Meaning of Term in AlgebraDate: 09/16/2007 at 12:22:14 From: Erin Subject: Terrible Terms When I was in middle school, I was taught that in algebra we do not write 2 * x because multiplication is used so much. Instead, we use 2x. However, as I went through the math course, they explained to me that something like 2x is considered one term, and that something like 2 + x is considered two terms. Because of this, each are treated differently. For example, in the problem 3(2x) all you need to do is multiply to the one term to get 6x. But in a problem like 3(2 + x), because there are two terms in parentheses, you need to distribute to get 6 + 3x. This is just one example of how a problem with one term and a problem with two terms are treated differently. My question is this: why is it that when you multiply or divide two numbers or variables they become one term (such as 2*x becoming 2x), but when you add or subtract two numbers or variables they are still two terms (such as 2 + x)? This was never really explained to me because all they told me was that it's easier to not put the multiplication dot (*) because it's used so much. But I know that's not the only reason because there are so many different properties and different ways to treat the terms, that there must be another reason then just for convenience. Can you help me with my curiosity on this? Thank you. Take the problem 2(xy). In 2(xy), since the value in parentheses is one term, the appropriate thing to simplify it to would be 2xy. This makes sense. Lets say xy is the area of a box that is x units wide and y units long. If we combine two of those kind of boxes together, we would have one rectangle with an an area of 2xy. But if we were to pretend that xy was two terms, and, using the two-term distribution rule, multiplied both x and y by 2, this would not make sense. That would be saying that 2 xy boxes placed together have a width of 2x and a length of 2y. This is not true because if 4 boxes were placed together, then the resulting box would be 2x wide and 2y long. Hence, if we distributed 2(xy) wrongly and got 2x2y, it would have to be simplified to 4xy. 4 boxes have an area of 4xy, and 2 boxes have an area of 2xy. x[] y x[][] 2y 2x [][] [][] 2y Now take the problem 2(x + y). Since x + y is two terms, if we were to distribute the 2 to both the x and the y, we would get 2x + 2y. This makes sense. Lets say x + y is the number of units x and the number of units y in one group. If we get two of these kind of groups, we would have double the number of units x and well as the number of units y. But if we were to pretend that x + y was one term, then we would simply the problem as 2x + y. This does not make sense. If we were to try to simplify it this way, we would be saying that if we had two groups, only the number of units x would double. This doesn't make sense since one group has both units x and units y, and if we were to double that group we would have double the units x as well as double the units y. One group: xxyy Two groups: xxyy xxyy Two groups would not be: xx xxyy My teacher always tells me "terms are terrible". You must treat them in different ways. In my opinion "terrible terms" are where most algebraic mistakes come from. I feel knowing more about how terms work will help me to understand and avoid these mistakes. An example of a "terrible term" algebraic mistake is that you cannot distribute an exponent among two terms. (x + y)^2 does not equal x^2 + y^2. But you can distribute an exponent over one term. (xy)^2 = x^2*y^2. Another example is that if the same term is added or subtracted in the numerator and denominator, they cannot be canceled out. (5 + 2)/(10 + 2) does not equal 5/10. But if the same term is multiplied in the numerator and denominator, they can be canceled out. (5*2)/(10*2)=5/10. Date: 09/16/2007 at 23:34:58 From: Doctor Peterson Subject: Re: Terrible Terms Hi, Erin. You've said a lot of good things; it sounds like you have a pretty good understanding of the essentials. I'll try saying a little more that might put things in perspective (which is what I think you want), and then we can discuss more if there are still gaps. The reason adding makes two terms is simply that we DEFINE a term as something that is added to other terms. In part due to the order of operations, we think of any expression as, ultimately, a sum: that is the last operation we do, so if there are any additions (that are not inside parentheses), they break up the expression into a sum, and the pieces are called terms. Many of the things you talk about are distributive properties. We can show that multiplication distributes over addition, (a+b)*c = ac + bc and that exponents distribute over multiplication, (ab)^c = a^c b^c But exponents do not distribute over addition, and multiplication does not distribute over another multiplication. So you simply have to keep in mind the rules that are true, and not do things that LOOK similar but are not valid. One help is to relate these properties to the order of operations (which I believe is a major reason that the order is what it is): exponentiation distributes over... multiplication distributes over ... addition Also, I believe that we leave out the symbol for multiplication not so much because it is common, but because it is powerful (higher in the order of operations). By writing ax + by with the a and the x close together, we make the order of operations look natural; we SEE ax as a single term, by as another, and the main operation in the expression as addition, a sum of terms. Does that help at all? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 10/02/2007 at 18:28:23 From: Erin Subject: Thank you (Terrible Terms) Hello. Thanks for answering my question. What you said about the order of operations really helps - that exponents distribute over multiplication, which distributes over addition. It helps make sense of it to me and puts things into better perspective. It seems like it has a lot to do with how we define math operations. It was really neat hearing back from a mathematician - I plan to major in math in college. Thanks again! :) |
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