I was asked recently why . Remember that if n is a positive whole number then . Clearly you can’t multiply 2 by itself 0 times 😕
The key, when extending properties of the number system, is to use definitions that work for every number. So, for example
which gives you the rule that
- to divide powers you subtract the indices (the small superscripted numbers)
This leads to
Similarly, for any positive real number.
And if the number is negative? Great care is needed in this case. For example, using only real numbers, but is not a real number. The problem arises because the general definition of a power is given by and is undefined if a is negative or 0. Using complex numbers (which helps with ) just makes things more complex 😕 – see Log of Complex Number
When teaching maths that you are familiar with, it is not easy to see why students struggle with it – indeed once the student has understood the problem, they can’t see why they had difficulty before! This means that you have to be very careful what you say in case an attentive student takes it literally.
How many times has a lecturer said “the integral of e is itself ” ? So this happens:
- Q. Find the value of
When the error was pointed out to the student, they responded with
- Why does integrate to ? integrates to and as surely the integral of is also ?
How would you respond to this?
Inspired by a posting on S.O.S. Mathematics CyberBoard
Most students will be familiar with the definition of the derivative of a real-valued function of a real variable defined on some interval (a,b):
- If then f is differentiable at if exists and the limit is denoted
It is also clear that for this to make sense must be defined at (and of course it is a well-known consequence of the definition that is also continuous at ). But what if is defined on but not at , can we do anything then? Yes, we can define a pseudo-derivative of provided is defined on a neighbourhood of :
This pseudo-derivative has similar properties to the derivative and indeed it has the same values where is differentiable but there are significant differences as the following exercises show:
- If is differentiable at show that
- If show that exists although does not
- If show that has a local maximum at 0 but
- Suppose is differentiable on , except at a point in , with for .
If exists show that