## Key Concepts

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

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

**commutative property of addition **two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[/latex]

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

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

**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, [latex]a+0=a[/latex]

**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, [latex]a\cdot 1=a[/latex]

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

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