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Identity of an Operation

Identity of an Operation

The quantity which, when combined with another quantity using an operation, leaves the quantity unchanged.

For example, the additive identity is 0 since x + 0 = 0 + x = x for any number x. The multiplicative identity is 1 since x·1 = 1·x = x for any number x.

 

Table showing identity elements: addition (0): a+0=0+a=a; multiplication (1): a·1=1·a=a; composition f(x)=x:...

 

 

See also

Zero, composition, identity function

Key Formula

a⋆e=e⋆a=aa \star e = e \star a = a
Where:
  • aa = Any number or element in the set
  • ⋆\star = The operation (such as addition, multiplication, etc.)
  • ee = The identity element for the operation \star

Example

Problem: Verify that 0 is the identity element for addition and that 1 is the identity element for multiplication, using the number 7.
Step 1: Test the additive identity by adding 0 to 7 from both sides.
7+0=7and0+7=77 + 0 = 7 \quad \text{and} \quad 0 + 7 = 7
Step 2: The number 7 is unchanged in both cases, confirming that 0 is the additive identity.
Step 3: Test the multiplicative identity by multiplying 7 by 1 from both sides.
7×1=7and1×7=77 \times 1 = 7 \quad \text{and} \quad 1 \times 7 = 7
Step 4: The number 7 is unchanged in both cases, confirming that 1 is the multiplicative identity.
Answer: 0 is the identity for addition and 1 is the identity for multiplication, since combining either identity with 7 using its respective operation returns 7.

Another Example

Problem: Is there an identity element for the operation of exponentiation, where a ⊛ b means a^b?
Step 1: For an identity element e on the right, we would need a^e = a for every number a. Test e = 1.
a1=afor all a✓a^1 = a \quad \text{for all } a \quad \checkmark
Step 2: For a true identity, we also need it to work on the left: e^a = a for every a. Test e = 1.
1a=1≠a(in general)1^a = 1 \neq a \quad \text{(in general)}
Step 3: Since the left-side condition fails, exponentiation has no (two-sided) identity element. This shows that not every operation has an identity.
Answer: Exponentiation has no two-sided identity element because no single value e satisfies both a^e = a and e^a = a for all a.

Frequently Asked Questions

Why isn't 0 the identity for multiplication?
Because multiplying any number by 0 gives 0, not the original number. For instance, 5 × 0 = 0, not 5. The identity must leave the original number unchanged, so the multiplicative identity is 1, since 5 × 1 = 5.
Does every mathematical operation have an identity element?
No. An identity element only exists if there is some value that leaves every input unchanged under the operation. For example, exponentiation and subtraction (in the usual sense) do not have two-sided identity elements. Whether an identity exists depends on the specific operation and the set of numbers involved.

Identity element vs. Inverse element

The identity element leaves a number unchanged under an operation (e.g., a+0=aa + 0 = a). The inverse element combines with a number to produce the identity (e.g., a+(−a)=0a + (-a) = 0). The identity asks 'what changes nothing?', while the inverse asks 'what undoes a given value?'. Every inverse is defined relative to the identity: you cannot define an inverse without first knowing what the identity is.

Why It Matters

Identity elements are foundational to algebra and equation-solving. When you solve x+5=12x + 5 = 12 by subtracting 5 from both sides, you rely on the fact that 5−5=05 - 5 = 0 (the additive identity) and that x+0=xx + 0 = x. The concept extends far beyond basic arithmetic — in matrix algebra the identity matrix plays the same role for matrix multiplication, and in abstract algebra, the existence of an identity element is one of the defining requirements of a mathematical group.

Common Mistakes

Mistake: Confusing the identity element with the inverse element.
Correction: The identity is the single special value that changes nothing (0 for addition, 1 for multiplication). The inverse is specific to each number and produces the identity when combined with that number (e.g., the additive inverse of 3 is −3 because 3 + (−3) = 0).
Mistake: Thinking 0 is 'the identity' for all operations.
Correction: Each operation has its own identity element — if one exists at all. For addition it is 0, but for multiplication it is 1. Always ask: what value leaves every number unchanged under this particular operation?

Related Terms