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Name: Bill
Status: student
Grade: 9-12
Country: USA
Date: N/A 

I have a question about terms. I am a sophomore in high school and I usually go about my business solving problems, but today I had some extra time and sat down and thought. What exactly is a term? This brought up some interesting situations that I would like to ask why they work, shown below. Ok, so my first question is what is a term. For example in 5x, are 5 and x separate terms or are they one term all together since they are being multiplied. The same question arises in 5x+6. Is this two terms, three terms, or one term of (5x+6)? This is where I really started to get confused. I know that I have always done this stuff and my questions may seem silly, but here I go. In 5x + 2x, I would usually say that the answer would be 7x by combining like terms, but if you considered them to be two different terms, like (5x) + (7x), then how would you add these terms if they were not the same. Here is another example of my confusion. If you have x/6= 5+x, I would first multiply 6 to both sides to get rid of the six on the left hand side of the equation. When I multiply the 6 to the right hand side of the equation, I would multiply it like this: 6*(5+x). Each side of the equation is a term, right? So when multiplying you have to distribute across the term. Well, what if you had x+6=12. In my mind, since each side of the equation is a term, I would think of it like this, (x+6)=(12). Here, each side is a term, and usually I would just subtract the six, but how can you do that since (x+6) and 12 are not like terms. I have always been taught that you cannot break up a term, such as in (x+10)/5, so why can you do it here? I know these questions may seem tedious and unnecessary, but I do not understand how we do what we do in math if we do not understand these basic principles of adding and multiplying.


Very good question.

I began realize that maybe the easiest way to separate terms is by the separation of the +'s and -'s; however, when equations include variables like x, y, z, etc, you have to group the x's and y's and z's, etc, TOGETHER. In the case of multiplication and division, 5x and 5/x are both single terms. In the case of complex equations like 5x/z, you must separate the x and z; therefore (5/x) * (1/z) are TWO terms of a product.

To simplify my explanation, I'll use the examples you have provided. 5x is a single term because you cannot "separate" the 5 and the x. It's simply the (x) term and in this case the 5x term. For 5x+6, you have an (x) term and a constant; therefore, 5x+6 has two terms. Using your example, 5x + 2x: here you have two values of "like" terms. Meaning, you have two (x) terms, and by the distributive property 5x + 2x = (5 + 2) x = 7x and, therefore you group the two (x) terms, resulting in one (x) term.

Hence x/6 = 5 + x, is x = 30 + 6x, and finally, 0 = 30 + (6-1)x, and therefore this equation has two terms: a constant term and one (x) term.

In short, terms are grouped values of similar notation. Group the x terms with the x's and the y terms with the y's...and so on... and do not the constants with other constants.

Be careful though. As mathematics become more complex, there are times when some variables are a "don't care" or "imaginary". Meaning, you could have in special circumstances when (5x+2)i, where i is an imaginary number, and hence you have ONE term that is "imaginary" and is simply (5x+2).

Hope that helps.
Alex Viray

Yours is not a silly question, nor is it simple, because you are dealing with some of the very fundamental questions of mathematics. Specifically: What do the "symbols" mean? How are the operations defined? Those two questions have a long and non-trivial history. Usually, I avoid recommending specific books, after all, we are not book sellers, but occasionally, a book that really addresses a question and is readable at the same time comes along, that it is worth breaking that general rule.

"Math through the Ages" by Berlinghoff and Gouvea is such an instance. What you will find is that mathematical symbols and operations as we know them, generically call these "terms", is a rather recent invention, developing between the 1600's and 1800's. Before then, algebra was expressed verbally. For example, the simple arithmetical statement: (5+6)-7 = 4, in modern notation, would have been expressed verbally, "When 7 is subtracted from the sum of 5 and 6, the result is 4." Not very efficient for such a simple arithmetical statement, and you can see how things could get very complicated for even a "simple" algebraic expression with powers and roots. You also must appreciate the understanding of unknowns, zero, negative numbers, irrational numbers, and imaginary (complex) numbers are all recent additions to the history of mathematics, as is the subject of your inquiry "terms".

Vince Calder

Fun question, but don't let it confuse you. A term is a grouping of numbers and symbols (variables), separated by an arithmetic sign (+ or -). And the variable is central to the whole concept!

When we see 5x + 3x both are terms (note the separating +) and both are terms of the variable x so they are “like terms”. The five and three are not considered terms in their own right because they are linked to (or modify) the x. Like terms may be added and subtracted.

So how come if we write x + 3 the 3 is a term? Well, not only the + sign a hint, but here we can say we wrote the expression in descending order and 3 is actually 3x0 !

Now as far as mathematical rigor is concerned, this may not be exactly proper, but I think it may make sense considering your question.

Hope this helps and thanks for a neat question!

Bob Avakian
B.S. Earth Sciences; M.S. Geophysics
Oklahoma State Univ. Inst. of Technology

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