Math Help

Several properties, also called principles or laws, are recognized as valid for the operations of addition, subtraction, multiplication, and division for all real numbers. Here are some of them. It’s not a bad idea to memorize these. You probably learned them in elementary school:

Additive Identity Element: When 0 is added to any real number a, the sum is always equal to a. The number 0 is said to be the additive identity element: a + 0 = a.

Multiplicative Identity Element: When any real number a is multiplied by 1, the product is always equal to a. The number is said to be the multiplicative identity element: a x 1 = a.

Additive Inverses: For every real number a, there exists a unique real number -a such that the sum of the two is equal to 0. The numbers a and -a are called additive inverse: a + (-a) = 0.

Multiplicative Inverses: For every nonzero real number a, there exists a unique real number 1/a such that the product of the two is equal to 1. The numbers a and 1/a are called multiplicative inverses: a x (1/a) = 1. The multiplicative inverse of a real number is also called its reciprocal.

Commutative Law For addition: When any two real numbers are added together, it does not matter in which order the sum is performed. The operation of addition is said to be commutative over the set of real numbers. For all real numbers a and b, the following equation is valid: a + b = b + a.

Commutative Law For Multiplication: When any two real numbers are multiplied by each other, it does not matter in which order the product is performed. The operation of multiplication, like addition, is commutative over the set of real numbers. For all real numbers a and b, the following equation is always true: a x b = b x a. A product can be written without the “times sign” (x) if, but only if, doing so does not result in an ambiguous or false statement. The above expression is often seen written this way: ab = ba.

If a = 3 and b = 52, however, it’s necessary to use the “times sign” and write this: 3 x 52 = 52 x 3. The reason becomes obvious if the above expression is written without using the “times sign.” This results in a false statement: 352 = 523.

Associative Law For Addition: When adding any three real numbers, it does not matter how the addends are grouped. The operation of addition is associative over the set of real numbers. For all real numbers a, b, and c, the following equation holds true: (a + b) + c = a + (b + c).

Associative Law For Multiplication: When multiplying any three real numbers, it does not matter how the multiplicands are grouped. Multiplication, like addition, is associative over the set of real numbers. For all real numbers a, b, and c, the following equation holds: (ab)c = a(bc).

Distributive Laws: For all real numbers a, b, and c, the following equation holds. The operation of multiplication is distributive with respect to addition: a(b + c) = ab + ac. This statement logically implies that multiplication is distributive with respect to subtraction, as well: a(b-c) = ab – ac. The distributive law can also be extended to division as long as there aren’t any denominators that end up being equal to zero. For all real numbers a, b, and c, where a ≠ 0, the following equations are valid:

(ab + ac)/a = ab/a + ac/a = b + c
(ab – ac)/a = ab/a – ac/a = b – c