In the past year or two, I’ve come to see more clearly one of my defining characteristics – not to the point where I can state it simply in a word or two, but enough to talk about how much it influences me. To put it simply, I love problem-solving. My favorite types of books growing up were mystery novels. I was deeply fascinated by math. I often forgot math formulas, but somehow could derive them when needed. When I first started studying the Bible, I was amazed by how things fit together and how a book with so many authors could have its words so in sync as if written by one. A potential enigma on its own, but something that makes so much sense when probed – like a dark cave just waiting to be explored. And, I love organizing (which is tied closely to problem-solving). Love sorting, categorizing, organizing, all of that.
Anyway, I brought this up because I recently saw someone on Facebook ask why 0!=1. Having forgotten myself, I looked around a bit and found some well-worded explanations to remind me (oddly enough, the person who explained why x^0 = 1 did a poor job at explaining why 0!=1, so that’s why I only include his explanation of x^0=1 and not his explanation as to why 0!=1).
Just thought I’d share in case anyone was as fascinated by numbers as I was. Oh, and I should also note. At UCLA, learning the application of these seemingly inapplicable truths was really quite exciting for me (yeah, I’m a nerd). But these days, I’m less interested in the application of mathematical truths in the world as we see it but the application of biblical truths in the areas we can’t see (i.e., the heart, the mind, the spirit). Anywho … here are the two well-worded explanations (in my opinion) for those nerds-at-heart:
Why x^0=1
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0! = 1 for reasons that are similar to why x^0 = 1. Both are defined that way. But there are reasons for these definitions; they are not arbitrary.
You cannot reason that x^0 = 1 by thinking of the meaning of powers as “repeated multiplications” because you cannot multiply x zero times. Similarly, you cannot reason out 0! just in terms of the meaning of factorial because you cannot multiply all the numbers from zero down to 1 to get 1.
Mathematicians *define* x^0 = 1 in order to make the laws of exponents work even when the exponents can no longer be thought of as repeated multiplication. For example, (x^3)(x^5) = x^8 because you can add exponents. In the same way (x^0)(x^2) should be equal to x^2 by adding exponents. But that means that x^0 must be 1 because when you multiply x^2 by it, the result is still x^2. Only x^0 = 1 makes sense here.
Why 0!=1
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In general any factorial number (call it n!), may be written,
n! = n x (n-1) x (n-2) x (n-3) x … x 2 x 1
This is the general definition of a factorial number.
If you want it in words; a factorial number is the product of all positive integers from 1 to the number under consideration.
The main place it is likely to be encountered is when considering those groups and arrangements of objects mentioned above.
So where does all this 0! stuff fit in?
Nobody has trouble in stating 2! = 2 x 1 , or even that 1! = 1, but 0! appears to make no sense.
It does however, have a value of 1. This is rather counter intuitive but arises directly from our general definition.
n! = n x (n-1) x (n-2) x (n-3) x … x 2 x 1
Notice this may be written,
n! = n x (n-1)! Still exactly the same definition.
If the left hand side (LHS) = the right hand side (RHS) then dividing both sides by n should leave them still equal, so it is still true to write,
n!/n = n x (n-1)!/n
The (n-1)! in the RHS is being both multiplied by n and divided by n. These cancel leaving,
n!/n = (n-1)! If you doubt this, try it with real numbers, e.g. 4!/4 = 3! or (4 x 3 x 2 x 1)/4 = 3 x 2 x 1 = 6
The equation we now have is,
n!/n = (n-1)!
It is still our original definition in are arranged form. For convenience I shall write it the other way round.
(n-1)! = n!/n
We also said that our factorial uses the positive integers 1 and above. Try the value of n=2 in our rearranged formula and we get,
(2-1)! = 2!/2 or 1! = 2×1/2
The RHS calculates to 1, so we have the statement 1!=1That is what we guessed intuitively above. It is now confirmed. But look what happens when we substitute the legitimate value of n=1 in our formula.
(1-1)! = 1!/1
Evaluating this statement gives
0! = 1!/1
We have just shown 1!=1 so the RHS is 1/1 or 1.
