Factorial zero; Power of zero; Infinity

[Lisasmith111]

Hello folks,

Although 0! = 1 can anyone explain why this is the case. Hopefully, in a simple way so that I can understand why this is the case.

Also why does anything to the power of zero = 1 eg 10^0 = 1 and 9^0 = 1.

Finally, we all act as though we understand infinity even though none of us really know what it is. Can anyone explain?

Kind regards

Lisa Smith

[MAS1]

Originally Posted by Lisasmith111

Hello folks,

Although 0! = 1 can anyone explain why this is the case. Hopefully, in a simple way so that I can understand why this is the case.

Also why does anything to the power of zero = 1 eg 10^0 = 1 and 9^0 = 1.

Finally, we all act as though we understand infinity even though none of us really know what it is. Can anyone explain?

Kind regards

Lisa Smith

1. 0!
By definition n! = n x (n - 1) x (n - 2) x (n - 3) x ... x 2 x 1, or you can say:
n! = n x (n - 1)!
Dividing both sides by n gives:
n!/n = (n - 1)!

Now lets try some numbers:
Let n = 2.
2!/2 = (2 - 1)!
(2 x 1)/2 = 1!
2/2 = 1!
1 = 1!

Let n = 1.
1!/1 = (1 - 1)!
1/1 = 0!
1 = 0!

2. Power of Zero
1 = 2/2
1 = 2^1 / 2^1
When dividing the same "base" number with exponents, we subtract the exponents. In this example our base number is 2.
So,
1 = 2^(1 - 1)
1 = 2^0

3. Infinity
Infinity is a fascinating subject. In fact some infinities are bigger than others. A wonderful book about infinity and one mathematician's (Georg Cantor) obsession with it is "The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity". It can be found on Amazon.com at

http://www.amazon.com/Mystery-Aleph-...1674569&sr=1-2

MAS1

[Lisasmith111]

Originally Posted by MAS1

1. 0!
By definition n! = n x (n - 1) x (n - 2) x (n - 3) x ... x 2 x 1, or you can say:
n! = n x (n - 1)!
Dividing both sides by n gives:
n!/n = (n - 1)!

Now lets try some numbers:
Let n = 2.
2!/2 = (2 - 1)!
(2 x 1)/2 = 1!
2/2 = 1!
1 = 1!

Let n = 1.
1!/1 = (1 - 1)!
1/1 = 0!
1 = 0!

2. Power of Zero
1 = 2/2
1 = 2^1 / 2^1
When dividing the same "base" number with exponents, we subtract the exponents. In this example our base number is 2.
So,
1 = 2^(1 - 1)
1 = 2^0

3. Infinity
Infinity is a fascinating subject. In fact some infinities are bigger than others. A wonderful book about infinity and one mathematician's (Georg Cantor) obsession with it is "The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity". It can be found on Amazon.com at

http://www.amazon.com/Mystery-Aleph-...1674569&sr=1-2

MAS1

Thank you Mas1 for taking the time to explain. I will certainly look at the references you mention.

Thanks again for your post!

Lisa Smith