I was meditating upon some math problems the other day, when I started to delve into the unknown for me. I was working from my bound booklet out in the shade, sitting camly beneath the mango tree, when I pondered a problem which I had difficiulty reasoning out. My meditations had brought me into unknown real as I stated before. The problem that I am having trouble reasoning out it what would 0^0 power equate. I also mulled over that which would ocuur when infinity was taken to the 0 power.
Please reply with suggestion. I may have much more meditation to do before i shall become truly enlightened.
Full Version: Math Help
I think that 00, like 0/0, could conceivably equal any number, which is why it's undefined.
Infinity to the zero should equal 1. But infinity divided by infinity doesn't necessarily equal one. The reason for this is that there are different sizes of infinities. Although they are both infinite, it makes sense to say that the set of all real numbers is larger than the set of integers. But one certain infinity divided by itself should be 1, so any infinity raised to the zeroth power should be 1.
Just my thoughts.
Infinity to the zero should equal 1. But infinity divided by infinity doesn't necessarily equal one. The reason for this is that there are different sizes of infinities. Although they are both infinite, it makes sense to say that the set of all real numbers is larger than the set of integers. But one certain infinity divided by itself should be 1, so any infinity raised to the zeroth power should be 1.
Just my thoughts.
Thats Nice. But my Meditations have achieved a possible answer to the question. Suppose that y=x^x. At this point x is undefined at 0. So, we will look at the linit as x→0. Then, the ln is taken of both sides, such that ln(y)= xln(x). Then, my meditations have told me, that i must use L'Hopital's famous rule. Behol, the limit as x→0 of ln(x)/(x^-1) = limit as x→0 of (1/x)/(-x^-2)= -x= 0. My meditations have not borught to to the infinity yet, but they have found another unexpected quandary. Find all primes p, q such that p^2 -2q^2=1.
p=3, q=2?
Hmm... I prefer math of the variety 2 + 2 = 4 or I'll even solve 20x^2 + 7x + 200 = 0 for you (he he, good old quadratic forumula), but I hope I never have to learn this stuff (oh, wait, Algebra 2, dangit!).
Limits are more like calculus.
For instance, approaching a removable discontinuality in a rational function, the limits are the same approaching from both sides. Meanwhile approaching an asymptote discontinuality from one side yieldsa limit of positive infinity while approaching it from the other side yields negative infinity.
I know that didn't help. Sorry.
For instance, approaching a removable discontinuality in a rational function, the limits are the same approaching from both sides. Meanwhile approaching an asymptote discontinuality from one side yieldsa limit of positive infinity while approaching it from the other side yields negative infinity.
I know that didn't help. Sorry.
QUOTE(Duckie @ May 18 2006, 08:07 PM)
I was meditating upon some math problems the other day, when I started to delve into the unknown for me. I was working from my bound booklet out in the shade, sitting camly beneath the mango tree, when I pondered a problem which I had difficiulty reasoning out. My meditations had brought me into unknown real as I stated before. The problem that I am having trouble reasoning out it what would 0^0 power equate. I also mulled over that which would ocuur when infinity was taken to the 0 power.
Please reply with suggestion. I may have much more meditation to do before i shall become truly enlightened.
Please reply with suggestion. I may have much more meditation to do before i shall become truly enlightened.
QUOTE
ou say that in trig you learned you can't raise 0 to the 0th power,
right? Well, did you also learn that you can't determine what 0 times
infinity is (or to put it another way, you can't determine what 0/0 is or
infinity /infinity is)? I don't know what you are studying now, but in
calculus you will learn that these are called INDETERMINATE FORMS.
That means we can't say what these expressions are unless we are given a
specific problem. For instance, say we were given a problem like finding
the limit as x goes to -1 of (3x^2 + 2x - 1)/(x^2 + x).
If we plug a -1 inside the limit, we get 0/0. That doesn't really tell us
anything. To figure out what 0/0 is in this problem, you have to use
something called L'hopital's rule which you will learn about in calculus.
As it turns out, in this problem, 0/0 = 4. Pretty neat, huh?
Anyway, the point of all of this is that 0^0 is just another indeterminite
form. Why is this, you might ask? Let the limit as x goes to 0 be 0 and
the limit as y goes to 0 be 0. Then the limit as x and y go to 0 of x^y is
0^0, if we plug in 0 for x and for y, right? But this is indeterminate.
Why? Well, x^y = e^ln(x^y), right? And, e^ln(x^y) = e^(ylnx).
As x goes to 0, lnx goes to negative infinity, right? So the limit as x and
y go to zero of ylnx is going to be 0 times negative infinity, which is
indeterminite. e raised to an indeterminate form is indeterminate, so,
x^y, itself, is indeterminate. So, we have 0^0 is indeterminate.
Does that make sense to you? I'm not sure how much you have worked
with natural logs, so if you have any questions about what I did, feel
free to write back.
On to your last question about infinity. What an excellent question! I
love thinking about concepts dealing with infinity as, it also seems, you
do. Mathematicians usually make a distinction between 2 different
"kinds" of infinity. The first kind of infinity is called COUNTABLY
infinite. The counting numbers, for instance, are countably infinite.
This means there exists a 1 to 1 correspondance between the counting
numbers and the natural numbers. The set of all rational numbers is
also countably infinite. Can you figure out why?
This is kind of tricky, so I'll give you a hint: you are looking for some
way to write down the rational numbers so that there is a specific order
in your arrangement...you are looking for an order such that if I were
to ask you what the 328th rational number in the order you selected was,
you could tell me (though it might be kind of painful!).
The second kind of infinity is called UNCOUNTABLY infinite. This
infinity is, in a way, bigger than the other infinity because it is impossible
to find a one to one correspondence with the natural numbers. In these
sets, there are simply not enough natural numbers to cover all of the
elements of the set. This is kind of a strange concept to get used to, I
think. The real numbers are an example of an uncountably infinite set
of numbers.
But, on to what you were asking about...there are an uncountably
infinite number of numbers between 1 and 2, and there are also an
uncountably infinite number of numbers between 1 and 50, but are there
more numbers between 1 and 50 than between 1 and 2. Thinking about
it in one way, I would agree with you, that there are more numbers
between 1 and 50 than there are numbers between 1 and 2. After all,
all the numbers between 1 and 2 are also between 1 and 50, but there
are lots more numbers betwen 1 and 50 that aren't between 1 and 2.
However, if you think about it in terms of the fact that there are lots and
lots of numbers between 1 and 2 -- an uncountably infinite amount, and
there is an uncountably infinite amount of numbers between 2 and 50,
and you add the two infinities, you are just going to get another
uncountably infinite amount. Is this infinity bigger than the original
infinities? What do you think? Remember that infinity is NOT a
number!!
I hope this helps you understand some of these concepts better, and I
hope also you keep thinking about these ideas and write us with any
other questions. I'm impressed you are thinking about these things in
high school! Keep it up.!!
http://mathforum.org/library/drmath/view/52400.htmlright? Well, did you also learn that you can't determine what 0 times
infinity is (or to put it another way, you can't determine what 0/0 is or
infinity /infinity is)? I don't know what you are studying now, but in
calculus you will learn that these are called INDETERMINATE FORMS.
That means we can't say what these expressions are unless we are given a
specific problem. For instance, say we were given a problem like finding
the limit as x goes to -1 of (3x^2 + 2x - 1)/(x^2 + x).
If we plug a -1 inside the limit, we get 0/0. That doesn't really tell us
anything. To figure out what 0/0 is in this problem, you have to use
something called L'hopital's rule which you will learn about in calculus.
As it turns out, in this problem, 0/0 = 4. Pretty neat, huh?
Anyway, the point of all of this is that 0^0 is just another indeterminite
form. Why is this, you might ask? Let the limit as x goes to 0 be 0 and
the limit as y goes to 0 be 0. Then the limit as x and y go to 0 of x^y is
0^0, if we plug in 0 for x and for y, right? But this is indeterminate.
Why? Well, x^y = e^ln(x^y), right? And, e^ln(x^y) = e^(ylnx).
As x goes to 0, lnx goes to negative infinity, right? So the limit as x and
y go to zero of ylnx is going to be 0 times negative infinity, which is
indeterminite. e raised to an indeterminate form is indeterminate, so,
x^y, itself, is indeterminate. So, we have 0^0 is indeterminate.
Does that make sense to you? I'm not sure how much you have worked
with natural logs, so if you have any questions about what I did, feel
free to write back.
On to your last question about infinity. What an excellent question! I
love thinking about concepts dealing with infinity as, it also seems, you
do. Mathematicians usually make a distinction between 2 different
"kinds" of infinity. The first kind of infinity is called COUNTABLY
infinite. The counting numbers, for instance, are countably infinite.
This means there exists a 1 to 1 correspondance between the counting
numbers and the natural numbers. The set of all rational numbers is
also countably infinite. Can you figure out why?
This is kind of tricky, so I'll give you a hint: you are looking for some
way to write down the rational numbers so that there is a specific order
in your arrangement...you are looking for an order such that if I were
to ask you what the 328th rational number in the order you selected was,
you could tell me (though it might be kind of painful!).
The second kind of infinity is called UNCOUNTABLY infinite. This
infinity is, in a way, bigger than the other infinity because it is impossible
to find a one to one correspondence with the natural numbers. In these
sets, there are simply not enough natural numbers to cover all of the
elements of the set. This is kind of a strange concept to get used to, I
think. The real numbers are an example of an uncountably infinite set
of numbers.
But, on to what you were asking about...there are an uncountably
infinite number of numbers between 1 and 2, and there are also an
uncountably infinite number of numbers between 1 and 50, but are there
more numbers between 1 and 50 than between 1 and 2. Thinking about
it in one way, I would agree with you, that there are more numbers
between 1 and 50 than there are numbers between 1 and 2. After all,
all the numbers between 1 and 2 are also between 1 and 50, but there
are lots more numbers betwen 1 and 50 that aren't between 1 and 2.
However, if you think about it in terms of the fact that there are lots and
lots of numbers between 1 and 2 -- an uncountably infinite amount, and
there is an uncountably infinite amount of numbers between 2 and 50,
and you add the two infinities, you are just going to get another
uncountably infinite amount. Is this infinity bigger than the original
infinities? What do you think? Remember that infinity is NOT a
number!!
I hope this helps you understand some of these concepts better, and I
hope also you keep thinking about these ideas and write us with any
other questions. I'm impressed you are thinking about these things in
high school! Keep it up.!!
QUOTE
It is commonly taught that any number to the zero power is 1, and zero to any power is 0. But if that is the case, what is zero to the zero power?
Well, it is undefined (since xy as a function of 2 variables is not continuous at the origin).
But if it could be defined, what "should" it be? 0 or 1?
Presentation Suggestions:
Take a poll to see what people think before you show them any of the reasons below.
The Math Behind the Fact:
We'll give several arguments to show that the answer "should" be 1.
* The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)n using the binomial theorem, i.e., 0n. But the alternating sum of the entries of every row except the top row is 0, since 0k=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)0=00 if it were defined, should be 1.
* The limit of xx as x tends to zero (from the right) is 1. In other words, if we want the xx function to be right continuous at 0, we should define it to be 1.
* The expression mn is the product of m with itself n times. Thus m0, the "empty product", should be 1 (no matter what m is).
* Another way to view the expression mn is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 02=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 00=1.
* Here's an aesthetic reason. A power series is often compactly expressed as
SUMn=0 to INFINITY an (x-c)n.
We desire this expression to evaluate to a0 when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a0 term (not as nice) or by defining 00=1.
http://www.math.hmc.edu/funfacts/ffiles/10005.3-5.shtmlWell, it is undefined (since xy as a function of 2 variables is not continuous at the origin).
But if it could be defined, what "should" it be? 0 or 1?
Presentation Suggestions:
Take a poll to see what people think before you show them any of the reasons below.
The Math Behind the Fact:
We'll give several arguments to show that the answer "should" be 1.
* The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)n using the binomial theorem, i.e., 0n. But the alternating sum of the entries of every row except the top row is 0, since 0k=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)0=00 if it were defined, should be 1.
* The limit of xx as x tends to zero (from the right) is 1. In other words, if we want the xx function to be right continuous at 0, we should define it to be 1.
* The expression mn is the product of m with itself n times. Thus m0, the "empty product", should be 1 (no matter what m is).
* Another way to view the expression mn is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 02=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 00=1.
* Here's an aesthetic reason. A power series is often compactly expressed as
SUMn=0 to INFINITY an (x-c)n.
We desire this expression to evaluate to a0 when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a0 term (not as nice) or by defining 00=1.
As to your second question
2 & 3
3 & 5
7 & 13
19 & 37
31 & 61
37 & 73
...
The list goes on...
Once you get up there 2^30402457-1 (its prime
)it stops happening becaus the next number will be more than double the last but until then there are tons of sets, I couldn't halp youif you wanted to find the largest set, you should direct our math questions to artofproblemsolving which way to many people I know are on.EDIT: Mango trees? In washington? WHERE?
I *heart* mangos.,
For the 00 thing and infinity thing, you pretty much repeated what I said. A bit more detailed, but, yeah. I'm glad you agree.
And for the prime number thing, you might want to read the question again.
And for the prime number thing, you might want to read the question again.
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