Ch 1-4 Identity and Equality Properties (1)

8/17/2019 Ch 1-4 Identity and Equality Properties (1)

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Identity and Equality Identity and Equality

Properties Properties


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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 resultfrom testing the truth or validity of something byexperiment or trial to establish a proof.

• Therefore, every mathematical problem from theeasiest to the more complex can be solved byfollowing step by step procedures that are

identified as mathematical properties.


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Additive Identity 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

!f a = " then " + 0 = 0 + " = ".


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Multiplicative Identity Property Multiplicative Identity Property

For any number a, a # = # a = a.

The product of any number and one is equal to

that number.

The number one is called the multiplicative

identity.

Example

!f a = $ then $ # = # $ = $.


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Multiplicative Property of Zero 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

!f a = $, then $ 0 = 0 $ = 0.


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For every non%zero number, a&b,

T'o numbers 'hose product is # are called

multiplicative inverses or reciprocals.

(ero has no reciprocal because any number times 0 is 0.

Example

34• 43

= 1

The fraction4

3 is the reciprocal of

3

4.

The two fractions are multiplicative inverses of each other.

Multiplicative Inverse Property Multiplicative Inverse Property

a

b •b

a =1


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Equality Properties Equality Properties

• !"uality Properties allow you to compute with expressions on bothsides of an e"uation by performing identical operations on both sides

of the e"ual sign. The basic rules to solving e"uations is this#

) Whatever you do to one side of an equation; You must perform the

same operation(s !ith the same number or e"pression on the other

side of the equals si#n$

• $eflexive Property of !"uality

• %ymmetric Property of !"uality

• Transitive Property of !"uality• %ubstitution Property of !"uality

• Addition Property of !"uality )

• Multiplication Property of !"uality )


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%efle"ive Property of Equality %efle"ive Property of Equality

For any number a, a = a.

The reflexive property of equality says that any

real number is equal to itself. *any mathematical statements and alebraic

properties are 'ritten in if&then form 'hendescribin the rules- or ivin an example.

The hypothesis is the part follo'in if , and theconclusion is the part follo'in then.

!f a = a then / = / then ". = "..

For any number a, a = a.

The reflexive property of equality says that anyreal number is equal to itself.

*any mathematical statements and alebraicproperties are 'ritten in if&then form 'hendescribin the rules- or ivin an example.


!f a = a then / = / then ". = "..


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'ymmetric Property of Equality 'ymmetric Property of Equality

For any numbers a and b, if a = b, then b = a.

The symmetric property of equality says that if onequantity equals a second quantity, then the second

quantity also equals the first.

*any mathematical statements and alebraicproperties are 'ritten in if&then form 'hendescribin the rules- or ivin an example.


!f #0 = / + 1 then / +1 = #0.

!f 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 onequantity equals a second quantity, then the second

quantity also equals the first. *any mathematical statements and alebraic

properties are 'ritten in if&then form 'hendescribin the rules- or ivin an example.


!f #0 = / + 1 then / +1 = #0.

!f a = b then b = a$


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ransitive Property of Equalityransitive 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. *any mathematical statements and alebraic properties are

'ritten in if&then form 'hen describin the rules- or ivin

an example.

The hypothesis is the part follo'in if , and the conclusion isthe part follo'in then.

!f 2 + 3 = # and # = / + ", then 2 + 3 = / + ".

!f 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.

*any mathematical statements and alebraic properties are

'ritten in if&then form 'hen describin the rules- or ivin

an example.

The hypothesis is the part follo'in if , and the conclusion is

the part follo'in then.

!f 2 + 3 = # and # = / + ", then 2 + 3 = / + ".

!f a = b and b = c , then a = c$


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'ubstitution Property of Equality 'ubstitution Property of Equality

If a & b, then a may be replaced by b in any expression.

The substitution property of e"uality says that a "uantity may besubstituted by its e"ual in any expression.

Many mathematical statements and algebraic properties arewritten in if-then form when describing the rule's( or giving anexample.

The hypothesis is the part following if , and the conclusion is the

part following then. If ) * + & * - since ) * + & /0 or * - & /0

Then we can substitute either simplification into the originalmathematical statement.

If a & b, then a may be replaced by b in any expression.

The substitution property of e"uality says that a "uantity may besubstituted by its e"ual in any expression.

Many mathematical statements and algebraic properties arewritten in if-then form when describing the rule's( or giving anexample.

The hypothesis is the part following if , and the conclusion is the

part following then. If ) * + & * - since ) * + & /0 or * - & /0

Then we can substitute either simplification into the originalmathematical statement.


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Addition Property of Equality Addition Property of Equality

If a & b, then a * c & b * c or a + (-c) = b + (-c)

The addition property of e"uality says that if you may add or

subtract e"ual "uantities to each side of the e"uation 1 still have

e"ual "uantities.

In if-then form#

If 2 & 2 then 2 * 3 & 2 * 3 or 2 * '43( & 2 * '43(.

5otice, that after adding 3 or 43 to both sides, the numbers

are still e"ual. This property will be very important when we

learn to solve e"uations6

If a & b, then a * c & b * c or a + (-c) = b + (-c)

The addition property of e"uality says that if you may add or

subtract e"ual "uantities to each side of the e"uation 1 still have

e"ual "uantities.

In if-then form#

If 2 & 2 then 2 * 3 & 2 * 3 or 2 * '43( & 2 * '43(.

5otice, that after adding 3 or 43 to both sides, the numbers

are still e"ual. This property will be very important when we



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Multiplication Property of Equality Multiplication Property of Equality

If a & b, then ac & bc

The multiplication property of e"uality says that if you may

multiply e"ual "uantities to each side of the e"uation 1 still

have e"ual "uantities.

In if-then form#

If 2 & 2 then 2 7 3 & 2 7 3.

5otice, that after multiplying 3 to both sides, the numbers are

still e"ual. This property will be very important when we


If a & b, then ac & bc

The multiplication property of e"uality says that if you may

multiply e"ual "uantities to each side of the e"uation 1 still

have e"ual "uantities.

In if-then form#

If 2 & 2 then 2 7 3 & 2 7 3.

5otice, that after multiplying 3 to both sides, the numbers are

still e"ual. This property will be very important when we


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Ch 1-4 Identity and Equality Properties (1)