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 OCTOBER 2005

The Dean’s Column
Some Amazing Number Relationships

By Dean Alfred Posamentier

We are accustomed to seeing numbers in charts and tables as on the sports or business pages of a newspaper. We use numbers continuously in our everyday life experiences, either to represent a quantity or to designate something such as a street, address, or page. We use numbers without ever taking the time to observe some of their unusual properties. That is, we don’t stop to smell the flowers as we walk through a garden, or as it is more commonly said: “take time to smell the roses.” Inspecting some of these unusual number properties provides us with a much deeper appreciation for these symbols that we all too often take for granted. Students too often are taught mathematics as a dry and required course of instruction. As teachers we have an obligation to make it interesting. To show some of the number oddities brings some new “life” to the subject. It will evoke a “gee whiz” response from students. That’s what you ought to strive for. Make them curious about the subject. Motivate them to “dig” further.

Who said numbers can’t form beautiful relationships! Showing your students some of these unique situations might give them the feeling that there is more to “numbers” than meets the eye. They should be encouraged not only to verify these relationships, but also to find others that can be considered “beautiful.”

Notice the consecutive exponents.

135=11+32+53
175=11+72+53
518=51+12+83
598=51+92+83

Now taken one place further we get:

1,306=11+32+03+64
1,676=11+62+73+64
2,427=21+42+23+74

The next ones are really amazing.  Notice the relationship between the exponents and the numbers*.

3,435=33+44+33+55
438,579,088=44+33+88+55+77+99+00+88+88

Now it’s up to the class to verify these and discover other beautiful relationships. When your students say “Wow!” you have achieved the first step to opening up a new dimension of number exploration.#
* In the second illustration you will notice that for convenience and for the sake of this unusual situation, we have considered 00 as though its value is 0, when in fact it is indeterminate.

Alfred Posamentier, Ph.D. is the Dean of the School of Education, City College of New York.