The Identity Property

The identity property says that certain special numbers can be combined with any other number without changing it at all. For addition, that number is 0 (the additive identity): a + 0 = a. For multiplication, that number is 1 (the multiplicative identity): a × 1 = a. These two humble numbers quietly hold the entire number system together – every later algebra technique that involves “undoing” an operation ultimately aims to get back to one of these two identities.

It took a genuinely long time in history for these two numbers to earn this special status. As covered earlier in this section, zero itself wasn't formally treated as a number with its own arithmetic rules until Brahmagupta's work in 628 CE – and it was only once zero was accepted as a real number that mathematicians could clearly state a rule as elegant as “adding it changes nothing.”

The Additive Identity: 0

For any real number a: a + 0 = a and 0 + a = a.

What is -37 + 0?

Adding 0 never changes a number: -37 + 0 = -37.

The Multiplicative Identity: 1

For any real number a: a × 1 = a and 1 × a = a.

What is 58 × 1?

Multiplying by 1 never changes a number: 58 × 1 = 58.

Real-Life Application

Equivalent fractions: multiplying by forms of 1 (like 2/2) keeps a fraction's value the same.
Starting a count: a running total begins at 0, the additive identity.
Unit conversions: multiplying by a conversion ratio that equals 1 (e.g. 100 cm / 1 m).

Key Takeaways

The additive identity is 0: a + 0 = a for every real number a.
The multiplicative identity is 1: a × 1 = a for every real number a.
These identities are the foundation for the inverse property, covered next.

Practice: The Identity Property

The Additive Identity