algebra vocabulary - chapter 2 Flashcards
Which property is represented by the following?
For all real numbers a and b:
a+b is a unique real number
ab is a unique real number
The sum and product of any two real numbers are also real numbers. Moreover, they are unique which means that there is one and only one possible answer then you add or multiply two real numbers.
Closure Properties
Which property is represented by the following?
For all real numbers a and b:
a+b=b+a and ab=ba
The order in which you add or multiply two numbers does not affect the result.
Commutative Properties
Which property is represented by the following?
For all real numbers a,b, and c:
(a+b)+c=a+(b+c)
(ab)c=a(bc)
When you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result.
Associative Properties
in the sum of a+b, what are a and b called?
terms
in the product “ab”, what are a and b called?
factors
which property says a=a?
reflexive property
which property says if a=b, then b=a?
symmetric property
which property says if a=b and b=c, then a=c
transitive property
which property says, there is unique real number 0 such that for every real number “a”, a+0=a and 0+a=a
identity property of addition
which property says that for every real number a, there is a unique real number -a such that a+(-a)=0 and (-a)+a=0
addition property of opposites
a number and it’s opposite which add up to zero are called what?
additive inverses
which property is illustrated by the following:
for all real numbers a and b:
-(a+b)=(-a)+(-b)
The opposite of a sum of real numbers is equal to the sum of the opposites of the numbers
Property of the Opposite of a Sum
Which property is illustrated by the following?
For all real numbers a and b, the “difference” a-b is defined by
a-b=a+(-b)
To subtract b, add the opposite of b.
Definition of Subtraction
For all real numbers a,b, and c:
a(b+c) = ab + ac AND (b+c)a=ba+ca a(b-c)= ab - ac AND (b-c)a= ba - ca
Distributive Properties
The properties of real numbers and equality guarantee that for all values of the variable x, 9x+5x and 14x represent the same number. Therefore, the two expressions are…
equivalent
Replacing an expression containing variables by an equivalent expression with as few terms as possible is called what?
for example…9x+5x is replaced by 14x
simplifying the expression
what is the identity ELEMENT for multiplication?
1
Which property is illustrated by the following?
There is a unique real number 1 such that for every real number “a”,
a x 1= a and 1 x a=a
Identity Property of Multiplication
The following illustrates which property? When one (or at least one) of the factors of a product is zero, the product itself is zero.
For every real number a:
a x 0 = 0 and 0 x a = 0
Multiplicative Property of Zero
Which property is illustrated by the following?
for every real number a:
a(-1) = -a AND (-1)a = -a
Multiplying any real number by -1 produces the opposite of the number
Multiplicative Property of -1
For all real numbers a and b:
(-a)(b) = -ab a(-b) = -ab (-a)(-b) = ab
Multiplicative Property of Opposites
Rules for Multiplication:
1a. If two numbers have the same sign, their product is _________.
1b. If two numbers have the opposite signs, their product is ________.
2a. The product of an even number of negative numbers is __________.
2b. The product of an odd number of negative numbers is __________.
1a. positive
1b. negative
2a. positive
2b. negative
What kind of integers do you have when you count by ones from any number in the set of integers?
For example, -2, -1,0,1,2
consecutive integers
what is it called when numbers are listed in order from least to greatest?
natural order
