Transcript Document

•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.
•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.
If a = 5 then 5 + 0 = 0 + 5 = 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.
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.
If a = 6 then 6  1 = 1  6 = 6
Multiplicative
Inverse Property
a
, where a, b  0, there is exactly
b
b
a b
one number such that   1
a
b a
For every nonzero number
Two numbers whose product is 1 are called multiplicative
inverses or reciprocals.
Zero has no reciprocal because any number times 0 is 0.
Given the fraction 3 ; then 3  4  3 4  12 1; the fraction 4 is the reciprocal .
4
4 3 4  3 12
3
Together t he two fractions are multiplica tive inverses that are equal to the product 1.
•Equality Properties allow you to compute with expressions on both sides of an
equation by performing identical operations on both sides of the equation. This
creates a balance to the mathematical problem and allows you to keep the equation
true and thus be referred to as a property. The basic rules to solving equations is
based on these properties. 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
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
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
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 8 + 4 = 12 and 12 = 7 + 5, then 8 + 4 = 7 + 5
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 8 + 4 = 7 + 5; since 8 + 4 = 12 or 7 + 5 = 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 add or
subtract equal quantities to each side of the equation you get
equal quantities.
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 6 = 6 ; then 6 +3 = 6 + 3 or 6 – 3 = 6 - 3
If a = b ; then a + c = b + c or a – c = b - c