Properties of Real Numbers The properties of real numbers help us simplify math expressions and help us better understand the concepts of algebra.

Commutative Property of Addition  a + b = b + a  Example:7 + 3 =  Two real numbers can be added in either order to achieve the same sum.

Commutative Property of Multiplication  a x b = b x a  Example:3 x 7 = 7 x 3  Two real numbers can be multiplied in either order to achieve the same product.

Associative Property of Multiplication  (a x b) x c = a x (b x c)  Example: (6 x 4) x 5 = 6 x (4 x 5)  When three real numbers are multiplied, it makes no difference which are multiplied first.  Notice how multiplying the 4 and 5 first makes completing the problem easier.

Associative Property of Addition  (a + b) + c = a + (b + c)  Example: ( ) + 7 = 29 + (13 + 7)  When three real numbers are added, it makes no difference which are added first.  Notice how adding the first makes completing the problem easier mentally.

Additive Identity Property  a + 0 = a  Example: = 9  The sum of zero and a real number equals the number itself.  Memory note: When you add zero to a number, that number will always keep its identity.

Multiplicative Identity Property  a x 1 = a  Example: 8 x 1 = 8  The product of one and a number equals the number itself.  Memory note: When you multiply any number by one, that number will keep its identity.

Additive Inverse Property  a + (-a) = 0  Example: 3 + (-3) = 0  The sum of a real number and its opposite is zero.

Multiplicative Inverse

Property of Zero (Multiplication)  When any number is multiplied with zero, the answer is zero.  98,756,432 X 0 = 0

Property of Opposites  a + (-a) = 0  If you added opposite #’s and ended with 0

Distributive Property  a(b + c) = ab + ac ora(b – c) = ab – ac  Example: 2(3 + 4) = (2 x 3) + (2 x 4)  or  2(3 - 4) = (2 x 3) - (2 x 4)  Distributive Property is the sum or difference of two expanded products.

Properties of Equality Addition property of equality If a = b, then a + c = b + c. Adding the same number to both sides of an equation does not change the equality of the equation. Subtraction property of equality If a = b, then a – c = b – c. Subtracting the same number from both sides of an equation does not change the equality of the equation. Multiplication property of equality If a = b and c ≠ 0, then a c = b c. Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation.

Transitive Property If a = b and b = c, then a = c If one quantity equals a second quantity and the second quantity equals a third quantity, then the first equals the third. If 1000 mm = 100 cm and 100 cm = 1 m, Then 1000 mm = 1m

Symmetric Property If a + b = c then c = a + b If one quantity equals a second quantity, then the second quantity equals the first. If 10 = 4 + 6, then = 10

Reflexive Property a = a a + b = a + b Any quantity is equal to itself. 7 = = 2 + 3

Justify Steps in Solving Equations  h/academy123_content/wl-book-demo/ph-374s.html h/academy123_content/wl-book-demo/ph-374s.html