Numbers can be represented in different ways. The kind of representation we choose can highlight or de-emphasise the properties of the numbers.
Studies about understanding mathematics discuss about two kinds of representations of a mathematical idea: (1) transparent representations and (2) opaque representations. A transparent representation has no more and no less meaning than the represented idea(s) or structure(s). An opaque representation emphasizes some aspects of the ideas or structures and de-emphasizes others.
Examples:
- Representing the number 784 as
emphasizes – makes transparent – that it is a perfect square, but de-emphasizes – leaves opaque – that it is divisible by 98.
- Representing the 784 as 13×60+4 makes it transparent that the remainder of 784 on dividing by 13 is 4, but leaves opaque its property of being a perfect square
- For a whole number k, 17k is a transparent representation for a multiple of 17, as this property is embedded or ‘can be seen’ in this form of the representation. However, it is impossible to determine whether 17k is a multiple of 3 by considering the representation alone. In this case we say that the representation is opaque with respect to divisibility by 3.
- An infinite non-repeating decimal representation (such as 0.010011000111. . .) is a transparent representation of an irrational number (that is, irrationality can be derived from this representation if the definition adopted is its being non-repeating, non-terminating decimal; It becomes an opaque representation for the definition of irrationals as numbers that cannot be expressed as quotient of two integers.)
- 2k+1 and 2k are transparent representations of odd and even numbers, respectively.
But what about prime numbers and irrational numbers in general? What are their representations? P for prime is not a representation. In the article Representing numbers: prime and irrational, Rina Zaskis argued these two numbers have something in common: they both cannot be represented. Don’t we say irrational numbers are those that cannot be represented as a quotient and prime numbers are those that cannot be represented as a product? The examples I listed above were from the same paper. The author used them to argue the importance of representations and how the absence of it can become a cognitive obstacle to understand the concept.
