| N Barber Poster Post Number: 19 |
I was recently reading a maths book containing various Algebraic equations, and one of them included 0!. I continued reading and it said that 0!=1, but to what I know that doesn't make sense. If X! means multiply by all numbers below X down to 1, how can 0! even exist and how does it equal 1?? |
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| colin peacock Regular poster Post Number: 51 |
Do you know the Gamma Function? |
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| N Barber Poster Post Number: 20 |
No I'm only in yr9UK and was reading the book in the library to pass time |
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| David Loeffler Veteran poster Post Number: 878 |
Alternatively, you can think about it this way. N! is the number of permutations of N objects - that is, the number of one-to-one functions from a set of size N to itself. There's only one function on the empty set - the empty function - and it's (vacuously) one-to-one. So 0! = 1. (I wouldn't get too hung up about this. Remember that functions and notations are invented by people for the convenience of people; ultimately the reason that 0! = 1 is that people find it convenient to extend the definition of the factorial by writing 0!=1, and it makes certain formulae easier to write down if you do so. It's not because there's some deep, subtle law of the universe ordaining that 0! has to be 1.) David |
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| N Barber Poster Post Number: 21 |
what is the gamma function and how does it relate to 0!=1 ? |
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| colin peacock Regular poster Post Number: 52 |
We define the Gamma Function to be: Now, consider Can you integrate this by parts to get Can you see how you go from here to defining the factorial in terms of this function? This result is called a recurrence relation. Also, you immediately have since as tends to infinity tends to 0 (since ) Post back if you require a fuller explanation. Colin |
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| Phil Freeman Regular poster Post Number: 73 |
Take a look at http://mathworld.wolfram.com/GammaFunction.html In Year 9 I don't think you'll know much about integral calculus, but here goes. The important bits are : gamma(n)=(n-1)! for integers n. and gamma(y) = integral from 0 to infinity of e^(-x).x^(y-1)dx. What the second bit means is unimportant, but putting y=1 gives gamma(1)=1 therefore 0!=1, although as David says, this is largely a convention. -Phil. |
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| Emma McCaughan Moderator Post Number: 996 |
You don't normally begin calculus until year 12, so don't worry if you don't understand what Colin and Phil are on about! Here's another argument for 0!=1. To get from 5! to 4!, you divide by 5; to get from 4! to 3!, you divide by 4, and so on. So to get from 1! to 0!, you divide by 1. The reason 0! exists is that we often have mathematical formulae which use factorials, and it is very useful if they still work for 0. For instance, if you've ever met Pascal's triangle , it turns out that the formula for the terms in the nth row is n!/((n-r)!r!). The first term is when r=0, and the next is when r=1, and so on. The formula works for all the terms, provided 0!=1. |
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| Sohaib Ali Poster Post Number: 6 |
Here is the proof for 0!=1 taken from mathematics olympiad handbook nCr = n(n-1)(n-2)...(n-r+1)/(n-r)!n! take r=n nCn=n(n-1)(n-2)...1/(n-n)!n! 1=n!/(n-n)!n! (n-n)!=n!/n! 0!=1 |
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| Yanqing Cheng Regular poster Post Number: 33 |
Another way: 3!=6=4!/4 2!=2=3!/3 1!=1=2!/2 If we follow this rule, 0! needs to be 1!/1=1 |
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| LORRRRRRRRIMORRRRRRRRE Poster Post Number: 8 |
Whatever anyone says, you can't really do anything with 0, as by definition it isn't supposed to be there ( 0 pencils=0 books = 0 everything) It is the pivot where everything is eveything and everything is nothing (in my opinion) 0 was created to 'fill the gaps' in the hundreds et cetera, now it is used to fill the gaps of logic o___0 . I do not really trust 0 or infinity at all...... (ps.. in year 12 at the mo, but haven't come across gamma function yet, where abouts in the syllabus is it? ) |
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| George Weatherill Frequent poster Post Number: 96 |
^ Anywhere from 1st year university to 3rd year university. There isn't much point in it being taught properly till you have a good grasp of complex numbers, residue theory and contour integration (since 99% of the Gamma function's properties are intertwined with such things), and it takes a while before all that is covered even in university. At Cambridge it's taught as part of "Further Complex Methods" in the 3rd year, but that also covers all the topics I just mentioned, and more. "you can't really do anything with 0" You can do plenty of things with zero, after all it's defined as the additive identity of the Field of Reals, and therefore is essential to the contruction of a proper system of numbers like the Reals. You just can't divide by it. I don't pretend to know much of this kind of things (Saxl's Groups, Rings and Modules last year wasn't exactly riveting for me....), but in "laymans terms" it can be shown as why we can't really do that operation. 0 = 0 right? How about 2x0 = 1x0. Both sides are still zero. I'll just cancel the zeros (ie divide by zero) 2 = 1. But thats wrong. Since I can use such an operation (dividing by zero) to prove something I know to be false, that means "dividing by zero" isn't a valid operation. I know to those pure mathematicians reading this post I'm probably WAY off the mark, but its a vaguely decent example (at least to me, a lowly applied mathematician) as to why such things aren't allowed. LORRRRRRRRIMORRRRRRRRE, if you plan on taking Mathematics as your degree, you'll end up studying at least some analysis, and that'll give a much more solid reason and explaination for the properties that certain numbers and operations have. |