2) What is 0! ? Why?
3) What is 0^0? Why?
3) What is 0^0? Why?
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1:
6/3 = 2 because 2 * 3 = 6
0/0 = ? because ? * 0 = 0
The problem is that any number makes this true. Since 0/0 doesn't have a unique value, it's not allowed at all.
#2:
For any number n, n! = n * (n-1)!
So it should be true that 1! = 1 * 0!
Since 1! = 1, 1 = 1 * 0!
and 0! must be equal to 1
#3:
For any non-zero number n, 0^n = 0
For any non-zero number, n^0 = 1
0^0 is a contradiction because we don't know whether it is 0 or 1. So it is not allowed.
6/3 = 2 because 2 * 3 = 6
0/0 = ? because ? * 0 = 0
The problem is that any number makes this true. Since 0/0 doesn't have a unique value, it's not allowed at all.
#2:
For any number n, n! = n * (n-1)!
So it should be true that 1! = 1 * 0!
Since 1! = 1, 1 = 1 * 0!
and 0! must be equal to 1
#3:
For any non-zero number n, 0^n = 0
For any non-zero number, n^0 = 1
0^0 is a contradiction because we don't know whether it is 0 or 1. So it is not allowed.
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1) 0/0 is undefined because division problems must have unique answers, and this one does not follow that rule.
Example: suppose 0/0 = 1. Then, as in ordinary arithmetic, you can multiply both sides by 0 and get a valid answer: 0 = 0·1.
Now suppose that 0/0 = 2. Multiply both sides by 0, giving 0 = 0·2, which is also true.
But 0/0 cannot equal 1 and 2 (and π, and -17, and...) at the same time, by the rules of arithmetic, so we say that 0/0 has no defined value.
2) In order for the mathematics of combination and permutation to work out properly, 0! must be defined as 1.
3) 0^0 needs to be equal to 1 for various branches of mathematics to be complete and useful. See this Wikipedia article section for a summary:
http://en.wikipedia.org/wiki/Exponentiat…
Example: suppose 0/0 = 1. Then, as in ordinary arithmetic, you can multiply both sides by 0 and get a valid answer: 0 = 0·1.
Now suppose that 0/0 = 2. Multiply both sides by 0, giving 0 = 0·2, which is also true.
But 0/0 cannot equal 1 and 2 (and π, and -17, and...) at the same time, by the rules of arithmetic, so we say that 0/0 has no defined value.
2) In order for the mathematics of combination and permutation to work out properly, 0! must be defined as 1.
3) 0^0 needs to be equal to 1 for various branches of mathematics to be complete and useful. See this Wikipedia article section for a summary:
http://en.wikipedia.org/wiki/Exponentiat…
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0 poses certain problems in math that calls for a definition to solve it.
0/0 is undefined because division by 0 is undefined. Think about 2/0.... you need to find how many times 0 fits into 2! Sounds crazy, but that is one way to see why division by 0 is not defined.
0! does not fit in factorial definition, so math guys agreed to 0! = 1.
0^0 is like division by 0: you want to multiply 0 zero times by it self.... creepy right, so it is undefined simply because there is no way to do it without fall into contradictions.
0/0 is undefined because division by 0 is undefined. Think about 2/0.... you need to find how many times 0 fits into 2! Sounds crazy, but that is one way to see why division by 0 is not defined.
0! does not fit in factorial definition, so math guys agreed to 0! = 1.
0^0 is like division by 0: you want to multiply 0 zero times by it self.... creepy right, so it is undefined simply because there is no way to do it without fall into contradictions.