Our experience with arithmetic leads us to believe that addition and multiplication are different operations in the sense that, given the same two numbers, they produce different answers. For example, given the constants 6 and 2, 6 + 2 = 8 while 6 x 2 = 12. Therefore, it seems reasonable to expect when we replace constant numbers, like 6 and 2, with variable numbers, like a and b, that addition and multiplication should continue to give different answers. That is, a + b and (a)(b) will typically represent unequal values.
There are only two operations with which we need concern ourselves: addition and multiplication. This will allow us to simply and quickly organize laws of algebra by associating them with one and only one of these two operations. We will study expressions that contain both constant and variable numbers that are "held together" by addition or multiplication.
Our experience with arithmetic leads us to believe that addition and multiplication are different operations in the sense that, given the same two numbers, they produce different answers. For example, given the constants 6 and 2, 6 + 2 = 8 while 6 x 2 = 12. Therefore, it seems reasonable to expect when we replace constant numbers, like 6 and 2, with variable numbers, like a and b, that addition and multiplication should continue to give different answers. That is, a + b and (a)(b) will typically represent unequal values.
To be successful in algebra, you will need to develop certain skills. You must learn to differentiate between the operations of addition and multiplication as they occur in algebraic expressions. For example, addition combines 9 with c in the expression 9 + c while multiplication combines the same two numbers in the expression 9c. As noted above, we expect that 9 + c and 9c represent unequal values.
Since addition and multiplication produce different answers, the algebra laws we employ for addition are different from those for multiplication. The good news is that we need only separate algebra laws into two groups: one for those we associate with the operation of addition and the other for those we associate with multiplication. The bad news is that it is easy (and incorrect!) to apply an "addition law" when the operation is multiplication or a "multiplication law" when the operation is addition!
To be successful in algebra, you will need to develop certain skills. You must learn to differentiate between the operations of addition and multiplication as they occur in algebraic expressions. For example, addition combines 9 with c in the expression 9 + c while multiplication combines the same two numbers in the expression 9c. As noted above, we expect that 9 + c and 9c represent unequal values.
Since addition and multiplication produce different answers, the algebra laws we employ for addition are different from those for multiplication. The good news is that we need only separate algebra laws into two groups: one for those we associate with the operation of addition and the other for those we associate with multiplication. The bad news is that it is easy (and incorrect!) to apply an "addition law" when the operation is multiplication or a "multiplication law" when the operation is addition!
