Power or Exponent?Date: 10/14/2003 at 10:35:28 From: Beth Subject: Exponents Our 5th grade textbook (McGraw Hill) defines "power" as "a number obtained by raising a base to an exponent". I have never seen this word defined in this way. I have always used the words "power" and "exponent" interchangeably. For example, 2 raised to the 3rd power, or 10 to the 6th power. Is the textbook's definition correct? I don't see how you can raise something to a power, and get the power as the answer. Date: 10/14/2003 at 12:54:12 From: Doctor Peterson Subject: Re: Exponents Hi, Beth. The book's terminology is correct; but the English usage here is awkward and is often misinterpreted, not only by students but even by textbook authors and lexicographers. That, of course, means that you can find many authorities for a different view than mine! In fact, my understanding of the terms may be a minority view; but I think it is correct. When we write 3 2 we say that 2 is the "base", 3 is the "exponent", and the whole thing is "a power of 2" (in particular, "the third power of 2"). The awkwardness comes from the fact that we call this expression "2 to the third power" or "2 raised to the power [of] 3". This SOUNDS as if we were saying that 3 was the "power" to which we raised 2, and as a result the word "power" is, as you point out, often used interchangeably with "exponent". But if you look closely at the phrase "raised to the third power", you see that we are not saying that 3 IS the power; rather, 3 IDENTIFIES which power you are talking about, the third one. We are raising the number 2 TO a power, changing it from its original "weak" form to a more "powerful" form; in fact, we have raised it to its third power, the third level it can reach. The power is the number it got to, not the number of steps it took to get there. I believe that the phrase started with "the third power of 2", which clearly names the result of the operation as a power; then moved to "2 raised to the third power", which means the same thing but emphasizes the operation of "raising" rather than the value; and then, when variable exponents were needed, had to be twisted around to "2 raised to the power of x" to avoid having to say "2 raised to the xth power". And at that point, it started sounding as if x was a power, though even here you can still see a distinction, in that the power is "of x", that is, belonging to, or associated with, x, not x itself. When people say "2 raised to the power x", that distinction is lost. So I can't blame people for getting confused, and I have to recognize that the term "power" is very commonly used to mean "exponent". But I think it is useful to retain a word that refers to the whole expression (just as we use "product" to refer to the result of multiplication); and the only word available is "power"! Can you think of an alternative, if we reserve "power" to mean the exponent? To support my theory, note that the first use of "power" in this sense meant the result of the operation, not the exponent: Earliest Uses of Some Words of Mathematics http://jeff560.tripod.com/p.html POWER appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "The power of a line, is the square of the same line." Here multiple powers are not yet in view, so no counting of powers is needed; the square is THE power. But the square is the result of the operation, clearly not the number 2. Similarly, see Math Words, and Some Other Words, of Interest http://www.pballew.net/arithme8.html#power Power: The word power comes from the French poeir and perhaps the earlier Latin word potere from which we get potent. Both words refer to ability or being able. In mathematics, power refers to the number arrived at by raising a number to an exponent. In the mathematical expression 3^2=9, three is the base, two is the exponent, and nine is the power. Students often refer to the exponent as the power, but this is not historically correct, although it has become so common, even among many teachers, that some dictionaries refer to the power as the exponent. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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