + 0 = -3.

The additive identity is the unique element 00 in the real number system such that for every number aa, the equation a+0=0+a=aa + 0 = 0 + a = a holds. More generally, in any algebraic structure with addition, the additive identity is the element that, when added to any element, yields that same element.

Key Formula

a+0=0+a=aa + 0 = 0 + a = a
Where:
  • aa = Any real number
  • 00 = The additive identity

Worked Example

Problem: Show that 0 is the additive identity for the number −12.
Add 0 on the right: Add 0 to −12.
−12+0=−12-12 + 0 = -12
Add 0 on the left: Add −12 to 0 to confirm the order doesn't matter.
0+(−12)=−120 + (-12) = -12
Answer: Both sums equal −12, confirming that adding 0 leaves the number unchanged.

Why It Matters

The additive identity is one of the foundational properties tested when you study real number axioms in algebra. Recognizing that adding 0 has no effect helps you simplify expressions and solve equations without accidentally changing values.

Common Mistakes

Mistake: Confusing the additive identity (0) with the multiplicative identity (1).
Correction: The additive identity is 0 because a+0=aa + 0 = a. The multiplicative identity is 1 because a×1=aa \times 1 = a. Each operation has its own identity element.