Last night, my brother and I were playing computer Jeopardy. The game was very close and we both bet it all on final Jeopardy. The category was math and the question was: “What is 6508 raised to the 0th power?”: I got it right, he got it wrong. The problem is that he questioned why “one” is really the correct answer. I have no idea other than I remembered this from high school because it seemed so counterintuitive.
He contends, and I cannot argue, that “zero” is obviously the correct answer. Can anyone explain. In plain English, why one is the correct answer. I am not really interested in esoteric proofs but rather a commonsense answer.
I think the problem comes from the fact that we are taught that X[sup]n[/sup] “means” X multiplied by iteslf n times. This isn’t precisely what it means, but is just a way to calculate the value. Otherwise, what would an expression such as X[sup]1/2[/sup] stand for; how can you multiply something by itself a half a time?
But I’ll have to let the mathematicians explain what it “means.”
“One” is the preferred answer amongst those who give a shit, mainly math geeks and those driven by alcohol to make silly bets on obscure subjects…
“Zero” is the ‘gut’ answer, and probably the one you’d receive most often if you took a poll or something. As an answer, it is just as wrong as any other answer except “one.”
By the way, is it just me, or does the fact that any number raised to the 0 power is equal to one seem just a bit irrelevant? I’ve found uses over the years for a lot of what I hated during the algebra and trig and calc classes, but I don’t recall ever making useful that particular mathematical quirk.
I’m certain it’s just as relevant as the square root of negative one being i (or j, depending what profession you are in), used by a relatively small group for important calculations but not really relevant to the rest of us.
Let me suggest a situation in which having the 0-th power of anything being equal to 1 is relevant. Suppose you put $1000 in a bank account earning 6% per year (and you are told that the account will be earning 6% forever). How much money will be in the account in 5 years? Answer: $1000 times (1.06 to the 5-th power) = $1338.23. How much money will be in the account in 10 years? Answer: $1000 times (1.06 to the 10-th power) = $1790.85. O.K., how much money is in the account right now? Answer: $1000 times (1.06 to the 0-th power) = $1000, since the 0-th power of 1.06, or of anything else, is 1.
Well, there’s a very good reason for imaginary numbers, but you’re right, they are not particularly relavent in every day life. As for exponents, the zeroth power dilemma can be looked at as a type of limit. For example.
So, 10^0 MUST be 1 in order to follow the pattern.
If you go into negatives, you get
10^-1 = 0.1
10^-2 = 0.01
10^-3 = 0.001
and so on. Exponents can thus be used to describe the “columns” in any numerical base. In base-10 (10^X) you get the ones place, the tens place, the hundreds place, the thousands place, and so on. This can be used to calculate the value of a number in any base.
In base 10, 6,547,324 = (6 x 10^6 = 6,000,000) + (5 x 10^5 = 500,000) + (4 x 10^4 = 40,000) + (7 x 10^3 = 7,000) + (3 x 10^2 = 300) + (2 x 10^1 = 20) + (4 x 10^0 = 4)
Now, say we have a number in binary. Since binary is base 2, the places are defined by 2^X.
1100110111
The rightmost place is 2^0 = 1, the ones place. The next is 2^1 = 2, the twos place. The next is 2^2 = 4, the fours place, and so on. The above number gives us:
(1 x 2^0 = 1) + (1 x 2^1 = 2) + (1 x 2^2 = 4) + (0 x 2^3 = 0) + (1 x 2^4 = 16) + (1 x 2^5 = 32) + (0 x 2^6 = 0) + (0 x 2^7 = 0) + (1 x 2^8 = 256) + (1 x 2^9 = 512) = 823 (in base ten, of course)
Try calculating the base-ten value of the base-eight number 3472325l. Or the binary value of the base-16 number 23FD0CD4. Loads of fun for the whole family.
I hope this makes the purpose of exponents clearer. When you’re using them, you’re really doing work with numbers in a system with base Z (in Z^X). This is why in 4^2, 4 is called the “base.”
This is what I remember from algebra class. Now I have to go fail calculus again
