Transcript Chapter 1-4: Identity and Equality Properties

Identity and Equality
Properties
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 + 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.
 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.
Multiplicative Inverse Property
 For every non-zero number, a/b,
a b
 1
b a
 Two numbers whose product is 1 are called
multiplicative inverses or reciprocals.
 Zero has no reciprocal because any number times 0 is
0.
3 4
 1
 Example:
4 3
4
3
The fraction
is the reciprocal of
.
3
4
The two fractions are multiplicative inverses of each other.
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.
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 may add or
subtract equal quantities to each side of the equation & still have
equal quantities.
 In if-then form:
 If 6 = 6 ; then 6 + 3 = 6 + 3 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!