When an equation is false for at least one value, it is called a conditional equation.
For example, 6x12 is conditional because it is false when x3 (and any number
other than 2). In other words, if at least one value can be found in which the equation
is false (or the right side is not equal to the left side) then the equation is called a con-
ditional equation.How do numbers associatewith each other?
Generally in mathematics, there are certain properties of operations that determine
how numbers associate with each other. Closureis a property of an operation that
reveals how numbers associate with each other; in particular, when two whole num-
bers are added, their sum will be a whole number. Closure as a property of multiplica-
tion occurs when two whole numbers are multiplied and their resulting product is a
whole number.An associative propertymeans that for a given operation that combines three
quantities (two at a time), the initial pairing of the quantities is arbitrary. For exam-
ple, when doing an addition operation, the numbers can be combined in two ways: (a
b) ca(bc). Thus, when adding the numbers 3, 4, and 5, this means that
they may be combined as (3 4) 5 12 or 3 (4 5) 12. Following the same
logic for multiplication, the associative law states that (ab) ca(bc). In
fact, in an associative operation, the parentheses that indicate what quantities are to
be first combined can be omitted; an example of the associative law for addition is 3
4 5 12, and for multiplication, 2 3 4 24. But not all operations are asso-
ciative. One good example is division: You can’t divide in the same way as you added or
multiplied above. For example, the result of dividing three numbers differs. The opera-
tion (96 12) 4 2 is not the same as 96 (12 4) 32.Like the associative property, the commutative propertyis another way of looking
at how numbers associate with each other in operations. In particular, this law holds
that for a given operation that combines two quantities, the order of the quantities is
arbitrary. For example, in addition, adding 4 5 can be written either as 4 5 9 or
5 4 9, or expressed as abba. When working on a multiplication opera-
tion, the same rule applies, as in abba. Again, not all operations are commu-
tative. For example, subtraction is not, as in the equation 6 3 3, which is not the
same as 3 6 3. Division also is not commutative. For instance 6 3 2 is not
the same as 3 6 1/2.The final property of an operation is the distributive property.In this rule, for any
two operations the first is distributive over the second. For example, multiplication is
distributive over addition; for any numbers a, b,and c, a(bc) (ab) (a
c). For the numbers 2, 3, and 4, you would have 2 (3 4) 14 or (2 3) (2 4)
14. Formally, there is a right and left distribution—left is listed above; right is (a
142 b) c(ac) (bc). In most cases, both are commonly referred to as distribu-