Identity and Equality Properties

Properties refer to rules that indicate a standard procedure or method to be followed. A proof is a demonstration of the truth of a statement in mathematics. Properties or rules in mathematics are the result from testing the truth or validity of something by experiment or trial to establish a proof. Therefore, every mathematical problem from the easiest to the more complex can be solved by following step by step procedures that are identified as mathematical properties.

Identity Properties Additive Identity Property Multiplicative Identity Property Multiplicative Identity Property of Zero Multiplicative Inverse Property

Additive Identity Property  For any number a, a + 0 = 0 + a = a.  The sum of any number and zero is equal to that number.  The number zero is called the additive identity.  Example: If a = 5 then = = 5.

Multiplicative Identity Property  For any number a, a  1 = 1  a = a.  The product of any number and one is equal to that number.  The number one is called the multiplicative identity.  Example: If a = 6 then 6  1 = 1  6 = 6.

Multiplicative Property of Zero  For any number a, a  0 = 0  a = 0.  The product of any number and zero is equal to zero.  Example: If a = 6, then 6  0 = 0  6 = 0.

 For every non-zero number, a/b,  Two numbers whose product is 1 are called multiplicative inverses or reciprocals.  Zero has no reciprocal because any number times 0 is 0.  Example: Multiplicative Inverse Property

Equality Properties Equality Properties allow you to compute with expressions on both sides of an equation by performing identical operations on both sides of the equal sign. The basic rules to solving equations is this: * Whatever you do to one side of an equation; You must perform the same operation(s) with the same number or expression on the other side of the equals sign. Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality Addition Property of Equality * Multiplication Property of Equality *

Reflexive Property of Equality  For any number a, a = a.  The reflexive property of equality says that any real number is equal to itself.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If a = a ; then 7 = 7; then 5.2 = 5.2.  For any number a, a = a.  The reflexive property of equality says that any real number is equal to itself.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If a = a ; then 7 = 7; then 5.2 = 5.2.

Symmetric Property of Equality  For any numbers a and b, if a = b, then b = a.  The symmetric property of equality says that if one quantity equals a second quantity, then the second quantity also equals the first.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If 10 = 7 + 3; then 7 +3 = 10.  If a = b then b = a.  For any numbers a and b, if a = b, then b = a.  The symmetric property of equality says that if one quantity equals a second quantity, then the second quantity also equals the first.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If 10 = 7 + 3; then 7 +3 = 10.  If a = b then b = a.

Transitive Property of Equality  For any numbers a, b and c, if a = b and b = c, then a = c.  The transitive property of equality says that if one quantity equals a second quantity, and the second quantity equals a third quantity, then the first and third quantities are equal.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If = 12 and 12 = 7 + 5, then =  If a = b and b = c, then a = c.  For any numbers a, b and c, if a = b and b = c, then a = c.  The transitive property of equality says that if one quantity equals a second quantity, and the second quantity equals a third quantity, then the first and third quantities are equal.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If = 12 and 12 = 7 + 5, then =  If a = b and b = c, then a = c.

Substitution Property of Equality  If a = b, then a may be replaced by b in any expression.  The substitution property of equality says that a quantity may be substituted by its equal in any expression.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If = 7 + 5; since = 12 or = 12;  Then we can substitute either simplification into the original mathematical statement.  If a = b, then a may be replaced by b in any expression.  The substitution property of equality says that a quantity may be substituted by its equal in any expression.  Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example.  The hypothesis is the part following if, and the conclusion is the part following then.  If = 7 + 5; since = 12 or = 12;  Then we can substitute either simplification into the original mathematical statement.

Addition Property of Equality  If a = b, then a + c = b + c or a + (-c) = b + (-c)  The addition property of equality says that if you may add or subtract equal quantities to each side of the equation & still have equal quantities.  In if-then form:  If 6 = 6 ; then = or 6 + (-3) = 6 + (-3). Notice, that after adding 3 or -3 to both sides, the numbers are still equal. This property will be very important when we learn to solve equations!  If a = b, then a + c = b + c or a + (-c) = b + (-c)  The addition property of equality says that if you may add or subtract equal quantities to each side of the equation & still have equal quantities.  In if-then form:  If 6 = 6 ; then = or 6 + (-3) = 6 + (-3). Notice, that after adding 3 or -3 to both sides, the numbers are still equal. This property will be very important when we learn to solve equations!

Multiplication Property of Equality If a = b, then ac = bc The multiplication property of equality says that if you may multiply equal quantities to each side of the equation & still have equal quantities. In if-then form:  If 6 = 6 ; then 6 * 3 = 6 * 3. Notice, that after multiplying 3 to both sides, the numbers are still equal. This property will be very important when we learn to solve equations! If a = b, then ac = bc The multiplication property of equality says that if you may multiply equal quantities to each side of the equation & still have equal quantities. In if-then form:  If 6 = 6 ; then 6 * 3 = 6 * 3. Notice, that after multiplying 3 to both sides, the numbers are still equal. This property will be very important when we learn to solve equations!