Key Concepts

associative property of addition the sum of three numbers may be grouped differently without affecting the result; in symbols, $a+\left(b+c\right)=\left(a+b\right)+c$

associative property of multiplication the product of three numbers may be grouped differently without affecting the result; in symbols, $a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c$

commutative property of addition two numbers may be added in either order without affecting the result; in symbols, $a+b=b+a$

commutative property of multiplication two numbers may be multiplied in any order without affecting the result; in symbols, $a\cdot b=b\cdot a$

distributive property the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$

identity property of addition there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, $a+0=a$

identity property of multiplication there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, $a\cdot 1=a$

inverse property of addition for every real number $a$, there is a unique number, called the additive inverse (or opposite), denoted $-a$, which, when added to the original number, results in the additive identity, 0; in symbols, $a+\left(-a\right)=0$

inverse property of multiplication for every non-zero real number $a$, there is a unique number, called the multiplicative inverse (or reciprocal), denoted $\dfrac{1}{a}$, which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, $a\cdot \dfrac{1}{a}=1$