I don’t understand. To my eyes, it looks like Thudlow Boink is saying, in the text I quoted, that I should start with the 5 x instead of the 10[sup]n[/sup], which - again, to my unskilled eyes - looks like putting multiplication in front of exponent in the order of operations.
No, he is not saying that you should start with the “5 x” instead of the “10[sup]n[/sup]”; perhaps you are misreading his “you used 10 as your starting point”. He is saying that to calculate 10[sup]n[/sup], you start from 1 (not 10) and multiply by 10 n times. To calculate 10[sup]-n[/sup], you start from 1 (not 10) and divide by 10 n times.
Thus:
5 x 10[sup]1[/sup] = 5 x (1 x 10) = 5 x 10 = 50
5 x 10[sup]0[/sup] = 5 x (1) = 5 x 1 = 5
5 x 10[sup]-1[/sup] = 5 x (1 / 10) = 5 x 0.1 = 0.5
5 x 10[sup]-2[/sup] = 5 x ((1 / 10) / 10) = 5 x 0.01 = 0.05
5 x 10[sup]-3[/sup] = 5 x (((1/10)/10)/10) = 5 x 0.001 = 0.005
Brilliant, thanks. I get it now. So is that always the rule with negative exponents? Divide by 1, [base] times?
In b[sup]n[/sup], n is the “exponent” or “power”, and b is the “base”. The rule, concerning integer exponents at least, is to start with 1, and multiply by the base b, doing so n many times. [If n is negative, then consider this to mean dividing by the base b the corresponding positive number of times]
So:
2[sup]3[/sup] = 1 x 2 x 2 x 2 = 8
2[sup]2[/sup] = 1 x 2 x 2 = 4
2[sup]1[/sup] = 1 x 2 = 2
2[sup]0[/sup] = 1 = 1
2[sup]-1[/sup] = 1 / 2 = 0.5
2[sup]-2[/sup] = (1 / 2) / 2 = 0.25
2[sup]-3[/sup] = ((1 / 2) / 2) / 2 = 0.125
The general rule for exponents is that, if a and b are positive integers, x[sup]a[/sup] * x[sup]b[/sup] = x[sup]a + b[/sup] (try a few examples to verify this). This is a very nice rule, and we’d like it to hold in general, so that influences how we define exponentiation with exponents that are not positive integers. For instance, with b = 0, we want that x[sup]a[/sup] * x[sup]0[/sup] = x[sup]a[/sup], which is fine as long as x[sup]0[/sup] = 1. Likewise, we want that x[sup]a[/sup] * x[sup]-a[/sup] = x[sup]0[/sup] = 1, so it follows that multiplying by x[sup]-a[/sup] must be the same as dividing by x[sup]a[/sup]. Make sense?
ETA: If you really want to make sure you understand this rule, tell me how we should define x[sup]1/2[/sup].
Excellent explanation, thank you.
Also a very nice explanation, thank you. Is it 1/2?
That’s odd… how come my superscripts were maintained through quoting but ultrafilter’s were not?
On edit: Oh, nevermind, you must’ve copied and pasted ultrafilter’s while "Reply"ing to mine. Yet another problem with the way sup tags work…
Yep. Did I get his question right?
If you are saying that x[sup]1/2[/sup] should equal 1/2 regardless of what x is, then, no, that is not correct. Remember what ultrafilter said, you want x[sup]a[/sup] times x[sup]b[/sup] to equal x[sup]a+b[/sup]. So, think about what this means for x[sup]1/2[/sup] in particular.
(It’s a little tricky, for a minor technical reason that may become apparent once you realize what the condition imposed is, but all in due time…)
Oh, I was thinking x=1 for some reason. I know why: I’m mathtarded. I have no idea what x[sup]1/2[/sup] is.
Maybe I’m shooting in the dark here, but is the answer X÷2?
Remember that x[sup]a[/sup] * x[sup]b[/sup] = x[sup]a + b[/sup]. Consider the case when a = b = 1/2.
Aw, don’t say that. Nobody’s intrinsically mathtarded or math-adept; it’s just a matter of practice and experience.
Here, I’ll walk you through it: We want x[sup]a[/sup] * x[sup]b[/sup] to equal x[sup]a+b[/sup], for all values of a and b, since it’s such a nice law that works out for the usual integer cases. In particular, what does that condition say when a and b are both 1/2?
ETA: Oh, ultrafilter beat me to it
Is it x=1?
We’re not solving for the value of x here; we’re trying to see what x[sup]1/2[/sup] should be, in terms of x. Let’s make it more concrete.
We want 3[sup]1/2[/sup] * 3[sup]1/2[/sup] to equal 3[sup]1/2 + 1/2[/sup]. What does that tell us that 3[sup]1/2[/sup] should equal?
0?
Well, you haven’t explained your reasoning (the question mark suggests perhaps a random guess?), but let’s see…
We want 3[sup]1/2[/sup] * 3[sup]1/2[/sup] to equal 3[sup]1/2 + 1/2[/sup]. If 3[sup]1/2[/sup] were 0, this would mean that 0 * 0 would have to equal 3[sup]1/2 + 1/2[/sup]. Is that the case?
[Incidentally, in case it’s not clear, the asterisk ***** stands for multiplication, whereas we’ve generally been using x not to stand for multiplication but as the name of a variable]
I was thinking 0[sup]1/2[/sup]*0[sup]1/2[/sup] = 0[sup]1/2[/sup]+[sup]1/2[/sup]
. . . I guess. I don’t even think I understand the question.
Is it -3[sup]1/2[/sup]?
glazed eyed, she scans the thread. From time to time a symbol looks familiar and she smiles, grunting softly to herself. Soon, she loses interest and goes back to examining the lint balls accumulating on the sofa’s throw blanket…