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Advice on using examples of ideas

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How do you teach a young child what a 'number' is? I doubt you would start with a theoretical discourse on the process of matching up sets of objects. You would start with actual examples of numbers. You might point to two cups and say `two cups' or point to two chairs and say 'two chairs' and so on. Once the child had understood some specific numbers, you might talk about a `number' as the word used to speak about any or all of the specific numbers.

Examples of real numbers

W we can convey the more sophisticated idea of a `real number' by considering a range of examples of real numbers. For instance, the following are all examples of real numbers:

-6, p , 188, 2.43, -0.177, Ö 2.

Real numbers include numbers which can be expressed as a ratio of two integers like -6, 188, 2.43, -0.177. For instance, 2.43=243/100 and -6=-6/1. Such numbers are called rational numbers.

And real numbers also include numbers like p or Ö 2 which cannot be expressed as a ratio of two integers and are thus called irrational numbers.

A range of examples

In general, a mathematical idea can be conveyed by considering a varied collection of examples of the idea. Indeed, one of the clearest indications of whether you understand a mathematical idea is whether you can point out a variety of examples of the idea. Any new example of the idea is now simply `one of those'. And it is also the case that creating a varied set of examples of an idea helps you to understand the idea.

A good set of examples will include simple examples and unusual examples, and it will be complemented by some non-examples. Once you have looked through these sections, try your hand at the exercises .

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Simple examples

Once you have been able to create at least one new example of the idea it is worth creating some simple examples. These kinds of examples are usually relatively easy to understand. And for that reason alone they are very useful. So what makes an example simple? This question is perhaps best answered by looking at some simple examples.

The real-valued function given by

f (x) = x

is certainly simple. This is the function that associates every real number with itself. By contrast, we can agree that the real-valued function

f(x) = x 6 - (2x -3) 3

is more complicated. After all, it takes some work to find out which real numbers are assigned to any given real numbers.

And what would constitute a simple real-world example of the speed of a body? We could take a jet fighter flying a distance of 1 mile in one hour. The mathematical example that matches this real-world example is of course the real number 1. But while this mathematical example is relatively simple, the real-world situation itself is certainly unusual. So for a simple example from the real world, I would be more likely to take a car travelling a sensible distance per hour.

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Typical examples

We can think of a typical example as one that displays all the characteristics of the idea. No simplifications are involved.

For instance, you could argue that the real-valued function given by f(x)=3x-1 is a typical example of a real-valued function. This at least avoids associating every real number with itself, as in the case of our simple example above. And for a typical example of the concept of speed, I would choose a person walking a distance of 3.1 miles per hour.

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Unusual examples

Our final stage in creating a range of examples is to find some unusual examples. Again, the easiest way to see what constitutes an unusual example is to look at some of them.

There is, for instance, plenty of scope for unusual real-valued functions. One of the most unusual I have come across is the real-valued function f which assigns to every rational number the number 1 and to every irrational number the number 0. More concisely we can define this function by the following rule:

[ 1 if x is rational

f (x) =

[ 0 if x is irrational

As far as unusual examples of real world concepts are concerned, each element involved in the concept can be unusual. Think, for instance, of unusual bodies. What about a tarantula travelling towards your toe a certain distance every second. Or take the distance travelled by a shaft of light per year. How tedious to stick to a car travelling a sensible distance in a comfortable unit of time!

Unusual examples test your understanding of an idea, to see if you really have understood what is going on.

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Non-examples

If the suggested example does not meet the definition of the idea, then we have what we can call a non-example of the idea. You will need to find the formal definition of the idea concerned and check to see whether the potential example meets every element of the definition.

Non-examples are particularly useful when they are almost, but not quite, an example. It is after all little use to say that the number two is not a real--valued function. That is obvious. It is, however, much more helpful to know that f(x)=x 2 ± 3 is a non-example of a real-valued function.

Non-examples help to clarify our understanding of what qualifies as an example.

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Exercises

1. Find a collection of examples for the following ideas. You should include the first example you can think of, two simple examples, two typical examples and an unusual example. Test each of your examples against a definition of the idea to make sure that you have genuinely found an example. In addition find a non-example for each idea.

(a) A set

(b) A vector

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2. Provide a set of examples for each of the following concepts from the real world. Further specify which mathematical idea models the concept from the real world and provide examples of the mathematical idea to match each of your examples of the real-world concept.

(a) An incline

(b) The tension in a string

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This content has been written by Peter Kahn, author of Studying Mathematics and its Applications.

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