### additive identity

number zero, which has the property that $a+0=a$ for any real number $a$

### additive inverse

for a real number $a$, the number $-a$. The sum of a number and its additive inverse is the additive identity, zero.

### associative property of addition

property stating that when adding three or more numbers, the sum does not change based on the way the numbers are grouped:

$a+(b+c)=(a+b)+c$

### associative property of multiplication

property stating that when multiplying three or more real numbers, the product does not change based on the way the numbers are grouped:

$a(bc)=(ab)c$

### commutative property of addition

property stating that when adding real numbers, the sum does not change based on the order of the numbers:

$a+b=b+a$

### commutative property of multiplication

property stating that when multiplying real numbers, the product does not change based on the order of the numbers:

$ab=ba$

### distributive property

property stating that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products:

$a(b+c)=ab+ac$

### identity property of addition

Property stating that the sum of zero and any number is the number itself:

$a+0=a$

### identity property of multiplication

property stating that the product of $1$ and any number is the given number:

$a\cdot1=a$

### multiplicative identity

number 1, which has the property $a\cdot1=a$ for any real number $a$

### multiplicative inverse

for a nonzero real number $a$, the number $\frac{1}{a}$. The product of a number and its multiplicative inverse is the multiplicative identity, 1.

### order of operations

set of rules indicating which calculations to perform first to simplify a mathematical expression