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:

1. Representing the number 784 as $28^2$ emphasizes – makes transparent – that it is a perfect square, but de-emphasizes – leaves opaque – that it is divisible by 98.
2. 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
3. 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.
4. 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.)
5. 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.

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### 2 Responses to “What can the representations of numbers tell us?”

Comments (2)
1.

They can give examples of primes say 5 but it’s hard to see from this representation the ‘definition’ or properties of a prime number. That’s what R. Zaskis meant. Thanks for the comment. You always make me think:-)

2.

Intriguing. So if I ask my 6th grade students to show me what a prime number looks like, they can’t. They can only show me what a prime number doesn’t look like.