How is 0! equal to 1? And what is natural logarithm and how is it's value evaluated?

Answer 1

Any factorial can be expressed x!

The next factorial is (x+1)! and is equal to (x+1)*x!

Hence:-

1!=1
2!=2*1!=2
3!=3*2!=6

But if x=y-1, then this can be expressed:-

y!=y*(y-1)!

Substituting y=1 leaves:-

1!=1*(1-1)!
1=1*0!
0!=1

Therefore 0!=1.

This seems a bit long winded so imagine it this way...

If you had three playing cards in your hand, you could arrange them 3! ways (=6)

If you had two you could arrange them 2! ways (=2)

But what if you had 0 playing cards?

Following the pattern, you could arrange them 0! ways.

Intuition would suggest there are no ways of arranging 0 cards but that would mean you would always have to have a playing card with youin order to arrange it some way.

In fact, you can arrange 0 playing cards 1 way and that way is not to have any at all.

Hence 0!=1 (Try http://www.murderousmaths.co.uk/ for more)


A natural logarithm is the number that you need to raise e to the power of in order to get the desired number i.e. it is the logarithm of a number in base e.

e is a constant used a lot in working out interest rates and is roughly equal to 2.718, however, e is irrational and transcendental hence it does not recur, terminate, cannot be expressed in the form a/b and is not the solution of any polynomial equation.

The actual formula for e is e=(1+1/x)^x as x tends to infinity.

In general, if log(e)(x)=y then e^y=x

There is a natural logarithm button on most scientific calculators and is usually shown by "ln" or "log(e)x"

There are some things unique about e, such as if y=e,dy/dx=e and most of these things can be found on http://en.wikipedia.org/wiki/E_(mathematical_constant)

Please pick me for best answer!!!

Answer 2

0!=1 for a number of reasons.
First the definition of ! needs to be such that 0!=1.
Observe,
n!=n*(n-1)!
so plugging in n=1 we see that 1!=1*(1-1)! = 1*0! =1*1 = 1

The second is that the mathematical version of factional is this function called the GAMMA function, this is basically a crazy integral, which for integer values, gives the factorial function. In fact GAMMA(n+1) = n! for any whole number n. The GAMMA function is more useful because the GAMMA function is not restricted to the integers, and we can find GAMMA of values such as pi. At any rate, in order for the two to match up 0!=1.

As to your second question, natural log (ln) is another mathematical function. To fully understand this we first must know what e is. e is a mathematical constant much like pi. e is irrational and transcendental. This basically means as a decimal e has no pattern, terminates, and is never the solution to a polynomial with integer or rational coefficients.
e is about 2.71828183....

e is actually related to the ! function as 1 + 1/2! + 1/3! +1/4!+ ... + 1/n! approaches e as n gets larger. In fact e is the sum of the infinite sum.

Now that you kind of know about e, we can talk about ln.

suppose e^x = 8, and you would like to know what x is, ln(8) would solve this question. Basically ln and e are inverses of each other, ln(e^x) = x and e^(ln(x)) = x.

So how can you compute ln(x) for a number? Use a calculator.

Hope this helps,

Answer 3

Natural logarithm is the "LN" key on your calculator. The symbol for it is "e". Its basically a logarithm to base 2.178... and so on, opposed to common logarithms to base 10. The number 2.178 comes from the sum of the series of 1+1/1+1/1*2 + 1/1*2*3... and so on. Hope that answers the natural log question.
NO clue how 0! is equal to 1. Try looking it up on Google.

Answer 4

Firstly, 0! (0 factorial) is equal to 1 purely by mathematical convention. There is no real arithmetic logic to being equal to 1.

Secondly, the natural logarithm of a number is the number to which e must be raised to get the original number. It is written as ln x (natural log of x).

For example, if ln x = y
then x = e^y

One way of calculating it is to use something like Taylor's expansion.

There's more on natural logs at http://en.wikipedia.org/wiki/Natural_logarithm

Answer 5

3!=3*2!
2!=2*1!
1!=1*0!

Natural logarithm is written "ln".

x=ln(5) is the solution to the equation e^x=5
e is a constant with the approximate value of 2.718281828.

When x approaches infinity this expression approaches e:

(1+x^-1)^x

Answer 6

By definition of a factorial, we have:
(p+1)! = (p+1)p!
Let p = 0. Then we have
(0+1)! = 1! = 1 = (0+1) 0! = (1) 0!
Dividing both sides by 1 gets us:
1 = 0!
QED

Answer 7

Although n! is (n)(n-1)(n-2)...(1), that is only valid for positive integers. 0! = 1 is a definition, and definitions do not require proof. Since ln(1) = 0, ln(0!) = 0, also.