# Properties of Zero

Pronunciation: /ˈpɹɑp.ər.tiz ʌv ˈzɪər.oʊ/ Explain

The number 0 has several important properties that are useful in algebra. Table 1 gives these properties.

Addition of 0 0 is the additive identity. Anything plus zero remains unchanged. a + 0 = 0 + a = a Additive Identity
Subtraction of 0 Anything minus zero remains unchanged. a - 0 = anone
Multiplication by 0 Anything times zero equals zero. a · 0 = 0 · a = 0 Property of Multiplication by Zero
Division by 0 Division by zero is undefined. n/a Division by 0
Exponent of 0 Any nonzero number raised to the 0 power equals 1. Zero raised to the 0 power is undefined. a0 = 1, a ≠ 0none
Zero factorial Zero factorial is defined to be 1. 0! = 1none
Idempotence Zero is idempotent under addition and multiplication. ( 0 + 0 = 0 ), ( 0 · 0 = 0 ). Idempotence
Table 1: Properties of zero.

McAdams, David E. Properties of Zero. 4/29/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/p/propertiesofzero.html.

### Revision History

4/29/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)