undefined or indeterminate forms
Created | Updated Jul 7, 2003
These little critters are the most misunderstood in mathematics. They are problematic as these expressions have no meaning or definite value. Undefined, or indeterminate forms as they are called, have many misconceptions about them. This entry may clear up some of the confusion.
Even roots of negative numbers
The even root of a negative number is not a real number. First lets examine the square root.
Let:
√-1=y
Rewriting
y^2=-1
For any value of y, y^2 is not negative, as any number squared is positive. Then there is no real value for y.
However, the square root of -1 can be given a value. Since (√-1)^2=-1 (if you assume that for any number √(x)^2=x), then we can add a number to our number system, called the imaginary unit, i. Then i is a solution to:
x^2 = -1
Imaginary numbers, multiples of i, can be mixed with real numbers, to give complex numbers. This is a whole new bedtime story.
Logarithms of 0, and logarithms of negative numbers
If you are unfamiliar with logarithms (abbreviated log), here's a quick refresher:
If:
a^x=y
(where a is any number)
Then:
a^y=x
is the logarithm of x to the base a, normally, written log and then a with a subscript.For brevity, common logarithms (where a=10) are written log x, and natural logarithms (where a=e, the exponential constant) are written ln x.
Then how is the logarithm of zero, and the logarithm of negative numbers undefined?
First, logs of zero.
Let:
log 0 = x
Rewriting:
10^x = 0
But for any value of x, the expression does not equal zero. Therefore there is no value for x, and the expression becomes undefined.
Similarly, for logs of negative numbers
Let:
log -1 = x
Rewriting:
10^x = -1
Again, for any value of x, the expression does not equal -1 or any negative number. Then there is no value for x, and the expression becomes undefined as well.
However, like the square roots of negative numbers, we can give the logarithm of negative numbers only a complex value.
We can write a complex number a+b*i as r*e^(i*x), where x is the argument and r is the modulus of the complex number (not really necessary to know for this discussion).
By Euler's famous equation, e^(i*pi)=-1, we have an expression in terms of e to a power, so we can define the natural logarithm (to the base e) of -1 to be (i*pi). Similarly ln -2 = ln(2)+i*pi (because ln(a*b)=ln(a)+ln(b), with a=2 and b=-1). And so on.
The 0^0 form
Zero to the power of zero is undefined.
k to the power of 3 is:
k * k * k
k to the power of 2 is:
k * k
k to the power of 1 is:
k
k to the power of 0 is:
1
And so on.
But 0 to the power of k is 0.
There is a conflict in definitions,
so 0^0 is undefined.
Well, there's certainly more undefined forms, but that's it for today. I'll add some more tomorrow perhaps.