All About Numbers
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.
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.
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.